Mock Version: 1.4.14
Mock Version: 1.4.14
ENTER ['do'](['bash', '--login', '-c', '/usr/bin/rpmbuild -bs --target riscv64 --nodeps /builddir/build/SPECS/gap-pkg-edim.spec'], chrootPath='/var/lib/mock/f31-build-89457-32073/root'env={'TERM': 'vt100', 'SHELL': '/bin/bash', 'HOME': '/builddir', 'HOSTNAME': 'mock', 'PATH': '/usr/bin:/bin:/usr/sbin:/sbin', 'PROMPT_COMMAND': 'printf "\\033]0;<mock-chroot>\\007"', 'PS1': '<mock-chroot> \\s-\\v\\$ ', 'LANG': 'en_US.UTF-8'}shell=Falselogger=<mockbuild.trace_decorator.getLog object at 0x1558101898>timeout=345600uid=986gid=135user='mockbuild'nspawn_args=[]unshare_net=TrueprintOutput=False)
Executing command: ['bash', '--login', '-c', '/usr/bin/rpmbuild -bs --target riscv64 --nodeps /builddir/build/SPECS/gap-pkg-edim.spec'] with env {'TERM': 'vt100', 'SHELL': '/bin/bash', 'HOME': '/builddir', 'HOSTNAME': 'mock', 'PATH': '/usr/bin:/bin:/usr/sbin:/sbin', 'PROMPT_COMMAND': 'printf "\\033]0;<mock-chroot>\\007"', 'PS1': '<mock-chroot> \\s-\\v\\$ ', 'LANG': 'en_US.UTF-8'} and shell False
Building target platforms: riscv64
Building for target riscv64
setting SOURCE_DATE_EPOCH=1564012800
Wrote: /builddir/build/SRPMS/gap-pkg-edim-1.3.3-6.fc31.src.rpm
Child return code was: 0
ENTER ['do'](['bash', '--login', '-c', '/usr/bin/rpmbuild -bb --target riscv64 --nodeps /builddir/build/SPECS/gap-pkg-edim.spec'], chrootPath='/var/lib/mock/f31-build-89457-32073/root'env={'TERM': 'vt100', 'SHELL': '/bin/bash', 'HOME': '/builddir', 'HOSTNAME': 'mock', 'PATH': '/usr/bin:/bin:/usr/sbin:/sbin', 'PROMPT_COMMAND': 'printf "\\033]0;<mock-chroot>\\007"', 'PS1': '<mock-chroot> \\s-\\v\\$ ', 'LANG': 'en_US.UTF-8'}shell=Falselogger=<mockbuild.trace_decorator.getLog object at 0x1558101898>timeout=345600uid=986gid=135user='mockbuild'nspawn_args=[]unshare_net=TrueprintOutput=False)
Executing command: ['bash', '--login', '-c', '/usr/bin/rpmbuild -bb --target riscv64 --nodeps /builddir/build/SPECS/gap-pkg-edim.spec'] with env {'TERM': 'vt100', 'SHELL': '/bin/bash', 'HOME': '/builddir', 'HOSTNAME': 'mock', 'PATH': '/usr/bin:/bin:/usr/sbin:/sbin', 'PROMPT_COMMAND': 'printf "\\033]0;<mock-chroot>\\007"', 'PS1': '<mock-chroot> \\s-\\v\\$ ', 'LANG': 'en_US.UTF-8'} and shell False
Building target platforms: riscv64
Building for target riscv64
setting SOURCE_DATE_EPOCH=1564012800
Executing(%prep): /bin/sh -e /var/tmp/rpm-tmp.iAYWin
+ umask 022
+ cd /builddir/build/BUILD
+ cd /builddir/build/BUILD
+ rm -rf EDIM-1.3.3
+ /usr/bin/bzip2 -dc /builddir/build/SOURCES/edim-1.3.3.tar.bz2
+ /usr/bin/tar -xof -
+ STATUS=0
+ '[' 0 -ne 0 ']'
+ cd EDIM-1.3.3
+ /usr/bin/chmod -Rf a+rX,u+w,g-w,o-w .
+ for fil in doc/edim.bib doc/edim.bbl
+ iconv -f iso8859-1 -t utf-8 doc/edim.bib
+ touch -r doc/edim.bib doc/edim.bib.utf8
+ mv -f doc/edim.bib.utf8 doc/edim.bib
+ for fil in doc/edim.bib doc/edim.bbl
+ iconv -f iso8859-1 -t utf-8 doc/edim.bbl
+ touch -r doc/edim.bbl doc/edim.bbl.utf8
+ mv -f doc/edim.bbl.utf8 doc/edim.bbl
+ RPM_EC=0
BUILDSTDERR: ++ jobs -p
+ exit 0
Executing(%build): /bin/sh -e /var/tmp/rpm-tmp.Z6abPm
+ umask 022
+ cd /builddir/build/BUILD
+ cd EDIM-1.3.3
+ export LC_ALL=C.UTF-8
+ LC_ALL=C.UTF-8
+ ./configure /usr/lib/gap
+ make -j4
+ ln -s /usr/lib/gap/doc ../../doc
+ mkdir ../pkg
+ ln -s ../EDIM-1.3.3 ../pkg
+ gap -l '/builddir/build/BUILD/EDIM-1.3.3/..;/usr/lib/gap'
Using /usr/lib/gap as GAP build directory.
Using config in /usr/lib/gap/sysinfo.gap-default64
Created ./Makefile-default64 with link from ./Makefile
if test ! -d bin;  then mkdir bin;  fi
if test ! -d bin/riscv64-redhat-linux-gnu-default64-kv3;  then mkdir -p bin/riscv64-redhat-linux-gnu-default64-kv3;  fi
/usr/lib/gap/bin/riscv64-redhat-linux-gnu-default64-kv3/gac -d -o bin/riscv64-redhat-linux-gnu-default64-kv3/ediv.so src/ediv.c
/bin/sh /usr/bin/libtool --mode=compile gcc -O2 -g -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -fexceptions -fstack-protector-strong -grecord-gcc-switches -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -fasynchronous-unwind-tables -fstack-clash-protection -o /tmp/gac41dXBPr/29894_ediv.lo -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS=/usr/lib/gap -c src/ediv.c
libtool: compile:  gcc -O2 -g -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -fexceptions -fstack-protector-strong -grecord-gcc-switches -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -fasynchronous-unwind-tables -fstack-clash-protection -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS=/usr/lib/gap -c src/ediv.c  -fPIC -DPIC -o /tmp/gac41dXBPr/.libs/29894_ediv.o
libtool: compile:  gcc -O2 -g -pipe -Wall -Werror=format-security -Wp,-D_FORTIFY_SOURCE=2 -Wp,-D_GLIBCXX_ASSERTIONS -fexceptions -fstack-protector-strong -grecord-gcc-switches -specs=/usr/lib/rpm/redhat/redhat-annobin-cc1 -fasynchronous-unwind-tables -fstack-clash-protection -I/usr/lib/gap/gen -I/usr/lib/gap/src -I/usr/lib/gap -DHAVE_CONFIG_H -DSYS_DEFAULT_PATHS=/usr/lib/gap -c src/ediv.c -o /tmp/gac41dXBPr/29894_ediv.o >/dev/null 2>&1
/bin/sh /usr/bin/libtool --mode=link gcc -module -avoid-version -rpath /usr/lib64 -Wl,-z,relro -Wl,--as-needed -o bin/riscv64-redhat-linux-gnu-default64-kv3/ediv.so /tmp/gac41dXBPr/29894_ediv.lo
libtool: link: gcc -shared  -fPIC -DPIC  /tmp/gac41dXBPr/.libs/29894_ediv.o    -Wl,-z -Wl,relro -Wl,--as-needed   -Wl,-soname -Wl,ediv.so -o bin/riscv64-redhat-linux-gnu-default64-kv3/.libs/ediv.so
libtool: link: ar cru bin/riscv64-redhat-linux-gnu-default64-kv3/.libs/ediv.a  /tmp/gac41dXBPr/29894_ediv.o
libtool: link: ranlib bin/riscv64-redhat-linux-gnu-default64-kv3/.libs/ediv.a
libtool: link: ( cd "bin/riscv64-redhat-linux-gnu-default64-kv3/.libs" && rm -f "ediv.la" && ln -s "../ediv.la" "ediv.la" )
rm -f /tmp/gac41dXBPr/29894_ediv.lo
 ┌───────┐   GAP 4.10.2 of 19-Jun-2019
 │  GAP  │   https://www.gap-system.org
 └───────┘   Architecture: riscv64-redhat-linux-gnu-default64-kv3
 Configuration:  gmp 6.1.2, readline
 Loading the library and packages ...
#I  autpgrp package is not available. Check that the name is correct
#I  and it is present in one of the GAP root directories (see '??RootPaths')
#I  alnuth package is not available. Check that the name is correct
#I  and it is present in one of the GAP root directories (see '??RootPaths')
#I  crisp package is not available. Check that the name is correct
#I  and it is present in one of the GAP root directories (see '??RootPaths')
#I  ctbllib package is not available. Check that the name is correct
#I  and it is present in one of the GAP root directories (see '??RootPaths')
#I  factint package is not available. Check that the name is correct
#I  and it is present in one of the GAP root directories (see '??RootPaths')
#I  fga package is not available. Check that the name is correct
#I  and it is present in one of the GAP root directories (see '??RootPaths')
#I  irredsol package is not available. Check that the name is correct
#I  and it is present in one of the GAP root directories (see '??RootPaths')
#I  laguna package is not available. Check that the name is correct
#I  and it is present in one of the GAP root directories (see '??RootPaths')
#I  polenta package is not available. Check that the name is correct
#I  and it is present in one of the GAP root directories (see '??RootPaths')
#I  polycyclic package is not available. Check that the name is correct
#I  and it is present in one of the GAP root directories (see '??RootPaths')
#I  resclasses package is not available. Check that the name is correct
#I  and it is present in one of the GAP root directories (see '??RootPaths')
#I  sophus package is not available. Check that the name is correct
#I  and it is present in one of the GAP root directories (see '??RootPaths')
#I  tomlib package is not available. Check that the name is correct
#I  and it is present in one of the GAP root directories (see '??RootPaths')
 Packages:   GAPDoc 1.6.3, PrimGrp 3.3.2, SmallGrp 1.3, TransGrp 2.0.4
 Try '??help' for help. See also '?copyright', '?cite' and '?authors'
gap> gap> gap> gap> gap> true
gap> gap> #I Composing XML document . . .
#I Parsing XML document . . .
#I Checking XML structure . . .
#I Text version (also produces labels for hyperlinks):
#I First run, collecting cross references, index, toc, bib and so on . . .
#I Table of contents complete.
#I Producing the index . . .
#I Reading bibliography data files . . . 
#I   BibTeX format: edim
#I   checking and translating to BibXMLext . . .
#I Writing bibliography . . .
#I Second run through document . . .
#I Producing simplified search strings and labels for hyperlinks . . .
#I Constructing LaTeX version and calling pdflatex:
#I Writing LaTeX file, 4 x pdflatex with bibtex and makeindex, 
#I Writing manual.six file ... 
#I Finally the HTML version . . .
#I First run, collecting cross references, index, toc, bib and so on . . .
#I Table of contents complete.
#I Producing the index . . .
#I Writing bibliography . . .
#I Second run through document . . .
#I - also HTML version for MathJax . . .
#I First run, collecting cross references, index, toc, bib and so on . . .
#I Table of contents complete.
#I Producing the index . . .
#I Writing bibliography . . .
#I Second run through document . . .
rec( Name := "edim", attributes := rec(  ), bibdata := "edim", 
  bibentries := [ <BibXMLext entry: D82>, <BibXMLext entry: HMM98>, 
      <BibXMLext entry: HS79>, <BibXMLext entry: L98> ], 
  bibkeys := [ "D82", "HMM98", "HS79", "L98" ], 
  biblabels := [ "Dix82", "HMM98", "HS79", "Lüb02" ], bibpath := dir("doc/"),
  bibstrings := [ [ 0, 0 ] ], 
  bibtext := "\n<p><a id=\"biBD82\" name=\"biBD82\"></a></p>\n<p class='BibEnt\
ry'>\n[<span class='BibKey'>Dix82</span>]   <b class='BibAuthor'>Dixon, J. D.<\
/b>,\n <i class='BibTitle'>Exact Solution of Linear Equations Using p-adic Exp\
ansions</i>,\n <span class='BibJournal'>Numer. Math.</span>,\n <em class='BibV\
olume'>40</em>\n (<span class='BibYear'>1982</span>),\n <span class='BibPages'\
>137--141</span>.\n</p>\n\n\n<p><a id=\"biBHMM98\" name=\"biBHMM98\"></a></p>\
\n<p class='BibEntry'>\n[<span class='BibKey'>HMM98</span>]   <b class='BibAut\
hor'>Havas, G., Majewski, B. S. and Matthews, K. R.</b>,\n <i class='BibTitle'\
>Extended gcd and Hermite normal form algorithms via lattice\n      basis redu\
ction</i>,\n <span class='BibJournal'>Experimental Mathematics</span>,\n <em c\
lass='BibVolume'>7</em>\n (<span class='BibYear'>1998</span>),\n <span class='\
BibPages'>125--135</span>.\n</p>\n\n\n<p><a id=\"biBHS79\" name=\"biBHS79\"></\
a></p>\n<p class='BibEntry'>\n[<span class='BibKey'>HS79</span>]   <b class='B\
ibAuthor'>Havas, G. and Sterling, L. S.</b>,\n <i class='BibTitle'>Integer mat\
rices and abelian groups</i>,\n  in  <i class='BibBooktitle'>Symbolic and alge\
braic computation</i>,\n <span class='BibPublisher'>Springer-Verlag, Berlin</s\
pan>,\n <span class='BibSeries'>Lecture Notes in Computer Science</span>,\n <e\
m class='BibVolume'>72</em>\n (<span class='BibYear'>1979</span>),\n <span cla\
ss='BibPages'>431--451</span>.\n</p>\n\n\n<p><a id=\"biBL98\" name=\"biBL98\">\
</a></p>\n<p class='BibEntry'>\n[<span class='BibKey'>Lüb02</span>]   <b clas\
s='BibAuthor'>Lübeck, F.</b>,\n <i class='BibTitle'>On the Computation of \n \
                 Elementary Divisors of Integer Matrices</i>,\n <span class='B\
ibJournal'>Journal of Symbolic Computation</span>,\n <em class='BibVolume'>33<\
/em>\n (<span class='BibYear'>2002</span>),\n <span class='BibPages'>57--65</s\
pan>.\n</p>\n\n", bookname := "edim", 
  chapsectlinks := 
    rec( 
      1 := "<div class=\"ChapSects\"><a href=\"chap1_mj.html#X7AA826067CC8C395\
\">1 <span class=\"Heading\">The <strong class=\"pkg\">EDIM</strong>-Package</\
span></a>\n<div class=\"ContSect\"><span class=\"tocline\"><span class=\"nocss\
\">&nbsp;</span><a href=\"chap1_mj.html#X84D3F5E77E3BF046\">1.1 <span class=\"\
Heading\">Installation of the <strong class=\"pkg\">EDIM</strong> package</spa\
n></a>\n</span>\n<div class=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><sp\
an class=\"nocss\">&nbsp;&nbsp;</span><a href=\"chap1_mj.html#X7AF83FEC7CD0311\
C\">1.1-1 InfoEDIM</a></span>\n</div></div>\n<div class=\"ContSect\"><span cla\
ss=\"tocline\"><span class=\"nocss\">&nbsp;</span><a href=\"chap1_mj.html#X818\
BE04687230849\">1.2 <span class=\"Heading\"><span class=\"SimpleMath\">\\(p\\)\
</span>-Parts of Elementary Divisors</span></a>\n</span>\n<div class=\"ContSSB\
lock\">\n<span class=\"ContSS\"><br /><span class=\"nocss\">&nbsp;&nbsp;</span\
><a href=\"chap1_mj.html#X813B0D73868CD751\">1.2-1 ElementaryDivisorsPPartRk</\
a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&nbsp;&nbsp;</sp\
an><a href=\"chap1_mj.html#X82EC4724865F4DF9\">1.2-2 ElementaryDivisorsPPartHa\
vasSterling</a></span>\n</div></div>\n<div class=\"ContSect\"><span class=\"to\
cline\"><span class=\"nocss\">&nbsp;</span><a href=\"chap1_mj.html#X80FF39C07E\
03D7EF\">1.3 <span class=\"Heading\">Inverse of Rational Matrices</span></a>\n\
</span>\n<div class=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><span class\
=\"nocss\">&nbsp;&nbsp;</span><a href=\"chap1_mj.html#X7A9656D47C4D2D16\">1.3-\
1 InverseRatMat</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"\
>&nbsp;&nbsp;</span><a href=\"chap1_mj.html#X8302B31E86B3AFDB\">1.3-2 Rational\
SolutionIntMat</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">\
&nbsp;&nbsp;</span><a href=\"chap1_mj.html#X86DA61D978D2D889\">1.3-3 ExponentS\
quareIntMatFullRank</a></span>\n</div></div>\n<div class=\"ContSect\"><span cl\
ass=\"tocline\"><span class=\"nocss\">&nbsp;</span><a href=\"chap1_mj.html#X7A\
6548FA7C837D27\">1.4 <span class=\"Heading\">All Elementary Divisors Using p-a\
dic Method</span></a>\n</span>\n<div class=\"ContSSBlock\">\n<span class=\"Con\
tSS\"><br /><span class=\"nocss\">&nbsp;&nbsp;</span><a href=\"chap1_mj.html#X\
7B6A3B8486872B0C\">1.4-1 ElementaryDivisorsSquareIntMatFullRank</a></span>\n<s\
pan class=\"ContSS\"><br /><span class=\"nocss\">&nbsp;&nbsp;</span><a href=\"\
chap1_mj.html#X821E30477A5DCE68\">1.4-2 ElementaryDivisorsIntMatDeterminant</a\
></span>\n</div></div>\n<div class=\"ContSect\"><span class=\"tocline\"><span \
class=\"nocss\">&nbsp;</span><a href=\"chap1_mj.html#X788047737FA04422\">1.5 <\
span class=\"Heading\">Gcd and Normal Forms Using LLL</span></a>\n</span>\n<di\
v class=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><span class=\"nocss\">&\
nbsp;&nbsp;</span><a href=\"chap1_mj.html#X799B1A5285D00859\">1.5-1 GcdexIntLL\
L</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&nbsp;&nbsp;<\
/span><a href=\"chap1_mj.html#X7C6AE6777B72F9D2\">1.5-2 HermiteIntMatLLL</a></\
span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&nbsp;&nbsp;</span><\
a href=\"chap1_mj.html#X862962B7878A284F\">1.5-3 HermiteIntMatLLLTrans</a></sp\
an>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&nbsp;&nbsp;</span><a \
href=\"chap1_mj.html#X8626F15179C09798\">1.5-4 SmithIntMatLLL</a></span>\n<spa\
n class=\"ContSS\"><br /><span class=\"nocss\">&nbsp;&nbsp;</span><a href=\"ch\
ap1_mj.html#X86094B1B7C87EBF6\">1.5-5 SmithIntMatLLLTrans</a></span>\n</div></\
div>\n<div class=\"ContSect\"><span class=\"tocline\"><span class=\"nocss\">&n\
bsp;</span><a href=\"chap1_mj.html#X832582857EB36B23\">1.6 <span class=\"Headi\
ng\">Utility Functions from the <strong class=\"pkg\">EDIM</strong>-package</s\
pan></a>\n</span>\n<div class=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><\
span class=\"nocss\">&nbsp;&nbsp;</span><a href=\"chap1_mj.html#X7EC844F97C469\
C03\">1.6-1 RatNumberFromModular</a></span>\n<span class=\"ContSS\"><br /><spa\
n class=\"nocss\">&nbsp;&nbsp;</span><a href=\"chap1_mj.html#X78518E1D81435762\
\">1.6-2 InverseIntMatMod</a></span>\n<span class=\"ContSS\"><br /><span class\
=\"nocss\">&nbsp;&nbsp;</span><a href=\"chap1_mj.html#X7CDF3D2081207080\">1.6-\
3 HadamardBoundIntMat</a></span>\n<span class=\"ContSS\"><br /><span class=\"n\
ocss\">&nbsp;&nbsp;</span><a href=\"chap1_mj.html#X7BAB977C7EB05067\">1.6-4 Ch\
eapFactorsInt</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&\
nbsp;&nbsp;</span><a href=\"chap1_mj.html#X8312EDA78209B4EA\">1.6-5 RankMod</a\
></span>\n</div></div>\n<div class=\"ContSect\"><span class=\"tocline\"><span \
class=\"nocss\">&nbsp;</span><a href=\"chap1_mj.html#X83FD1AAB7EB2E934\">1.7 <\
span class=\"Heading\">InverseRatMat - the Algorithm</span></a>\n</span>\n<div\
 class=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><span class=\"nocss\">&n\
bsp;&nbsp;</span><a href=\"chap1_mj.html#X791ED7D97F87FDFD\">1.7-1 <span class\
=\"Heading\">Rank of Integer Matrix</span></a>\n</span>\n</div></div>\n</div>\
\n" ), 
  chapsectlinkstext := 
    rec( 
      1 := "<div class=\"ChapSects\"><a href=\"chap1_mj.html#X7AA826067CC8C395\
\">1 <span class=\"Heading\">The <strong class=\"pkg\">EDIM</strong>-Package</\
span></a>\n<div class=\"ContSect\"><span class=\"tocline\"><span class=\"nocss\
\">&nbsp;</span><a href=\"chap1_mj.html#X84D3F5E77E3BF046\">1.1 <span class=\"\
Heading\">Installation of the <strong class=\"pkg\">EDIM</strong> package</spa\
n></a>\n</span>\n<div class=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><sp\
an class=\"nocss\">&nbsp;&nbsp;</span><a href=\"chap1_mj.html#X7AF83FEC7CD0311\
C\">1.1-1 InfoEDIM</a></span>\n</div></div>\n<div class=\"ContSect\"><span cla\
ss=\"tocline\"><span class=\"nocss\">&nbsp;</span><a href=\"chap1_mj.html#X818\
BE04687230849\">1.2 <span class=\"Heading\"><span class=\"SimpleMath\">\\(p\\)\
</span>-Parts of Elementary Divisors</span></a>\n</span>\n<div class=\"ContSSB\
lock\">\n<span class=\"ContSS\"><br /><span class=\"nocss\">&nbsp;&nbsp;</span\
><a href=\"chap1_mj.html#X813B0D73868CD751\">1.2-1 ElementaryDivisorsPPartRk</\
a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&nbsp;&nbsp;</sp\
an><a href=\"chap1_mj.html#X82EC4724865F4DF9\">1.2-2 ElementaryDivisorsPPartHa\
vasSterling</a></span>\n</div></div>\n<div class=\"ContSect\"><span class=\"to\
cline\"><span class=\"nocss\">&nbsp;</span><a href=\"chap1_mj.html#X80FF39C07E\
03D7EF\">1.3 <span class=\"Heading\">Inverse of Rational Matrices</span></a>\n\
</span>\n<div class=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><span class\
=\"nocss\">&nbsp;&nbsp;</span><a href=\"chap1_mj.html#X7A9656D47C4D2D16\">1.3-\
1 InverseRatMat</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\"\
>&nbsp;&nbsp;</span><a href=\"chap1_mj.html#X8302B31E86B3AFDB\">1.3-2 Rational\
SolutionIntMat</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">\
&nbsp;&nbsp;</span><a href=\"chap1_mj.html#X86DA61D978D2D889\">1.3-3 ExponentS\
quareIntMatFullRank</a></span>\n</div></div>\n<div class=\"ContSect\"><span cl\
ass=\"tocline\"><span class=\"nocss\">&nbsp;</span><a href=\"chap1_mj.html#X7A\
6548FA7C837D27\">1.4 <span class=\"Heading\">All Elementary Divisors Using p-a\
dic Method</span></a>\n</span>\n<div class=\"ContSSBlock\">\n<span class=\"Con\
tSS\"><br /><span class=\"nocss\">&nbsp;&nbsp;</span><a href=\"chap1_mj.html#X\
7B6A3B8486872B0C\">1.4-1 ElementaryDivisorsSquareIntMatFullRank</a></span>\n<s\
pan class=\"ContSS\"><br /><span class=\"nocss\">&nbsp;&nbsp;</span><a href=\"\
chap1_mj.html#X821E30477A5DCE68\">1.4-2 ElementaryDivisorsIntMatDeterminant</a\
></span>\n</div></div>\n<div class=\"ContSect\"><span class=\"tocline\"><span \
class=\"nocss\">&nbsp;</span><a href=\"chap1_mj.html#X788047737FA04422\">1.5 <\
span class=\"Heading\">Gcd and Normal Forms Using LLL</span></a>\n</span>\n<di\
v class=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><span class=\"nocss\">&\
nbsp;&nbsp;</span><a href=\"chap1_mj.html#X799B1A5285D00859\">1.5-1 GcdexIntLL\
L</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&nbsp;&nbsp;<\
/span><a href=\"chap1_mj.html#X7C6AE6777B72F9D2\">1.5-2 HermiteIntMatLLL</a></\
span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&nbsp;&nbsp;</span><\
a href=\"chap1_mj.html#X862962B7878A284F\">1.5-3 HermiteIntMatLLLTrans</a></sp\
an>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&nbsp;&nbsp;</span><a \
href=\"chap1_mj.html#X8626F15179C09798\">1.5-4 SmithIntMatLLL</a></span>\n<spa\
n class=\"ContSS\"><br /><span class=\"nocss\">&nbsp;&nbsp;</span><a href=\"ch\
ap1_mj.html#X86094B1B7C87EBF6\">1.5-5 SmithIntMatLLLTrans</a></span>\n</div></\
div>\n<div class=\"ContSect\"><span class=\"tocline\"><span class=\"nocss\">&n\
bsp;</span><a href=\"chap1_mj.html#X832582857EB36B23\">1.6 <span class=\"Headi\
ng\">Utility Functions from the <strong class=\"pkg\">EDIM</strong>-package</s\
pan></a>\n</span>\n<div class=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><\
span class=\"nocss\">&nbsp;&nbsp;</span><a href=\"chap1_mj.html#X7EC844F97C469\
C03\">1.6-1 RatNumberFromModular</a></span>\n<span class=\"ContSS\"><br /><spa\
n class=\"nocss\">&nbsp;&nbsp;</span><a href=\"chap1_mj.html#X78518E1D81435762\
\">1.6-2 InverseIntMatMod</a></span>\n<span class=\"ContSS\"><br /><span class\
=\"nocss\">&nbsp;&nbsp;</span><a href=\"chap1_mj.html#X7CDF3D2081207080\">1.6-\
3 HadamardBoundIntMat</a></span>\n<span class=\"ContSS\"><br /><span class=\"n\
ocss\">&nbsp;&nbsp;</span><a href=\"chap1_mj.html#X7BAB977C7EB05067\">1.6-4 Ch\
eapFactorsInt</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&\
nbsp;&nbsp;</span><a href=\"chap1_mj.html#X8312EDA78209B4EA\">1.6-5 RankMod</a\
></span>\n</div></div>\n<div class=\"ContSect\"><span class=\"tocline\"><span \
class=\"nocss\">&nbsp;</span><a href=\"chap1_mj.html#X83FD1AAB7EB2E934\">1.7 <\
span class=\"Heading\">InverseRatMat - the Algorithm</span></a>\n</span>\n<div\
 class=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><span class=\"nocss\">&n\
bsp;&nbsp;</span><a href=\"chap1_mj.html#X791ED7D97F87FDFD\">1.7-1 <span class\
=\"Heading\">Rank of Integer Matrix</span></a>\n</span>\n</div></div>\n</div>\
\n" ), 
  chaptitle := 
    rec( 
      1 := "<span class=\"Heading\">The <strong class=\"pkg\">EDIM</strong>-Pa\
ckage</span>" ), 
  content := 
    [ 
      rec( content := "xml version=\"1.0\" encoding=\"UTF-8\"", 
          count := [ 0, 0, 0, 1 ], name := "XMLPI", root := ~ ), 
      rec( content := "Book SYSTEM \"gapdoc.dtd\"", count := [ 0, 0, 0, 1 ], 
          name := "XMLDOCTYPE", root := ~ ), 
      rec( 
          content := "\n\n  manual.xml          EDIM mini-package documentatio\
n        Frank Lübeck\n\n  Copyright (C) 2000-2018, Lehrstuhl D für Mathemat\
ik, RWTH-Aachen\n\n", count := [ 0, 0, 0, 1 ], name := "XMLCOMMENT", 
          root := ~ ), 
      rec( attributes := rec( Name := "edim" ), 
          content := 
            [ 
              rec( attributes := rec(  ), 
                  content := 
                    [ 
                      rec( attributes := rec(  ), 
                          content := 
                            [ 
                              rec( attributes := rec(  ), 
                                  content := 
                                    [ 
                                      rec( content := "EDIM", 
                                          count := [ 0, 0, 0, 1 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ) ], 
                                  count := [ 0, 0, 0, 1 ], mathmode := "Text",
                                  name := "Package", next := 300, root := ~, 
                                  start := 277, stop := 299 ) ], 
                          count := [ 0, 0, 0, 1 ], name := "Title", 
                          next := 308, root := ~, start := 270, stop := 307 ),
                      rec( attributes := rec(  ), 
                          content := 
                            [ 
                              rec( 
                                  content := "Elementary Divisors and Integer M\
atrices", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", 
                                  root := ~ ) ], count := [ 0, 0, 0, 1 ], 
                          name := "Subtitle", next := 374, root := ~, 
                          start := 313, stop := 373 ), 
                      rec( attributes := rec(  ), 
                          content := 
                            [ 
                              rec( content := "\n      Version 1.3.3\n\n    ", 
                                  count := [ 0, 0, 0, 1 ], mathmode := "Text",
                                  name := "PCDATA", root := ~ ) ], 
                          count := [ 0, 0, 0, 1 ], name := "Version", 
                          next := 424, root := ~, start := 379, stop := 423 ),
                      rec( attributes := rec(  ), 
                          content := 
                            [ 
                              rec( content := "Frank Lübeck\n", 
                                  count := [ 0, 0, 0, 1 ], mathmode := "Text",
                                  name := "PCDATA", root := ~ ), 
                              rec( attributes := rec(  ), 
                                  content := 
                                    [ 
                                      rec( 
                                          content := "\nLehrstuhl D für Mathem\
atik\nRWTH Aachen\nPontdriesch 14/16\n52062 Aachen\nGermany\n", 
                                          count := [ 0, 0, 0, 2 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ) ], 
                                  count := [ 0, 0, 0, 2 ], name := "Address", 
                                  next := 550, root := ~, start := 451, 
                                  stop := 549 ), 
                              rec( content := "\n    ", 
                                  count := [ 0, 0, 0, 3 ], mathmode := "Text",
                                  name := "PCDATA", root := ~ ), 
                              rec( attributes := rec(  ), 
                                  content := 
                                    [ 
                                      rec( 
                                          content := "Frank.Luebeck@Math.RWTH-A\
achen.De", count := [ 0, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", 
                                          root := ~ ) ], 
                                  count := [ 0, 0, 0, 3 ], mathmode := "Text",
                                  name := "Email", next := 603, root := ~, 
                                  start := 555, stop := 602 ), 
                              rec( content := "\n    ", 
                                  count := [ 0, 0, 0, 3 ], mathmode := "Text",
                                  name := "PCDATA", root := ~ ), 
                              rec( attributes := rec(  ), 
                                  content := 
                                    [ 
                                      rec( 
                                          content := "http://www.math.rwth-aach\
en.de/~Frank.Luebeck", count := [ 0, 0, 0, 3 ], mathmode := "Text", 
                                          name := "PCDATA", root := ~ ) ], 
                                  count := [ 0, 0, 0, 3 ], mathmode := "Text",
                                  name := "Homepage", next := 680, root := ~, 
                                  start := 608, stop := 679 ), 
                              rec( content := "\n    ", 
                                  count := [ 0, 0, 0, 3 ], mathmode := "Text",
                                  name := "PCDATA", root := ~ ) ], 
                          count := [ 0, 0, 0, 1 ], name := "Author", 
                          next := 694, root := ~, start := 429, stop := 693 ),
                      rec( attributes := rec(  ), 
                          content := 
                            [ 
                              rec( content := "January 2018", 
                                  count := [ 0, 0, 0, 3 ], mathmode := "Text",
                                  name := "PCDATA", root := ~ ) ], 
                          count := [ 0, 0, 0, 3 ], name := "Date", 
                          next := 724, root := ~, start := 699, stop := 723 ),
                      rec( attributes := rec(  ), 
                          content := 
                            [ 
                              rec( content := " ", count := [ 0, 0, 1, 1 ], 
                                  mathmode := "Text", name := "PCDATA", 
                                  root := ~ ), 
                              rec( attributes := rec(  ), 
                                  content := 
                                    [ 
                                      rec( content := "License", 
                                          count := [ 0, 0, 1, 1 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ) ], 
                                  count := [ 0, 0, 1, 1 ], mathmode := "Text",
                                  name := "Index", next := 763, root := ~, 
                                  start := 741, stop := 762 ), 
                              rec( 
                                  content := "\n              © 2000-2018 by \
Frank Lübeck ", count := [ 0, 0, 1, 1 ], mathmode := "Text", 
                                  name := "PCDATA", root := ~ ), 
                              rec( attributes := rec(  ), content := 0, 
                                  count := [ 0, 0, 1, 2 ], mathmode := "Text",
                                  name := "P", next := 821, root := ~, 
                                  start := 817, stop := 820 ), 
                              rec( content := "\n              ", 
                                  count := [ 0, 0, 1, 2 ], mathmode := "Text",
                                  name := "PCDATA", root := ~ ), 
                              rec( attributes := rec(  ), 
                                  content := 
                                    [ 
                                      rec( content := "EDIM", 
                                          count := [ 0, 0, 1, 2 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ) ], 
                                  count := [ 0, 0, 1, 2 ], mathmode := "Text",
                                  name := "Package", next := 859, root := ~, 
                                  start := 836, stop := 858 ), 
                              rec( 
                                  content := "  \n              is free softwa\
re; you can redistribute it\n              and/or modify it under the terms of\
 the ", count := [ 0, 0, 1, 2 ], mathmode := "Text", name := "PCDATA", 
                                  root := ~ ), 
                              rec( 
                                  attributes := 
                                    rec( Text := "GNU General Public License" 
                                     ), 
                                  content := 
                                    [ 
                                      rec( 
                                          content := "http://www.fsf.org/licens\
es/gpl.html", count := [ 0, 0, 1, 2 ], mathmode := "Text", name := "PCDATA", 
                                          root := ~ ) ], 
                                  count := [ 0, 0, 1, 2 ], mathmode := "Text",
                                  name := "URL", next := 1067, root := ~, 
                                  start := 972, stop := 1066 ), 
                              rec( 
                                  content := "              as published by th\
e Free Software Foundation;\n              either version 2 of the License, or\
 (at your option) any\n              later version.\n    ", 
                                  count := [ 0, 0, 1, 2 ], mathmode := "Text",
                                  name := "PCDATA", root := ~ ) ], 
                          count := [ 0, 0, 1, 1 ], name := "Copyright", 
                          next := 1243, root := ~, start := 729, stop := 1242 
                         ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", 
                  name := "TitlePage", next := 1257, root := ~, start := 254, 
                  stop := 1256 ), 
              rec( 
                  content := "  TableOfContents  . . . . . .  generate a table\
 of contents   ", count := [ 0, 0, 1, 2 ], mathmode := "Text", 
                  name := "XMLCOMMENT", root := ~ ), 
              rec( attributes := rec(  ), content := 0, 
                  count := [ 0, 0, 2, 1 ], mathmode := "Text", 
                  name := "TableOfContents", next := 1348, root := ~, 
                  start := 1330, stop := 1347 ), 
              rec( content := "  The only chapter of the book  ", 
                  count := [ 0, 0, 2, 1 ], mathmode := "Text", 
                  name := "XMLCOMMENT", root := ~ ), 
              rec( attributes := rec(  ), 
                  content := 
                    [ 
                      rec( content := "\n", count := [ 0, 0, 2, 1 ], 
                          mathmode := "Text", name := "PCDATA", root := ~ ), 
                      rec( 
                          content := " \n\n  main.xml            EDIM package \
documentation        Frank Lübeck\n\n         Copyright (C) 1999-2004, Lehrst\
uhl D für Mathematik, RWTH-Aachen\n\nThis is the main chapter of the document\
ation.\n\n", count := [ 0, 0, 2, 1 ], mathmode := "Text", 
                          name := "XMLCOMMENT", root := ~ ), 
                      rec( content := "\n\n", count := [ 0, 0, 2, 1 ], 
                          mathmode := "Text", name := "PCDATA", root := ~ ), 
                      rec( attributes := rec( Label := "Chap-EDIM" ), 
                          content := 
                            [ 
                              rec( content := "\n", count := [ 1, 0, 0, 1 ], 
                                  mathmode := "Text", name := "PCDATA", 
                                  root := ~ ), 
                              rec( attributes := rec(  ), 
                                  content := 
                                    [ 
                                      rec( content := "The ", 
                                          count := [ 1, 0, 0, 2 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "EDIM", 
                                                  count := [ 1, 0, 0, 2 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 0, 0, 2 ],
                                          mathmode := "Text", 
                                          name := "Package", next := 1669, 
                                          root := ~, start := 1646, 
                                          stop := 1668 ), 
                                      rec( content := "-Package", 
                                          count := [ 1, 0, 0, 2 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ) ], 
                                  count := [ 1, 0, 0, 2 ], mathmode := "Text",
                                  name := "Heading", next := 1687, root := ~, 
                                  start := 1633, stop := 1686 ), 
                              rec( content := "\n", count := [ 1, 0, 0, 3 ], 
                                  mathmode := "Text", name := "PCDATA", 
                                  root := ~ ), 
                              rec( attributes := rec(  ), 
                                  content := 
                                    [ 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "EDIM", 
                                                  count := [ 1, 0, 0, 3 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 0, 0, 3 ],
                                          mathmode := "Text", 
                                          name := "Package", next := 1718, 
                                          root := ~, start := 1695, 
                                          stop := 1717 ) ], 
                                  count := [ 1, 0, 0, 3 ], mathmode := "Text",
                                  name := "Index", next := 1726, root := ~, 
                                  start := 1688, stop := 1725 ), 
                              rec( content := "\n\n", count := [ 1, 0, 0, 3 ],
                                  mathmode := "Text", name := "PCDATA", 
                                  root := ~ ), 
                              rec( attributes := rec(  ), 
                                  content := 
                                    [ 
                                      rec( 
                                          content := "(Elementary   Divisors   \
and   Integer    Matrices,  by    Frank\nLübeck)", count := [ 1, 0, 0, 3 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ) ], 
                                  count := [ 1, 0, 0, 3 ], mathmode := "Text",
                                  name := "Emph", next := 1814, root := ~, 
                                  start := 1728, stop := 1813 ), 
                              rec( attributes := rec(  ), content := 0, 
                                  count := [ 1, 0, 0, 4 ], mathmode := "Text",
                                  name := "P", next := 1818, root := ~, 
                                  start := 1814, stop := 1817 ), 
                              rec( 
                                  content := "\n\nThis  chapter describes  the\
  functions defined  in  the ", count := [ 1, 0, 0, 4 ], mathmode := "Text", 
                                  name := "PCDATA", root := ~ ), 
                              rec( attributes := rec(  ), 
                                  content := 
                                    [ 
                                      rec( content := "GAP", 
                                          count := [ 1, 0, 0, 4 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ) ], 
                                  count := [ 1, 0, 0, 4 ], mathmode := "Text",
                                  name := "Package", next := 23, root := ~, 
                                  start := 1, stop := 22 ), 
                              rec( content := "4  package\n", 
                                  count := [ 1, 0, 0, 4 ], mathmode := "Text",
                                  name := "PCDATA", root := ~ ), 
                              rec( attributes := rec(  ), 
                                  content := 
                                    [ 
                                      rec( content := "EDIM", 
                                          count := [ 1, 0, 0, 4 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ) ], 
                                  count := [ 1, 0, 0, 4 ], mathmode := "Text",
                                  name := "Package", next := 1916, root := ~, 
                                  start := 1893, stop := 1915 ), 
                              rec( 
                                  content := ".  The main  functions implement\
  variants of  an\nalgorithm for computing for a given prime ", 
                                  count := [ 1, 0, 0, 4 ], mathmode := "Text",
                                  name := "PCDATA", root := ~ ), 
                              rec( attributes := rec(  ), 
                                  content := 
                                    [ 
                                      rec( content := "p", 
                                          count := [ 1, 0, 0, 4 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ) ], 
                                  count := [ 1, 0, 0, 4 ], mathmode := "Text",
                                  name := "M", next := 2016, root := ~, 
                                  start := 2008, stop := 2015 ), 
                              rec( content := " the ", 
                                  count := [ 1, 0, 0, 4 ], mathmode := "Text",
                                  name := "PCDATA", root := ~ ), 
                              rec( attributes := rec(  ), 
                                  content := 
                                    [ 
                                      rec( content := "p", 
                                          count := [ 1, 0, 0, 4 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ) ], 
                                  count := [ 1, 0, 0, 4 ], mathmode := "Text",
                                  name := "M", next := 2029, root := ~, 
                                  start := 2021, stop := 2028 ), 
                              rec( 
                                  content := "-parts of\nthe  elementary divis\
ors  of  an integer  matrix.  These algorithms  use\na  ", 
                                  count := [ 1, 0, 0, 4 ], mathmode := "Text",
                                  name := "PCDATA", root := ~ ), 
                              rec( attributes := rec(  ), 
                                  content := 
                                    [ 
                                      rec( content := "p", 
                                          count := [ 1, 0, 0, 4 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ) ], 
                                  count := [ 1, 0, 0, 4 ], mathmode := "Text",
                                  name := "M", next := 2123, root := ~, 
                                  start := 2115, stop := 2122 ), 
                              rec( 
                                  content := "-adic  method  and  are  describ\
ed by  the  author  in  ", count := [ 1, 0, 0, 4 ], mathmode := "Text", 
                                  name := "PCDATA", root := ~ ), 
                              rec( attributes := rec( Key := "L98" ), 
                                  content := 0, count := [ 1, 0, 0, 4 ], 
                                  mathmode := "Text", name := "Cite", 
                                  next := 2196, root := ~, start := 2179, 
                                  stop := 2195 ), 
                              rec( content := " (see ", 
                                  count := [ 1, 0, 0, 4 ], mathmode := "Text",
                                  name := "PCDATA", root := ~ ), 
                              rec( 
                                  attributes := 
                                    rec( Func := "ElementaryDivisorsPPartRk" )
                                    , content := 0, count := [ 1, 0, 0, 4 ], 
                                  mathmode := "Text", name := "Ref", 
                                  next := 2241, root := ~, start := 2202, 
                                  stop := 2240 ), 
                              rec( content := ").", count := [ 1, 0, 0, 4 ], 
                                  mathmode := "Text", name := "PCDATA", 
                                  root := ~ ), 
                              rec( attributes := rec(  ), content := 0, 
                                  count := [ 1, 0, 0, 5 ], mathmode := "Text",
                                  name := "P", next := 2247, root := ~, 
                                  start := 2243, stop := 2246 ), 
                              rec( 
                                  content := "\n\nThese functions were already\
 applied to  integer matrices of dimension\ngreater  than  ", 
                                  count := [ 1, 0, 0, 5 ], mathmode := "Text",
                                  name := "PCDATA", root := ~ ), 
                              rec( attributes := rec(  ), 
                                  content := 
                                    [ 
                                      rec( content := "11000", 
                                          count := [ 1, 0, 0, 5 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ) ], 
                                  count := [ 1, 0, 0, 5 ], mathmode := "Text",
                                  name := "M", next := 2347, root := ~, 
                                  start := 2335, stop := 2346 ), 
                              rec( 
                                  content := " (which had  many   non-trivial \
 elementary\ndivisors which were products of small primes).", 
                                  count := [ 1, 0, 0, 5 ], mathmode := "Text",
                                  name := "PCDATA", root := ~ ), 
                              rec( attributes := rec(  ), content := 0, 
                                  count := [ 1, 0, 0, 6 ], mathmode := "Text",
                                  name := "P", next := 2441, root := ~, 
                                  start := 2437, stop := 2440 ), 
                              rec( 
                                  content := "\n\nFurthermore  there are  func\
tions  for finding  the biggest elementary\ndivisor of an invertible integer m\
atrix and  the inverse of a rational\ninvertible matrix  (see ", 
                                  count := [ 1, 0, 0, 6 ], mathmode := "Text",
                                  name := "PCDATA", root := ~ ), 
                              rec( 
                                  attributes := 
                                    rec( 
                                      Func := "ExponentSquareIntMatFullRank" )
                                    , content := 0, count := [ 1, 0, 0, 6 ], 
                                  mathmode := "Text", name := "Ref", 
                                  next := 2651, root := ~, start := 2609, 
                                  stop := 2650 ), 
                              rec( content := " and\n", 
                                  count := [ 1, 0, 0, 6 ], mathmode := "Text",
                                  name := "PCDATA", root := ~ ), 
                              rec( 
                                  attributes := rec( Func := "InverseRatMat" )
                                    , content := 0, count := [ 1, 0, 0, 6 ], 
                                  mathmode := "Text", name := "Ref", 
                                  next := 2684, root := ~, start := 2656, 
                                  stop := 2683 ), 
                              rec( content := ").   These   algorithms use  ",
                                  count := [ 1, 0, 0, 6 ], mathmode := "Text",
                                  name := "PCDATA", root := ~ ), 
                              rec( attributes := rec(  ), 
                                  content := 
                                    [ 
                                      rec( content := "p", 
                                          count := [ 1, 0, 0, 6 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ) ], 
                                  count := [ 1, 0, 0, 6 ], mathmode := "Text",
                                  name := "M", next := 2721, root := ~, 
                                  start := 2713, stop := 2720 ), 
                              rec( 
                                  content := "-adic\napproximations, explained\
 in ", count := [ 1, 0, 0, 6 ], mathmode := "Text", name := "PCDATA", 
                                  root := ~ ), 
                              rec( 
                                  attributes := 
                                    rec( Sect := "Sect-InvRatMatAlg" ), 
                                  content := 0, count := [ 1, 0, 0, 6 ], 
                                  mathmode := "Text", name := "Ref", 
                                  next := 2787, root := ~, start := 2756, 
                                  stop := 2786 ), 
                              rec( content := ".", count := [ 1, 0, 0, 6 ], 
                                  mathmode := "Text", name := "PCDATA", 
                                  root := ~ ), 
                              rec( attributes := rec(  ), content := 0, 
                                  count := [ 1, 0, 0, 7 ], mathmode := "Text",
                                  name := "P", next := 2792, root := ~, 
                                  start := 2788, stop := 2791 ), 
                              rec( 
                                  content := "\n\nFinally we distribute  imple\
mentations  of some other   algorithms for\nfinding elementary  divisors or no\
rmal  forms  of integer  matrices: A\n", count := [ 1, 0, 0, 7 ], 
                                  mathmode := "Text", name := "PCDATA", 
                                  root := ~ ), 
                              rec( attributes := rec(  ), 
                                  content := 
                                    [ 
                                      rec( content := "p", 
                                          count := [ 1, 0, 0, 7 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ) ], 
                                  count := [ 1, 0, 0, 7 ], mathmode := "Text",
                                  name := "M", next := 2944, root := ~, 
                                  start := 2936, stop := 2943 ), 
                              rec( 
                                  content := "-modular  algorithm  by    Havas\
  and   Sterling from    ", count := [ 1, 0, 0, 7 ], mathmode := "Text", 
                                  name := "PCDATA", root := ~ ), 
                              rec( attributes := rec( Key := "HS79" ), 
                                  content := 0, count := [ 1, 0, 0, 7 ], 
                                  mathmode := "Text", name := "Cite", 
                                  next := 3019, root := ~, start := 3001, 
                                  stop := 3018 ), 
                              rec( content := "  (see ", 
                                  count := [ 1, 0, 0, 7 ], mathmode := "Text",
                                  name := "PCDATA", root := ~ ), 
                              rec( 
                                  attributes := 
                                    rec( 
                                      Func := "ElementaryDivisorsPPartHavasSte\
rling" ), content := 0, count := [ 1, 0, 0, 7 ], mathmode := "Text", 
                                  name := "Ref", next := 3076, root := ~, 
                                  start := 3026, stop := 3075 ), 
                              rec( 
                                  content := ")\nand LLL-based  algorithms for\
 extended   greatest  common divisors  of\nintegers (see ", 
                                  count := [ 1, 0, 0, 7 ], mathmode := "Text",
                                  name := "PCDATA", root := ~ ), 
                              rec( attributes := rec( Func := "GcdexIntLLL" ),
                                  content := 0, count := [ 1, 0, 0, 7 ], 
                                  mathmode := "Text", name := "Ref", 
                                  next := 3188, root := ~, start := 3163, 
                                  stop := 3187 ), 
                              rec( 
                                  content := ") and  for Hermite normal forms\\
nof integer matrices with  (very nice) transforming matrices  (see ", 
                                  count := [ 1, 0, 0, 7 ], mathmode := "Text",
                                  name := "PCDATA", root := ~ ), 
                              rec( 
                                  attributes := 
                                    rec( Func := "HermiteIntMatLLL" ), 
                                  content := 0, count := [ 1, 0, 0, 7 ], 
                                  mathmode := "Text", name := "Ref", 
                                  next := 3316, root := ~, start := 3286, 
                                  stop := 3315 ), 
                              rec( content := ").", count := [ 1, 0, 0, 7 ], 
                                  mathmode := "Text", name := "PCDATA", 
                                  root := ~ ), 
                              rec( attributes := rec(  ), content := 0, 
                                  count := [ 1, 0, 0, 8 ], mathmode := "Text",
                                  name := "P", next := 3322, root := ~, 
                                  start := 3318, stop := 3321 ), 
                              rec( content := "\n\nBy   default  the   ", 
                                  count := [ 1, 0, 0, 8 ], mathmode := "Text",
                                  name := "PCDATA", root := ~ ), 
                              rec( attributes := rec(  ), 
                                  content := 
                                    [ 
                                      rec( content := "EDIM", 
                                          count := [ 1, 0, 0, 8 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ) ], 
                                  count := [ 1, 0, 0, 8 ], mathmode := "Text",
                                  name := "Package", next := 3367, root := ~, 
                                  start := 3344, stop := 3366 ), 
                              rec( 
                                  content := "   is  automatically   loaded\nb\
y   ", count := [ 1, 0, 0, 8 ], mathmode := "Text", name := "PCDATA", 
                                  root := ~ ), 
                              rec( attributes := rec(  ), 
                                  content := 
                                    [ 
                                      rec( content := "GAP", 
                                          count := [ 1, 0, 0, 8 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ) ], 
                                  count := [ 1, 0, 0, 8 ], mathmode := "Text",
                                  name := "Package", next := 23, root := ~, 
                                  start := 1, stop := 22 ), 
                              rec( 
                                  content := "   when   it   is  installed.   \
If   the   automatic   loading\nis   disabled  in   your  installation   you  \
must   load  the   package\nwith   ", count := [ 1, 0, 0, 8 ], 
                                  mathmode := "Text", name := "PCDATA", 
                                  root := ~ ), 
                              rec( attributes := rec(  ), 
                                  content := 
                                    [ 
                                      rec( 
                                          content := "RequirePackage(\"edim\");\
", count := [ 1, 0, 0, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                     ], count := [ 1, 0, 0, 8 ], 
                                  mathmode := "Text", name := "C", 
                                  next := 3580, root := ~, start := 3550, 
                                  stop := 3579 ), 
                              rec( 
                                  content := "   before  its   functions   bec\
ome\navailable.", count := [ 1, 0, 0, 8 ], mathmode := "Text", 
                                  name := "PCDATA", root := ~ ), 
                              rec( attributes := rec(  ), content := 0, 
                                  count := [ 1, 0, 0, 9 ], mathmode := "Text",
                                  name := "P", next := 3630, root := ~, 
                                  start := 3626, stop := 3629 ), 
                              rec( 
                                  content := "\n\nPlease,            send     \
        me            an            e-mail\n(", count := [ 1, 0, 0, 9 ], 
                                  mathmode := "Text", name := "PCDATA", 
                                  root := ~ ), 
                              rec( attributes := rec(  ), 
                                  content := 
                                    [ 
                                      rec( 
                                          content := "Frank.Luebeck@Math.RWTH-A\
achen.De", count := [ 1, 0, 0, 9 ], mathmode := "Text", name := "PCDATA", 
                                          root := ~ ) ], 
                                  count := [ 1, 0, 0, 9 ], mathmode := "Text",
                                  name := "Email", next := 3752, root := ~, 
                                  start := 3704, stop := 3751 ), 
                              rec( 
                                  content := ")  if  you  have  any\nquestions\
,  remarks, suggestions,  etc. concerning  this mini-package.\nAlso, I would l\
ike to hear about applications of this package.", count := [ 1, 0, 0, 9 ], 
                                  mathmode := "Text", name := "PCDATA", 
                                  root := ~ ), 
                              rec( attributes := rec(  ), content := 0, 
                                  count := [ 1, 0, 0, 10 ], 
                                  mathmode := "Text", name := "P", 
                                  next := 3911, root := ~, start := 3907, 
                                  stop := 3910 ), 
                              rec( content := "\n\nFrank Lübeck \n\n", 
                                  count := [ 1, 0, 0, 10 ], 
                                  mathmode := "Text", name := "PCDATA", 
                                  root := ~ ), 
                              rec( 
                                  attributes := rec( Label := "Sect-Install" )
                                    , 
                                  content := 
                                    [ 
                                      rec( content := "\n", 
                                          count := [ 1, 1, 0, 1 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  content := "Installation of t\
he ", count := [ 1, 1, 0, 2 ], mathmode := "Text", name := "PCDATA", 
                                                  root := ~ ), 
                                              rec( attributes := rec(  ), 
                                                  content := 
                                                    [ 
                                                      rec( content := "EDIM", 
                                                          count := 
                                                            [ 1, 1, 0, 2 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ) ], 
                                                  count := [ 1, 1, 0, 2 ], 
                                                  mathmode := "Text", 
                                                  name := "Package", 
                                                  next := 4012, root := ~, 
                                                  start := 3989, stop := 4011 
                                                 ), 
                                              rec( content := " package", 
                                                  count := [ 1, 1, 0, 2 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 1, 0, 2 ],
                                          mathmode := "Text", 
                                          name := "Heading", next := 4030, 
                                          root := ~, start := 3960, 
                                          stop := 4029 ), 
                                      rec( 
                                          content := "\n\nTo install this pack\
age first unpack it inside some GAP root directory\ninto the subdirectory ", 
                                          count := [ 1, 1, 0, 3 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "pkg/edim", 
                                                  count := [ 1, 1, 0, 3 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 1, 0, 3 ],
                                          mathmode := "Text", name := "F", 
                                          next := 4140, root := ~, 
                                          start := 4125, stop := 4139 ), 
                                      rec( content := " (see \n", 
                                          count := [ 1, 1, 0, 3 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( 
                                          attributes := 
                                            rec( BookName := "Reference", 
                                              Sect := "Installing a GAP Packag\
e" ), content := 0, count := [ 1, 1, 0, 3 ], mathmode := "Text", 
                                          name := "Ref", next := 4206, 
                                          root := ~, start := 4147, 
                                          stop := 4205 ), 
                                      rec( content := "). Then\nthe ", 
                                          count := [ 1, 1, 0, 3 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "EDIM", 
                                                  count := [ 1, 1, 0, 3 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 1, 0, 3 ],
                                          mathmode := "Text", 
                                          name := "Package", next := 4241, 
                                          root := ~, start := 4218, 
                                          stop := 4240 ), 
                                      rec( 
                                          content := " package can already be \
loaded and used.\nBut we strongly recommend to compile a kernel function as we\
ll during\ninstallation, otherwise \nthe function    ", 
                                          count := [ 1, 1, 0, 3 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( 
                                          attributes := 
                                            rec( 
                                              Func := "ElementaryDivisorsPPart\
RkExpSmall" ), content := 0, count := [ 1, 1, 0, 3 ], mathmode := "Text", 
                                          name := "Ref", next := 4440, 
                                          root := ~, start := 4393, 
                                          stop := 4439 ), 
                                      rec( 
                                          content := " \nwill not be available\
.", count := [ 1, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ )
                                        , 
                                      rec( attributes := rec(  ), 
                                          content := 0, 
                                          count := [ 1, 1, 0, 4 ], 
                                          mathmode := "Text", name := "P", 
                                          next := 4468, root := ~, 
                                          start := 4464, stop := 4467 ), 
                                      rec( 
                                          content := "\n\nTo install the kerne\
l function  go to  the  directory  ", count := [ 1, 1, 0, 4 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "pkg/edim", 
                                                  count := [ 1, 1, 0, 4 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 1, 0, 4 ],
                                          mathmode := "Text", name := "F", 
                                          next := 4540, root := ~, 
                                          start := 4525, stop := 4539 ), 
                                      rec( 
                                          content := " to\nwhich the package w\
as extracted and call", count := [ 1, 1, 0, 4 ], mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 0, 
                                          count := [ 1, 1, 0, 5 ], 
                                          mathmode := "Text", name := "P", 
                                          next := 4588, root := ~, 
                                          start := 4584, stop := 4587 ), 
                                      rec( content := "\n\n", 
                                          count := [ 1, 1, 0, 5 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  content := "/bin/sh ./configu\
re [path] [CONFIGNAME=...]", count := [ 1, 1, 0, 5 ], mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 1, 0, 5 ],
                                          mathmode := "Text", name := "C", 
                                          next := 4640, root := ~, 
                                          start := 4590, stop := 4639 ), 
                                      rec( attributes := rec(  ), 
                                          content := 0, 
                                          count := [ 1, 1, 0, 6 ], 
                                          mathmode := "Text", name := "P", 
                                          next := 4644, root := ~, 
                                          start := 4640, stop := 4643 ), 
                                      rec( content := "\n\nwhere ", 
                                          count := [ 1, 1, 0, 6 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "path", 
                                                  count := [ 1, 1, 0, 6 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 1, 0, 6 ],
                                          mathmode := "Text", name := "C", 
                                          next := 4663, root := ~, 
                                          start := 4652, stop := 4662 ), 
                                      rec( 
                                          content := "  is a path  to  the mai\
n ", count := [ 1, 1, 0, 6 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                         ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "GAP", 
                                                  count := [ 1, 1, 0, 6 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 1, 0, 6 ],
                                          mathmode := "Text", 
                                          name := "Package", next := 4711, 
                                          root := ~, start := 4689, 
                                          stop := 4710 ), 
                                      rec( 
                                          content := " root\ndirectory (if not\
 given, the default ", count := [ 1, 1, 0, 6 ], mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "../..", 
                                                  count := [ 1, 1, 0, 6 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 1, 0, 6 ],
                                          mathmode := "Text", name := "F", 
                                          next := 4766, root := ~, 
                                          start := 4754, stop := 4765 ), 
                                      rec( 
                                          content := " is assumed). \nIf you h\
ave installed several GAP kernels you can compile the corresponding\n", 
                                          count := [ 1, 1, 0, 6 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "EDIM", 
                                                  count := [ 1, 1, 0, 6 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 1, 0, 6 ],
                                          mathmode := "Text", 
                                          name := "Package", next := 4880, 
                                          root := ~, start := 4857, 
                                          stop := 4879 ), 
                                      rec( 
                                          content := " kernel function by spec\
ifying the ", count := [ 1, 1, 0, 6 ], mathmode := "Text", name := "PCDATA", 
                                          root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "CONFIGNAME", 
                                                  count := [ 1, 1, 0, 6 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 1, 0, 6 ],
                                          mathmode := "Text", name := "C", 
                                          next := 4932, root := ~, 
                                          start := 4915, stop := 4931 ), 
                                      rec( 
                                          content := "\nthat was used to confi\
gure that kernel.", count := [ 1, 1, 0, 6 ], mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 0, 
                                          count := [ 1, 1, 0, 7 ], 
                                          mathmode := "Text", name := "P", 
                                          next := 4976, root := ~, 
                                          start := 4972, stop := 4975 ), 
                                      rec( content := "\n\nAfterwards call ", 
                                          count := [ 1, 1, 0, 7 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "make", 
                                                  count := [ 1, 1, 0, 7 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 1, 0, 7 ],
                                          mathmode := "Text", name := "C", 
                                          next := 5005, root := ~, 
                                          start := 4994, stop := 5004 ), 
                                      rec( 
                                          content := "  to  compile  a binary \
file. ", count := [ 1, 1, 0, 7 ], mathmode := "Text", name := "PCDATA", 
                                          root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 0, 
                                          count := [ 1, 1, 0, 8 ], 
                                          mathmode := "Text", name := "P", 
                                          next := 5039, root := ~, 
                                          start := 5035, stop := 5038 ), 
                                      rec( 
                                          content := "\n\nIf you have installe\
d several GAP kernels repeat these two steps for each \nof them, using the var\
ious values of ", count := [ 1, 1, 0, 8 ], mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "CONFIGNAME", 
                                                  count := [ 1, 1, 0, 8 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 1, 0, 8 ],
                                          mathmode := "Text", name := "C", 
                                          next := 5170, root := ~, 
                                          start := 5153, stop := 5169 ), 
                                      rec( 
                                          content := ".\n\nYou  can   run  a t\
est of    the installation  by  typing ", count := [ 1, 1, 0, 8 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "make test", 
                                                  count := [ 1, 1, 0, 8 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 1, 0, 8 ],
                                          mathmode := "Text", name := "C", 
                                          next := 5247, root := ~, 
                                          start := 5231, stop := 5246 ), 
                                      rec( content := ".\n\n", 
                                          count := [ 1, 1, 0, 8 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  attributes := 
                                                    rec( Name := "InfoEDIM" ),
                                                  content := 0, 
                                                  count := [ 1, 1, 1, 2 ], 
                                                  mathmode := "Text", 
                                                  name := "InfoClass", 
                                                  next := 5293, root := ~, 
                                                  start := 5265, stop := 5292 
                                                 ), 
                                              rec( attributes := rec(  ), 
                                                  content := 
                                                    [ 
                                                      rec( 
                                                          content := "    \nThi\
s is  an ", count := [ 1, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "I\
nfo", count := [ 1, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 1, 1, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "K", 
                                                          next := 5337, 
                                                          root := ~, 
                                                          start := 5326, 
                                                          stop := 5336 ), 
                                                      rec( 
                                                          content := " class f\
or the ", count := [ 1, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "E\
DIM", count := [ 1, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 1, 1, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "Package", 
                                                          next := 5375, 
                                                          root := ~, 
                                                          start := 5352, 
                                                          stop := 5374 ), 
                                                      rec( 
                                                          content := "-package\
.\nBy ", count := [ 1, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "S\
etInfoLevel(InfoEDIM, 1);", count := [ 1, 1, 1, 3 ], mathmode := "Text", 
                                                                  name := "PCD\
ATA", root := ~ ) ], count := [ 1, 1, 1, 3 ], mathmode := "Text", 
                                                          name := "C", 
                                                          next := 5421, 
                                                          root := ~, 
                                                          start := 5388, 
                                                          stop := 5420 ), 
                                                      rec( 
                                                          content := " you can\
 switch on the printing of\nsome information     during      the    computatio\
ns     of    certain\n", count := [ 1, 1, 1, 3 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "E\
DIM", count := [ 1, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 1, 1, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "Package", 
                                                          next := 5550, 
                                                          root := ~, 
                                                          start := 5527, 
                                                          stop := 5549 ), 
                                                      rec( 
                                                          content := "-functio\
ns.  \n", count := [ 1, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", 
                                                          root := ~ ) ], 
                                                  count := [ 1, 1, 1, 3 ], 
                                                  mathmode := "Text", 
                                                  name := "Description", 
                                                  next := 5578, root := ~, 
                                                  start := 5296, stop := 5577 
                                                 ) ], count := [ 1, 1, 1, 1 ],
                                          mathmode := "Text", 
                                          name := "ManSection", next := 5592, 
                                          root := ~, start := 5250, 
                                          stop := 5591 ), 
                                      rec( content := "\n\n", 
                                          count := [ 1, 1, 1, 3 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ) ], 
                                  count := [ 1, 1, 0, 1 ], mathmode := "Text",
                                  name := "Section", next := 5604, root := ~, 
                                  start := 3929, stop := 5603 ), 
                              rec( content := "\n\n", count := [ 1, 1, 1, 3 ],
                                  mathmode := "Text", name := "PCDATA", 
                                  root := ~ ), 
                              rec( 
                                  content := " ###############################\
############################# ", count := [ 1, 1, 1, 3 ], mathmode := "Text", 
                                  name := "XMLCOMMENT", root := ~ ), 
                              rec( content := "\n", count := [ 1, 1, 1, 3 ], 
                                  mathmode := "Text", name := "PCDATA", 
                                  root := ~ ), 
                              rec( 
                                  attributes := rec( Label := "Sect-PPElDiv" )
                                    , 
                                  content := 
                                    [ 
                                      rec( content := "\n", 
                                          count := [ 1, 2, 0, 1 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( attributes := rec(  ), 
                                                  content := 
                                                    [ 
                                                      rec( content := "p", 
                                                          count := 
                                                            [ 1, 2, 0, 2 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ) ], 
                                                  count := [ 1, 2, 0, 2 ], 
                                                  mathmode := "Text", 
                                                  name := "M", next := 5724, 
                                                  root := ~, start := 5716, 
                                                  stop := 5723 ), 
                                              rec( 
                                                  content := "-Parts of Elemen\
tary Divisors", count := [ 1, 2, 0, 2 ], mathmode := "Text", name := "PCDATA",
                                                  root := ~ ) ], 
                                          count := [ 1, 2, 0, 2 ], 
                                          mathmode := "Text", 
                                          name := "Heading", next := 5763, 
                                          root := ~, start := 5707, 
                                          stop := 5762 ), 
                                      rec( 
                                          content := "\n\nHere we explain the \
main functions of the package.\n\n", count := [ 1, 2, 0, 3 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  attributes := 
                                                    rec( Arg := "A, p[, rk ]",
                                                      Name := "ElementaryDivis\
orsPPartRk" ), content := 0, count := [ 1, 2, 1, 2 ], mathmode := "Text", 
                                                  name := "Func", 
                                                  next := 5894, root := ~, 
                                                  start := 5830, stop := 5893 
                                                 ), 
                                              rec( 
                                                  attributes := 
                                                    rec( Arg := "A, p, rk", 
                                                      Name := "ElementaryDivis\
orsPPartRkI" ), content := 0, count := [ 1, 2, 1, 4 ], mathmode := "Text", 
                                                  name := "Func", 
                                                  next := 5957, root := ~, 
                                                  start := 5895, stop := 5956 
                                                 ), 
                                              rec( 
                                                  attributes := 
                                                    rec( Arg := "A, p, rk", 
                                                      Name := "ElementaryDivis\
orsPPartRkII" ), content := 0, count := [ 1, 2, 1, 6 ], mathmode := "Text", 
                                                  name := "Func", 
                                                  next := 6021, root := ~, 
                                                  start := 5958, stop := 6020 
                                                 ), 
                                              rec( 
                                                  attributes := 
                                                    rec( 
                                                      Arg := "A, p, rk, exp", 
                                                      Name := "ElementaryDivis\
orsPPartRkExp" ), content := 0, count := [ 1, 2, 1, 8 ], mathmode := "Text", 
                                                  name := "Func", 
                                                  next := 6091, root := ~, 
                                                  start := 6022, stop := 6090 
                                                 ), 
                                              rec( 
                                                  attributes := 
                                                    rec( 
                                                      Arg := "A, p, rk, exp, i\
l", Name := "ElementaryDivisorsPPartRkExpSmall" ), content := 0, 
                                                  count := [ 1, 2, 1, 10 ], 
                                                  mathmode := "Text", 
                                                  name := "Func", 
                                                  next := 6170, root := ~, 
                                                  start := 6092, stop := 6169 
                                                 ), 
                                              rec( attributes := rec(  ), 
                                                  content := 
                                                    [ 
                                                      rec( 
                                                          content := "  These f\
unctions  return  a list  ", count := [ 1, 2, 1, 11 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "[\
m_1, m_2,  \\ldots,\nm_r]", count := [ 1, 2, 1, 11 ], mathmode := "Text", 
                                                                  name := "PCD\
ATA", root := ~ ) ], count := [ 1, 2, 1, 11 ], mathmode := "Text", 
                                                          name := "M", 
                                                          next := 6251, 
                                                          root := ~, 
                                                          start := 6220, 
                                                          stop := 6250 ), 
                                                      rec( 
                                                          content := " where "
                                                            , 
                                                          count := 
                                                            [ 1, 2, 1, 11 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "m\
_i", count := [ 1, 2, 1, 11 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 2, 1, 11 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 6268, 
                                                          root := ~, 
                                                          start := 6258, 
                                                          stop := 6267 ), 
                                                      rec( 
                                                          content := " is the \
number of nonzero elementary divisors\nof ", count := [ 1, 2, 1, 11 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "A\
", count := [ 1, 2, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 2, 1, 11 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 6325, 
                                                          root := ~, 
                                                          start := 6317, 
                                                          stop := 6324 ), 
                                                      rec( 
                                                          content := "  divisi\
ble by  ", count := [ 1, 2, 1, 11 ], mathmode := "Text", name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  attributes :=\
 rec(  ), content := [ rec( content := "p", count := [ 1, 2, 1, 11 ], mathmode\
 := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 2, 1, 11 ], 
                                                                  mathmode := \
"Text", name := "A", next := 6352, root := ~, start := 6344, stop := 6351 ), 
                                                              rec( 
                                                                  content := "\
^i", count := [ 1, 2, 1, 11 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 2, 1, 11 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 6358, 
                                                          root := ~, 
                                                          start := 6341, 
                                                          stop := 6357 ), 
                                                      rec( 
                                                          content := " (see  "
                                                            , 
                                                          count := 
                                                            [ 1, 2, 1, 11 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec( 
                                                              BookName := "ref\
", Func := "ElementaryDivisorsMat" ), content := 0, count := [ 1, 2, 1, 11 ], 
                                                          mathmode := "Text", 
                                                          name := "Ref", 
                                                          next := 6417, 
                                                          root := ~, 
                                                          start := 6365, 
                                                          stop := 6416 ), 
                                                      rec( 
                                                          content := " for  a \
definition  of the  elementary\ndivisors).", count := [ 1, 2, 1, 11 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 2, 1, 12 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 6470, 
                                                          root := ~, 
                                                          start := 6466, 
                                                          stop := 6469 ), 
                                                      rec( 
                                                          content := "\n\nThe \
   algorithms   for   these  functions  are     described in ", 
                                                          count := 
                                                            [ 1, 2, 1, 12 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec( Key := "L98" 
                                                             ), content := 0, 
                                                          count := 
                                                            [ 1, 2, 1, 12 ], 
                                                          mathmode := "Text", 
                                                          name := "Cite", 
                                                          next := 6554, 
                                                          root := ~, 
                                                          start := 6537, 
                                                          stop := 6553 ), 
                                                      rec( content := ".", 
                                                          count := 
                                                            [ 1, 2, 1, 12 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 2, 1, 13 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 6559, 
                                                          root := ~, 
                                                          start := 6555, 
                                                          stop := 6558 ), 
                                                      rec( content := "\n\n", 
                                                          count := 
                                                            [ 1, 2, 1, 13 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "A\
", count := [ 1, 2, 1, 13 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 2, 1, 13 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 6569, 
                                                          root := ~, 
                                                          start := 6561, 
                                                          stop := 6568 ), 
                                                      rec( 
                                                          content := " must be\
 a matrix with integer entries,  ", count := [ 1, 2, 1, 13 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "p\
", count := [ 1, 2, 1, 13 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 2, 1, 13 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 6618, 
                                                          root := ~, 
                                                          start := 6610, 
                                                          stop := 6617 ), 
                                                      rec( 
                                                          content := " a prime\
, and\n", count := [ 1, 2, 1, 13 ], mathmode := "Text", name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "r\
k", count := [ 1, 2, 1, 13 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 2, 1, 13 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 6641, 
                                                          root := ~, 
                                                          start := 6632, 
                                                          stop := 6640 ), 
                                                      rec( 
                                                          content := " the  ra\
nk of  ", count := [ 1, 2, 1, 13 ], mathmode := "Text", name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "A\
", count := [ 1, 2, 1, 13 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 2, 1, 13 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 6664, 
                                                          root := ~, 
                                                          start := 6656, 
                                                          stop := 6663 ), 
                                                      rec( 
                                                          content := " (as rat\
ional  matrix).  In  the first\nversion of the command ", count := 
                                                            [ 1, 2, 1, 13 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "r\
k", count := [ 1, 2, 1, 13 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 2, 1, 13 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 6735, 
                                                          root := ~, 
                                                          start := 6726, 
                                                          stop := 6734 ), 
                                                      rec( 
                                                          content := " is comp\
uted, if it is not given. ", count := [ 1, 2, 1, 13 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 2, 1, 14 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 6773, 
                                                          root := ~, 
                                                          start := 6769, 
                                                          stop := 6772 ), 
                                                      rec( 
                                                          content := "\n\nThe \
first   version of the  command delegates  its  job  to the fourth\nversion by\
 trying growing values for ", count := [ 1, 2, 1, 14 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "e\
xp", count := [ 1, 2, 1, 14 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 2, 1, 14 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 6893, 
                                                          root := ~, 
                                                          start := 6883, 
                                                          stop := 6892 ), 
                                                      rec( 
                                                          content := ", see be\
low. ", count := [ 1, 2, 1, 14 ], mathmode := "Text", name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 2, 1, 15 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 6910, 
                                                          root := ~, 
                                                          start := 6906, 
                                                          stop := 6909 ), 
                                                      rec( 
                                                          content := "\n\nThe \
second  and third versions implement  the main algorithm described\nin   ", 
                                                          count := 
                                                            [ 1, 2, 1, 15 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec( Key := "L98" 
                                                             ), content := 0, 
                                                          count := 
                                                            [ 1, 2, 1, 15 ], 
                                                          mathmode := "Text", 
                                                          name := "Cite", 
                                                          next := 7009, 
                                                          root := ~, 
                                                          start := 6988, 
                                                          stop := 7008 ), 
                                                      rec( 
                                                          content := "    and \
  a    variation.      Here     ", count := [ 1, 2, 1, 15 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec( 
                                                              Func := "Element\
aryDivisorsPPartRkII" ), content := 0, count := [ 1, 2, 1, 15 ], 
                                                          mathmode := "Text", 
                                                          name := "Ref", 
                                                          next := 7090, 
                                                          root := ~, 
                                                          start := 7049, 
                                                          stop := 7089 ), 
                                                      rec( 
                                                          content := " has a b\
it  more overhead, but can\nbe   advantageous    because  the  intermediate   \
 entries  during the\ncomputation can be much smaller.", count := 
                                                            [ 1, 2, 1, 15 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 2, 1, 16 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 7232, 
                                                          root := ~, 
                                                          start := 7228, 
                                                          stop := 7231 ), 
                                                      rec( 
                                                          content := "\n\nIn t\
he fourth form ", count := [ 1, 2, 1, 16 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "e\
xp", count := [ 1, 2, 1, 16 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 2, 1, 16 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 7263, 
                                                          root := ~, 
                                                          start := 7253, 
                                                          stop := 7262 ), 
                                                      rec( 
                                                          content := " must be\
 an  upper bound for the  highest\npower of ", count := [ 1, 2, 1, 16 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "p\
", count := [ 1, 2, 1, 16 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 2, 1, 16 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 7322, 
                                                          root := ~, 
                                                          start := 7314, 
                                                          stop := 7321 ), 
                                                      rec( 
                                                          content := "  appear\
ing  in an  elementary divisor  of  ", count := [ 1, 2, 1, 16 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "A\
", count := [ 1, 2, 1, 16 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 2, 1, 16 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 7374, 
                                                          root := ~, 
                                                          start := 7366, 
                                                          stop := 7373 ), 
                                                      rec( 
                                                          content := ".\nThis \
 information  allows    reduction   of  matrix   entries   modulo\n", 
                                                          count := 
                                                            [ 1, 2, 1, 16 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  attributes :=\
 rec(  ), content := [ rec( content := "p", count := [ 1, 2, 1, 16 ], mathmode\
 := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 2, 1, 16 ], 
                                                                  mathmode := \
"Text", name := "A", next := 7458, root := ~, start := 7450, stop := 7457 ), 
                                                              rec( 
                                                                  content := "\
^{", count := [ 1, 2, 1, 16 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ),
                                                              rec( 
                                                                  attributes :\
= rec(  ), content := [ rec( content := "exp", count := [ 1, 2, 1, 16 ], mathm\
ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 2, 1, 16 ], 
                                                                  mathmode := \
"Text", name := "A", next := 7470, root := ~, start := 7460, stop := 7469 ), 
                                                              rec( 
                                                                  content := "\
}", count := [ 1, 2, 1, 16 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 2, 1, 16 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 7475, 
                                                          root := ~, 
                                                          start := 7447, 
                                                          stop := 7474 ), 
                                                      rec( 
                                                          content := " during \
the computation. ", count := [ 1, 2, 1, 16 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 2, 1, 17 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 7504, 
                                                          root := ~, 
                                                          start := 7500, 
                                                          stop := 7503 ), 
                                                      rec( 
                                                          content := "\n\nIf "
                                                            , 
                                                          count := 
                                                            [ 1, 2, 1, 17 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "e\
xp", count := [ 1, 2, 1, 17 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 2, 1, 17 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 7519, 
                                                          root := ~, 
                                                          start := 7509, 
                                                          stop := 7518 ), 
                                                      rec( 
                                                          content := " is too \
 small or the given ", count := [ 1, 2, 1, 17 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "r\
k", count := [ 1, 2, 1, 17 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 2, 1, 17 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 7556, 
                                                          root := ~, 
                                                          start := 7547, 
                                                          stop := 7555 ), 
                                                      rec( 
                                                          content := "  is not\
 correct the\nfunction returns `fail'. ", count := [ 1, 2, 1, 17 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 2, 1, 18 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 7606, 
                                                          root := ~, 
                                                          start := 7602, 
                                                          stop := 7605 ), 
                                                      rec( 
                                                          content := "\n\nAs l\
ong as ", count := [ 1, 2, 1, 18 ], mathmode := "Text", name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  attributes :=\
 rec(  ), content := [ rec( content := "p", count := [ 1, 2, 1, 18 ], mathmode\
 := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 2, 1, 18 ], 
                                                                  mathmode := \
"Text", name := "A", next := 7630, root := ~, start := 7622, stop := 7629 ), 
                                                              rec( 
                                                                  content := "\
^{", count := [ 1, 2, 1, 18 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ),
                                                              rec( 
                                                                  attributes :\
= rec(  ), content := [ rec( content := "exp", count := [ 1, 2, 1, 18 ], mathm\
ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 2, 1, 18 ], 
                                                                  mathmode := \
"Text", name := "A", next := 7642, root := ~, start := 7632, stop := 7641 ), 
                                                              rec( 
                                                                  content := "\
}", count := [ 1, 2, 1, 18 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 2, 1, 18 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 7647, 
                                                          root := ~, 
                                                          start := 7619, 
                                                          stop := 7646 ), 
                                                      rec( 
                                                          content := "  is sma\
ller than ", count := [ 1, 2, 1, 18 ], mathmode := "Text", name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "2\
^{28}", count := [ 1, 2, 1, 18 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 2, 1, 18 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 7678, 
                                                          root := ~, 
                                                          start := 7665, 
                                                          stop := 7677 ), 
                                                      rec( 
                                                          content := "\nand ",
                                                          count := 
                                                            [ 1, 2, 1, 18 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  attributes :=\
 rec(  ), content := [ rec( content := "p", count := [ 1, 2, 1, 18 ], mathmode\
 := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 2, 1, 18 ], 
                                                                  mathmode := \
"Text", name := "A", next := 7694, root := ~, start := 7686, stop := 7693 ), 
                                                              rec( 
                                                                  content := "\
^{", count := [ 1, 2, 1, 18 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ),
                                                              rec( 
                                                                  attributes :\
= rec(  ), content := [ rec( content := "exp", count := [ 1, 2, 1, 18 ], mathm\
ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 2, 1, 18 ], 
                                                                  mathmode := \
"Text", name := "A", next := 7706, root := ~, start := 7696, stop := 7705 ), 
                                                              rec( 
                                                                  content := "\
 +  2}", count := [ 1, 2, 1, 18 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 2, 1, 18 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 7716, 
                                                          root := ~, 
                                                          start := 7683, 
                                                          stop := 7715 ), 
                                                      rec( 
                                                          content := " is smal\
ler than ", count := [ 1, 2, 1, 18 ], mathmode := "Text", name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "2\
^{31}", count := [ 1, 2, 1, 18 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 2, 1, 18 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 7746, 
                                                          root := ~, 
                                                          start := 7733, 
                                                          stop := 7745 ), 
                                                      rec( 
                                                          content := " we\nuse\
  internally a kernel  function which can also  be used directly in\nthe  fift\
h form  of the command.   There ", count := [ 1, 2, 1, 18 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "i\
l", count := [ 1, 2, 1, 18 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 2, 1, 18 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 7871, 
                                                          root := ~, 
                                                          start := 7862, 
                                                          stop := 7870 ), 
                                                      rec( 
                                                          content := "  can be\
 ", count := [ 1, 2, 1, 18 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                         ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "0\
", count := [ 1, 2, 1, 18 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 2, 1, 18 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 7888, 
                                                          root := ~, 
                                                          start := 7880, 
                                                          stop := 7887 ), 
                                                      rec( content := " or\n",
                                                          count := 
                                                            [ 1, 2, 1, 18 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "1\
", count := [ 1, 2, 1, 18 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 2, 1, 18 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 7900, 
                                                          root := ~, 
                                                          start := 7892, 
                                                          stop := 7899 ), 
                                                      rec( 
                                                          content := " where i\
n the  second case some  information is printed during\nthe computation.", 
                                                          count := 
                                                            [ 1, 2, 1, 18 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 2, 1, 19 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 7983, 
                                                          root := ~, 
                                                          start := 7979, 
                                                          stop := 7982 ), 
                                                      rec( 
                                                          content := "\n\nThis\
 last form  of the  function was  already  succesfully applied  to\ndense matr\
ices of rank up to ", count := [ 1, 2, 1, 19 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "1\
1000", count := [ 1, 2, 1, 19 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 2, 1, 19 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 8097, 
                                                          root := ~, 
                                                          start := 8085, 
                                                          stop := 8096 ), 
                                                      rec( content := ".", 
                                                          count := 
                                                            [ 1, 2, 1, 19 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 2, 1, 20 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 8102, 
                                                          root := ~, 
                                                          start := 8098, 
                                                          stop := 8101 ), 
                                                      rec( 
                                                          content := "\n\nNote\
 that you have to compile a  file (see ", count := [ 1, 2, 1, 20 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec( 
                                                              Sect := "Sect-In\
stall" ), content := 0, count := [ 1, 2, 1, 20 ], mathmode := "Text", 
                                                          name := "Ref", 
                                                          next := 8173, 
                                                          root := ~, 
                                                          start := 8147, 
                                                          stop := 8172 ), 
                                                      rec( 
                                                          content := ")\nwhile\
  installing this  package, if   you  want  to  have this  kernel\nfunction av\
ailable.", count := [ 1, 2, 1, 20 ], mathmode := "Text", name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 2, 1, 21 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 8269, 
                                                          root := ~, 
                                                          start := 8265, 
                                                          stop := 8268 ), 
                                                      rec( content := " \n\n",
                                                          count := 
                                                            [ 1, 2, 1, 21 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "\\
ngap> ReadPackage(\"edim\",  \"tst/mat\");\nReading 242x242 integer matrix 'ma\
t' with elementary divisors 'eldiv'.\ntrue\ngap> ElementaryDivisorsPPartRkI(ma\
t, 2, 242); time; # mat has full rank\n[ 94, 78, 69, 57, 23, 23, 9, 2, 2, 0 ]\
\n490\ngap> ElementaryDivisorsPPartRkExpSmall(mat, 2, 242, 10, 0); time;\n[ 94\
, 78, 69, 57, 23, 23, 9, 2, 2, 0 ]\n10\n", count := [ 1, 2, 1, 22 ], 
                                                                  name := "PCD\
ATA", root := ~ ) ], count := [ 1, 2, 1, 22 ], mathmode := "Text", 
                                                          name := "Example", 
                                                          next := 8629, 
                                                          root := ~, 
                                                          start := 8272, 
                                                          stop := 8628 ), 
                                                      rec( content := "\n", 
                                                          count := 
                                                            [ 1, 2, 1, 23 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ) ], 
                                                  count := [ 1, 2, 1, 11 ], 
                                                  mathmode := "Text", 
                                                  name := "Description", 
                                                  next := 8644, root := ~, 
                                                  start := 6172, stop := 8643 
                                                 ) ], count := [ 1, 2, 1, 1 ],
                                          mathmode := "Text", 
                                          name := "ManSection", next := 8658, 
                                          root := ~, start := 5817, 
                                          stop := 8657 ), 
                                      rec( content := "\n\n", 
                                          count := [ 1, 2, 1, 23 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( 
                                          content := " #######################\
##################################### ", count := [ 1, 2, 1, 23 ], 
                                          mathmode := "Text", 
                                          name := "XMLCOMMENT", root := ~ ), 
                                      rec( content := "\n", 
                                          count := [ 1, 2, 1, 23 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  attributes := 
                                                    rec( Arg := "A, p, d", 
                                                      Comm := "another modular\
 algorithm", Name := "ElementaryDivisorsPPartHavasSterling" ), content := 0, 
                                                  count := [ 1, 2, 2, 2 ], 
                                                  mathmode := "Text", 
                                                  name := "Func", 
                                                  next := 8854, root := ~, 
                                                  start := 8743, stop := 8853 
                                                 ), 
                                              rec( attributes := rec(  ), 
                                                  content := 
                                                    [ 
                                                      rec( 
                                                          content := " \nFor an\
 integer  matrix  ", count := [ 1, 2, 2, 3 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "A\
", count := [ 1, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 2, 2, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 8903, 
                                                          root := ~, 
                                                          start := 8895, 
                                                          stop := 8902 ), 
                                                      rec( 
                                                          content := " and a  \
prime ", count := [ 1, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "p\
", count := [ 1, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 2, 2, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 8925, 
                                                          root := ~, 
                                                          start := 8917, 
                                                          stop := 8924 ), 
                                                      rec( 
                                                          content := "   this \
function\nreturns a list ", count := [ 1, 2, 2, 3 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "[\
m_1, m_2,  \\ldots, m_r]", count := [ 1, 2, 2, 3 ], mathmode := "Text", 
                                                                  name := "PCD\
ATA", root := ~ ) ], count := [ 1, 2, 2, 3 ], mathmode := "Text", 
                                                          name := "M", 
                                                          next := 8988, 
                                                          root := ~, 
                                                          start := 8957, 
                                                          stop := 8987 ), 
                                                      rec( 
                                                          content := " where "
                                                            , 
                                                          count := 
                                                            [ 1, 2, 2, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "m\
_i", count := [ 1, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 2, 2, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 9005, 
                                                          root := ~, 
                                                          start := 8995, 
                                                          stop := 9004 ), 
                                                      rec( 
                                                          content := " is the\\
nnumber   of nonzero  elementary   divisors of   ", count := [ 1, 2, 2, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "A\
", count := [ 1, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 2, 2, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 9069, 
                                                          root := ~, 
                                                          start := 9061, 
                                                          stop := 9068 ), 
                                                      rec( 
                                                          content := "  divisi\
ble by\n", count := [ 1, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  attributes :=\
 rec(  ), content := [ rec( content := "p", count := [ 1, 2, 2, 3 ], mathmode \
:= "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 2, 2, 3 ], 
                                                                  mathmode := \
"Text", name := "A", next := 9095, root := ~, start := 9087, stop := 9094 ), 
                                                              rec( 
                                                                  content := "\
^i", count := [ 1, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 2, 2, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 9101, 
                                                          root := ~, 
                                                          start := 9084, 
                                                          stop := 9100 ), 
                                                      rec( content := ". ", 
                                                          count := 
                                                            [ 1, 2, 2, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 2, 2, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 9107, 
                                                          root := ~, 
                                                          start := 9103, 
                                                          stop := 9106 ), 
                                                      rec( 
                                                          content := "\n\nAn u\
pper bound ", count := [ 1, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "d\
", count := [ 1, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 2, 2, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 9132, 
                                                          root := ~, 
                                                          start := 9124, 
                                                          stop := 9131 ), 
                                                      rec( 
                                                          content := " for the\
 highest power of ", count := [ 1, 2, 2, 4 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "p\
", count := [ 1, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 2, 2, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 9166, 
                                                          root := ~, 
                                                          start := 9158, 
                                                          stop := 9165 ), 
                                                      rec( 
                                                          content := " appeari\
ng in\nan  elementary  divisor of  ", count := [ 1, 2, 2, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "A\
", count := [ 1, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 2, 2, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 9216, 
                                                          root := ~, 
                                                          start := 9208, 
                                                          stop := 9215 ), 
                                                      rec( 
                                                          content := " must be\
   given. Smaller ", count := [ 1, 2, 2, 4 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "d\
", count := [ 1, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 2, 2, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 9250, 
                                                          root := ~, 
                                                          start := 9242, 
                                                          stop := 9249 ), 
                                                      rec( 
                                                          content := "\nimprov\
e the performance of the algorithm considerably.", count := [ 1, 2, 2, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 2, 2, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 9309, 
                                                          root := ~, 
                                                          start := 9305, 
                                                          stop := 9308 ), 
                                                      rec( 
                                                          content := "\n\nThis\
 is an implementation of  the modular algorithm described in ", 
                                                          count := 
                                                            [ 1, 2, 2, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec( 
                                                              Key := "HS79" ),
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 2, 2, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "Cite", 
                                                          next := 9394, 
                                                          root := ~, 
                                                          start := 9376, 
                                                          stop := 9393 ), 
                                                      rec( content := ". ", 
                                                          count := 
                                                            [ 1, 2, 2, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 2, 2, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 9400, 
                                                          root := ~, 
                                                          start := 9396, 
                                                          stop := 9399 ), 
                                                      rec( 
                                                          content := "\n\nWe a\
dded a slight improvement: we divide the considered submatrices by\nthe ", 
                                                          count := 
                                                            [ 1, 2, 2, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "p\
", count := [ 1, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 2, 2, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 9485, 
                                                          root := ~, 
                                                          start := 9477, 
                                                          stop := 9484 ), 
                                                      rec( 
                                                          content := "-part  o\
f the greatest common divisor  of all entries (and\nlower the  ", 
                                                          count := 
                                                            [ 1, 2, 2, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "d\
", count := [ 1, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 2, 2, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 9563, 
                                                          root := ~, 
                                                          start := 9555, 
                                                          stop := 9562 ), 
                                                      rec( 
                                                          content := " appropr\
iately).  This  reduces  the size  of   the\nentries and often shortens the pi\
vot search.", count := [ 1, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 2, 2, 7 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 9663, 
                                                          root := ~, 
                                                          start := 9659, 
                                                          stop := 9662 ), 
                                                      rec( content := "\n\n", 
                                                          count := 
                                                            [ 1, 2, 2, 7 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "\\
ngap> ReadPackage(\"edim\",  \"tst/mat\");\nReading 242x242 integer matrix 'ma\
t' with elementary divisors 'eldiv'.\ntrue\ngap> ElementaryDivisorsPPartHavasS\
terling(mat, 2, 10); time;\n[ 94, 78, 69, 57, 23, 23, 9, 2, 2 ]\n1260\n", 
                                                                  count := 
                                                                    [ 1, 2, 
                                                                      2, 8 ], 
                                                                  name := "PCD\
ATA", root := ~ ) ], count := [ 1, 2, 2, 8 ], mathmode := "Text", 
                                                          name := "Example", 
                                                          next := 9901, 
                                                          root := ~, 
                                                          start := 9665, 
                                                          stop := 9900 ), 
                                                      rec( content := "\n", 
                                                          count := 
                                                            [ 1, 2, 2, 9 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ) ], 
                                                  count := [ 1, 2, 2, 3 ], 
                                                  mathmode := "Text", 
                                                  name := "Description", 
                                                  next := 9916, root := ~, 
                                                  start := 8856, stop := 9915 
                                                 ) ], count := [ 1, 2, 2, 1 ],
                                          mathmode := "Text", 
                                          name := "ManSection", next := 9930, 
                                          root := ~, start := 8730, 
                                          stop := 9929 ), 
                                      rec( content := "\n", 
                                          count := [ 1, 2, 2, 9 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ) ], 
                                  count := [ 1, 2, 0, 1 ], mathmode := "Text",
                                  name := "Section", next := 9941, root := ~, 
                                  start := 5676, stop := 9940 ), 
                              rec( content := "\n\n", count := [ 1, 2, 2, 9 ],
                                  mathmode := "Text", name := "PCDATA", 
                                  root := ~ ), 
                              rec( 
                                  content := " ###############################\
############################# ", count := [ 1, 2, 2, 9 ], mathmode := "Text", 
                                  name := "XMLCOMMENT", root := ~ ), 
                              rec( content := "\n\n", count := [ 1, 2, 2, 9 ],
                                  mathmode := "Text", name := "PCDATA", 
                                  root := ~ ), 
                              rec( 
                                  attributes := 
                                    rec( Label := "Sect-InvRatMat" ), 
                                  content := 
                                    [ 
                                      rec( content := "\n", 
                                          count := [ 1, 3, 0, 1 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  content := "Inverse of Ration\
al Matrices", count := [ 1, 3, 0, 2 ], mathmode := "Text", name := "PCDATA", 
                                                  root := ~ ) ], 
                                          count := [ 1, 3, 0, 2 ], 
                                          mathmode := "Text", 
                                          name := "Heading", next := 10094, 
                                          root := ~, start := 10047, 
                                          stop := 10093 ), 
                                      rec( content := "\n\n", 
                                          count := [ 1, 3, 0, 3 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  attributes := 
                                                    rec( Arg := "A[, p ]", 
                                                      Comm := "inverse of a ra\
tional matrix", Name := "InverseRatMat" ), content := 0, 
                                                  count := [ 1, 3, 1, 2 ], 
                                                  mathmode := "Text", 
                                                  name := "Func", 
                                                  next := 10199, root := ~, 
                                                  start := 10109, 
                                                  stop := 10198 ), 
                                              rec( attributes := rec(  ), 
                                                  content := 
                                                    [ 
                                                      rec( 
                                                          content := "  \nThis \
 function returns  the inverse  of  an invertible matrix over the\nrational nu\
mbers.", count := [ 1, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 3, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 10309, 
                                                          root := ~, 
                                                          start := 10305, 
                                                          stop := 10308 ), 
                                                      rec( 
                                                          content := "\n\nIt f\
irst  computes the  inverse  modulo some  prime ", count := [ 1, 3, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "p\
", count := [ 1, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 3, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 10371, 
                                                          root := ~, 
                                                          start := 10363, 
                                                          stop := 10370 ), 
                                                      rec( 
                                                          content := ", comput\
es\nfrom this  a  ", count := [ 1, 3, 1, 4 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "p\
", count := [ 1, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 3, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 10404, 
                                                          root := ~, 
                                                          start := 10396, 
                                                          stop := 10403 ), 
                                                      rec( 
                                                          content := "-adic ap\
proximation  to  the inverse and finally\nconstructs  the     rational    entr\
ies from      their  ", count := [ 1, 3, 1, 4 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "p\
", count := [ 1, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 3, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 10518, 
                                                          root := ~, 
                                                          start := 10510, 
                                                          stop := 10517 ), 
                                                      rec( 
                                                          content := "-adic\na\
pproximations. See  section  ", count := [ 1, 3, 1, 4 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec( 
                                                              Sect := "Sect-In\
vRatMatAlg" ), content := 0, count := [ 1, 3, 1, 4 ], mathmode := "Text", 
                                                          name := "Ref", 
                                                          next := 10585, 
                                                          root := ~, 
                                                          start := 10554, 
                                                          stop := 10584 ), 
                                                      rec( 
                                                          content := "  for mo\
re\ndetails.", count := [ 1, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 3, 1, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 10608, 
                                                          root := ~, 
                                                          start := 10604, 
                                                          stop := 10607 ), 
                                                      rec( 
                                                          content := "\n\nThis\
    seems    to    be   better    than    ", count := [ 1, 3, 1, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "G\
AP", count := [ 1, 3, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 3, 1, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "Package", 
                                                          next := 10678, 
                                                          root := ~, 
                                                          start := 10656, 
                                                          stop := 10677 ), 
                                                      rec( 
                                                          content := "'s\nstan\
dard    Gauß   algorithm    (", count := [ 1, 3, 1, 5 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "A\
^-1", count := [ 1, 3, 1, 5 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 3, 1, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "C", 
                                                          next := 10726, 
                                                          root := ~, 
                                                          start := 10715, 
                                                          stop := 10725 ), 
                                                      rec( 
                                                          content := ")   alre\
ady    for   small\nmatrices.   (Try,  e.g.,   ", count := [ 1, 3, 1, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "R\
andomMat(20,20,[-10000..10000])", count := [ 1, 3, 1, 5 ], mathmode := "Text",
                                                                  name := "PCD\
ATA", root := ~ ) ], count := [ 1, 3, 1, 5 ], mathmode := "Text", 
                                                          name := "C", 
                                                          next := 10819, 
                                                          root := ~, 
                                                          start := 10780, 
                                                          stop := 10818 ), 
                                                      rec( 
                                                          content := "  or\n",
                                                          count := 
                                                            [ 1, 3, 1, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "R\
andomMat(100,100)", count := [ 1, 3, 1, 5 ], mathmode := "Text", name := "PCDA\
TA", root := ~ ) ], count := [ 1, 3, 1, 5 ], mathmode := "Text", name := "C", 
                                                          next := 10849, 
                                                          root := ~, 
                                                          start := 10824, 
                                                          stop := 10848 ), 
                                                      rec( content := ".)", 
                                                          count := 
                                                            [ 1, 3, 1, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 3, 1, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 10855, 
                                                          root := ~, 
                                                          start := 10851, 
                                                          stop := 10854 ), 
                                                      rec( 
                                                          content := "\n\nThe \
 optional argument ", count := [ 1, 3, 1, 6 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "p\
", count := [ 1, 3, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 3, 1, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 10888, 
                                                          root := ~, 
                                                          start := 10880, 
                                                          stop := 10887 ), 
                                                      rec( 
                                                          content := " should \
 be  a prime such that ", count := [ 1, 3, 1, 6 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "A\
", count := [ 1, 3, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 3, 1, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 10927, 
                                                          root := ~, 
                                                          start := 10919, 
                                                          stop := 10926 ), 
                                                      rec( 
                                                          content := "\nmodulo\
  ", count := [ 1, 3, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                         ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "p\
", count := [ 1, 3, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 3, 1, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 10944, 
                                                          root := ~, 
                                                          start := 10936, 
                                                          stop := 10943 ), 
                                                      rec( 
                                                          content := " is  inv\
ertible  (default is  ", count := [ 1, 3, 1, 6 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  attributes :=\
 rec(  ), content := [ rec( content := "p", count := [ 1, 3, 1, 6 ], mathmode \
:= "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 3, 1, 6 ], 
                                                                  mathmode := \
"Text", name := "A", next := 10985, root := ~, start := 10977, stop := 10984 )
                                                                , 
                                                              rec( 
                                                                  content := "\
=251", count := [ 1, 3, 1, 6 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 3, 1, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 10993, 
                                                          root := ~, 
                                                          start := 10974, 
                                                          stop := 10992 ), 
                                                      rec( 
                                                          content := "). If\n"
                                                            , 
                                                          count := 
                                                            [ 1, 3, 1, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "A\
", count := [ 1, 3, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 3, 1, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 11007, 
                                                          root := ~, 
                                                          start := 10999, 
                                                          stop := 11006 ), 
                                                      rec( 
                                                          content := "   is   \
not  invertible   modulo   ", count := [ 1, 3, 1, 6 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "p\
", count := [ 1, 3, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 3, 1, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 11050, 
                                                          root := ~, 
                                                          start := 11042, 
                                                          stop := 11049 ), 
                                                      rec( 
                                                          content := "  then "
                                                            , 
                                                          count := 
                                                            [ 1, 3, 1, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "p\
", count := [ 1, 3, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 3, 1, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 11065, 
                                                          root := ~, 
                                                          start := 11057, 
                                                          stop := 11064 ), 
                                                      rec( 
                                                          content := "  is\nau\
tomatically replaced by the next prime.", count := [ 1, 3, 1, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 3, 1, 7 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 11115, 
                                                          root := ~, 
                                                          start := 11111, 
                                                          stop := 11114 ), 
                                                      rec( content := "\n", 
                                                          count := 
                                                            [ 1, 3, 1, 7 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ) ], 
                                                  count := [ 1, 3, 1, 3 ], 
                                                  mathmode := "Text", 
                                                  name := "Description", 
                                                  next := 11130, root := ~, 
                                                  start := 10201, 
                                                  stop := 11129 ) ], 
                                          count := [ 1, 3, 1, 1 ], 
                                          mathmode := "Text", 
                                          name := "ManSection", 
                                          next := 11144, root := ~, 
                                          start := 10096, stop := 11143 ), 
                                      rec( content := "\n\n", 
                                          count := [ 1, 3, 1, 7 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( 
                                          content := " #######################\
##################################### ", count := [ 1, 3, 1, 7 ], 
                                          mathmode := "Text", 
                                          name := "XMLCOMMENT", root := ~ ), 
                                      rec( content := "\n", 
                                          count := [ 1, 3, 1, 7 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  attributes := 
                                                    rec( 
                                                      Arg := "A, v[, p[, invA \
] ]", Comm := "rational solution of linear system over integers", 
                                                      Name := "RationalSolutio\
nIntMat" ), content := 0, count := [ 1, 3, 2, 2 ], mathmode := "Text", 
                                                  name := "Func", 
                                                  next := 11360, root := ~, 
                                                  start := 11229, 
                                                  stop := 11359 ), 
                                              rec( attributes := rec(  ), 
                                                  content := 
                                                    [ 
                                                      rec( 
                                                          content := "  \nThis \
 function returns the solution ", count := [ 1, 3, 2, 3 ], mathmode := "Text",
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "x\
", count := [ 1, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 3, 2, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 11422, 
                                                          root := ~, 
                                                          start := 11414, 
                                                          stop := 11421 ), 
                                                      rec( 
                                                          content := " of the \
system of linear\nequations ", count := [ 1, 3, 2, 3 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "x\
 ", count := [ 1, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ )
                                                                , 
                                                              rec( 
                                                                  attributes :\
= rec(  ), content := [ rec( content := "A", count := [ 1, 3, 2, 3 ], mathmode\
 := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 3, 2, 3 ], 
                                                                  mathmode := \
"Text", name := "A", next := 11470, root := ~, start := 11462, stop := 11469 )
                                                                , 
                                                              rec( 
                                                                  content := "\
 = ", count := [ 1, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ),
                                                              rec( 
                                                                  attributes :\
= rec(  ), content := [ rec( content := "v", count := [ 1, 3, 2, 3 ], mathmode\
 := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 3, 2, 3 ], 
                                                                  mathmode := \
"Text", name := "A", next := 11481, root := ~, start := 11473, stop := 11480 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 3, 2, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 11485, 
                                                          root := ~, 
                                                          start := 11457, 
                                                          stop := 11484 ), 
                                                      rec( content := ".", 
                                                          count := 
                                                            [ 1, 3, 2, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 3, 2, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 11490, 
                                                          root := ~, 
                                                          start := 11486, 
                                                          stop := 11489 ), 
                                                      rec( 
                                                          content := "\n\nHere\
, ", count := [ 1, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                         ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "A\
", count := [ 1, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 3, 2, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 11506, 
                                                          root := ~, 
                                                          start := 11498, 
                                                          stop := 11505 ), 
                                                      rec( 
                                                          content := " must be\
 a matrix with integer entries which is invertible\nover the rationals and ", 
                                                          count := 
                                                            [ 1, 3, 2, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "v\
", count := [ 1, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 3, 2, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 11596, 
                                                          root := ~, 
                                                          start := 11588, 
                                                          stop := 11595 ), 
                                                      rec( 
                                                          content := " must be\
 a vector with integer entries of\nthe appropriate length.", 
                                                          count := 
                                                            [ 1, 3, 2, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 3, 2, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 11665, 
                                                          root := ~, 
                                                          start := 11661, 
                                                          stop := 11664 ), 
                                                      rec( 
                                                          content := "\n\nThe \
 optional arguments  are a  prime ", count := [ 1, 3, 2, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "p\
", count := [ 1, 3, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 3, 2, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 11713, 
                                                          root := ~, 
                                                          start := 11705, 
                                                          stop := 11712 ), 
                                                      rec( 
                                                          content := "  such t\
hat  ", count := [ 1, 3, 2, 5 ], mathmode := "Text", name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  attributes :=\
 rec(  ), content := [ rec( content := "A", count := [ 1, 3, 2, 5 ], mathmode \
:= "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 3, 2, 5 ], 
                                                                  mathmode := \
"Text", name := "A", next := 11737, root := ~, start := 11729, stop := 11736 )
                                                                , 
                                                              rec( 
                                                                  content := "\
\n\\pmod{p}", count := [ 1, 3, 2, 5 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 3, 2, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 11750, 
                                                          root := ~, 
                                                          start := 11726, 
                                                          stop := 11749 ), 
                                                      rec( 
                                                          content := " is inve\
rtible  (if not given, ", count := [ 1, 3, 2, 5 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "p\
 =  251", count := [ 1, 3, 2, 5 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 3, 2, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 11796, 
                                                          root := ~, 
                                                          start := 11781, 
                                                          stop := 11795 ), 
                                                      rec( 
                                                          content := " is assu\
med)\nand the inverse ", count := [ 1, 3, 2, 5 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "i\
nvA", count := [ 1, 3, 2, 5 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 3, 2, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 11836, 
                                                          root := ~, 
                                                          start := 11825, 
                                                          stop := 11835 ), 
                                                      rec( content := " of ", 
                                                          count := 
                                                            [ 1, 3, 2, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  attributes :=\
 rec(  ), content := [ rec( content := "A", count := [ 1, 3, 2, 5 ], mathmode \
:= "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 3, 2, 5 ], 
                                                                  mathmode := \
"Text", name := "A", next := 11851, root := ~, start := 11843, stop := 11850 )
                                                                , 
                                                              rec( 
                                                                  content := "\
 \\pmod{p}", count := [ 1, 3, 2, 5 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 3, 2, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 11864, 
                                                          root := ~, 
                                                          start := 11840, 
                                                          stop := 11863 ), 
                                                      rec( content := ".", 
                                                          count := 
                                                            [ 1, 3, 2, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 3, 2, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 11869, 
                                                          root := ~, 
                                                          start := 11865, 
                                                          stop := 11868 ), 
                                                      rec( 
                                                          content := "\n\nThe \
solution is computed via ", count := [ 1, 3, 2, 6 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "p\
", count := [ 1, 3, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 3, 2, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 11908, 
                                                          root := ~, 
                                                          start := 11900, 
                                                          stop := 11907 ), 
                                                      rec( 
                                                          content := "-adic ap\
proximation as explained in \n", count := [ 1, 3, 2, 6 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec( 
                                                              Sect := "Sect-In\
vRatMatAlg" ), content := 0, count := [ 1, 3, 2, 6 ], mathmode := "Text", 
                                                          name := "Ref", 
                                                          next := 11976, 
                                                          root := ~, 
                                                          start := 11945, 
                                                          stop := 11975 ), 
                                                      rec( content := ".", 
                                                          count := 
                                                            [ 1, 3, 2, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 3, 2, 7 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 11981, 
                                                          root := ~, 
                                                          start := 11977, 
                                                          stop := 11980 ), 
                                                      rec( content := "\n", 
                                                          count := 
                                                            [ 1, 3, 2, 7 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ) ], 
                                                  count := [ 1, 3, 2, 3 ], 
                                                  mathmode := "Text", 
                                                  name := "Description", 
                                                  next := 11996, root := ~, 
                                                  start := 11362, 
                                                  stop := 11995 ) ], 
                                          count := [ 1, 3, 2, 1 ], 
                                          mathmode := "Text", 
                                          name := "ManSection", 
                                          next := 12010, root := ~, 
                                          start := 11216, stop := 12009 ), 
                                      rec( content := "\n\n", 
                                          count := [ 1, 3, 2, 7 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( 
                                          content := " #######################\
##################################### ", count := [ 1, 3, 2, 7 ], 
                                          mathmode := "Text", 
                                          name := "XMLCOMMENT", root := ~ ), 
                                      rec( content := "\n\n", 
                                          count := [ 1, 3, 2, 7 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  attributes := 
                                                    rec( 
                                                      Arg := " A[, p[, nr ] ]"
                                                        , Comm := "", 
                                                      Name := "ExponentSquareI\
ntMatFullRank" ), content := 0, count := [ 1, 3, 3, 2 ], mathmode := "Text", 
                                                  name := "Func", 
                                                  next := 12181, root := ~, 
                                                  start := 12096, 
                                                  stop := 12180 ), 
                                              rec( attributes := rec(  ), 
                                                  content := 
                                                    [ 
                                                      rec( 
                                                          content := "\nThis fu\
nction  returns the biggest   elementary  divisor of  a  square\ninteger matri\
x ", count := [ 1, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                         ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "A\
", count := [ 1, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 3, 3, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 12292, 
                                                          root := ~, 
                                                          start := 12284, 
                                                          stop := 12291 ), 
                                                      rec( 
                                                          content := " of full\
 rank.", count := [ 1, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 3, 3, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 12310, 
                                                          root := ~, 
                                                          start := 12306, 
                                                          stop := 12309 ), 
                                                      rec( 
                                                          content := "\n\nFor \
 such a  matrix ", count := [ 1, 3, 3, 4 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "A\
", count := [ 1, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 3, 3, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 12340, 
                                                          root := ~, 
                                                          start := 12332, 
                                                          stop := 12339 ), 
                                                      rec( 
                                                          content := "   the  \
 least  common  multiple of    the\ndenominators of all entries of the inverse\
 matrix ", count := [ 1, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  attributes :=\
 rec(  ), content := [ rec( content := "A", count := [ 1, 3, 3, 4 ], mathmode \
:= "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 3, 3, 4 ], 
                                                                  mathmode := \
"Text", name := "A", next := 12444, root := ~, start := 12436, stop := 12443 )
                                                                , 
                                                              rec( 
                                                                  content := "\
^{-1}", count := [ 1, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 3, 3, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 12453, 
                                                          root := ~, 
                                                          start := 12433, 
                                                          stop := 12452 ), 
                                                      rec( 
                                                          content := "\nis exa\
ctly the biggest elementary divisor of ", count := [ 1, 3, 3, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "A\
", count := [ 1, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 3, 3, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 12507, 
                                                          root := ~, 
                                                          start := 12499, 
                                                          stop := 12506 ), 
                                                      rec( content := ".", 
                                                          count := 
                                                            [ 1, 3, 3, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 3, 3, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 12512, 
                                                          root := ~, 
                                                          start := 12508, 
                                                          stop := 12511 ), 
                                                      rec( 
                                                          content := "\n  \nTh\
is function  is  implemented  by   a  slight modification   of  ", 
                                                          count := 
                                                            [ 1, 3, 3, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec( 
                                                              Func := "Inverse\
RatMat" ), content := 0, count := [ 1, 3, 3, 5 ], mathmode := "Text", 
                                                          name := "Ref", 
                                                          next := 12609, 
                                                          root := ~, 
                                                          start := 12582, 
                                                          stop := 12608 ), 
                                                      rec( 
                                                          content := ".   The \
 third   argument ", count := [ 1, 3, 3, 5 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "n\
r", count := [ 1, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 3, 3, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 12644, 
                                                          root := ~, 
                                                          start := 12635, 
                                                          stop := 12643 ), 
                                                      rec( 
                                                          content := "   tells\
  the\nfunction  to return the least common   multiple of the first ", 
                                                          count := 
                                                            [ 1, 3, 3, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "n\
r", count := [ 1, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 3, 3, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 12728, 
                                                          root := ~, 
                                                          start := 12719, 
                                                          stop := 12727 ), 
                                                      rec( 
                                                          content := "\nrows  \
of the rational  inverse matrix  only.  Very  often the function\nwill    alre\
ady    return   the  biggest    elementary    divisor with\n", 
                                                          count := 
                                                            [ 1, 3, 3, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  attributes :=\
 rec(  ), content := [ rec( content := "nr", count := [ 1, 3, 3, 5 ], mathmode\
 := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 3, 3, 5 ], 
                                                                  mathmode := \
"Text", name := "A", next := 12883, root := ~, start := 12874, stop := 12882 )
                                                                , 
                                                              rec( 
                                                                  content := "\
=2", count := [ 1, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 3, 3, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 12889, 
                                                          root := ~, 
                                                          start := 12871, 
                                                          stop := 12888 ), 
                                                      rec( content := " or ", 
                                                          count := 
                                                            [ 1, 3, 3, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "3\
", count := [ 1, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 3, 3, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 12901, 
                                                          root := ~, 
                                                          start := 12893, 
                                                          stop := 12900 ), 
                                                      rec( 
                                                          content := "  (and t\
he command without this argument\nwould spend most time in checking, that this\
 is correct).", count := [ 1, 3, 3, 5 ], mathmode := "Text", name := "PCDATA",
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 3, 3, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 13003, 
                                                          root := ~, 
                                                          start := 12999, 
                                                          stop := 13002 ), 
                                                      rec( 
                                                          content := "\n\nThe \
optional argument  ", count := [ 1, 3, 3, 6 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "p\
", count := [ 1, 3, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 3, 3, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 13036, 
                                                          root := ~, 
                                                          start := 13028, 
                                                          stop := 13035 ), 
                                                      rec( 
                                                          content := " should \
be  a prime such  that ", count := [ 1, 3, 3, 6 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "A\
", count := [ 1, 3, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 3, 3, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 13075, 
                                                          root := ~, 
                                                          start := 13067, 
                                                          stop := 13074 ), 
                                                      rec( 
                                                          content := "\nmodulo\
 ", count := [ 1, 3, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ )
                                                        , 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "p\
", count := [ 1, 3, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 3, 3, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 13091, 
                                                          root := ~, 
                                                          start := 13083, 
                                                          stop := 13090 ), 
                                                      rec( 
                                                          content := " is  inv\
ertible  (default  is  ", count := [ 1, 3, 3, 6 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  attributes :=\
 rec(  ), content := [ rec( content := "p", count := [ 1, 3, 3, 6 ], mathmode \
:= "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 3, 3, 6 ], 
                                                                  mathmode := \
"Text", name := "A", next := 13133, root := ~, start := 13125, stop := 13132 )
                                                                , 
                                                              rec( 
                                                                  content := "\
=251", count := [ 1, 3, 3, 6 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 3, 3, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 13141, 
                                                          root := ~, 
                                                          start := 13122, 
                                                          stop := 13140 ), 
                                                      rec( 
                                                          content := "). If\n"
                                                            , 
                                                          count := 
                                                            [ 1, 3, 3, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "A\
", count := [ 1, 3, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 3, 3, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 13155, 
                                                          root := ~, 
                                                          start := 13147, 
                                                          stop := 13154 ), 
                                                      rec( 
                                                          content := "     is \
not  invertible   modulo   ", count := [ 1, 3, 3, 6 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "p\
", count := [ 1, 3, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 3, 3, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 13198, 
                                                          root := ~, 
                                                          start := 13190, 
                                                          stop := 13197 ), 
                                                      rec( 
                                                          content := "  then "
                                                            , 
                                                          count := 
                                                            [ 1, 3, 3, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "p\
", count := [ 1, 3, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 3, 3, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 13213, 
                                                          root := ~, 
                                                          start := 13205, 
                                                          stop := 13212 ), 
                                                      rec( 
                                                          content := "  is\nau\
tomatically replaced by the next prime.", count := [ 1, 3, 3, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 3, 3, 7 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 13263, 
                                                          root := ~, 
                                                          start := 13259, 
                                                          stop := 13262 ), 
                                                      rec( content := "\n\n", 
                                                          count := 
                                                            [ 1, 3, 3, 7 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "\\
ngap> ReadPackage(\"edim\",  \"tst/mat\");\nReading 242x242 integer matrix 'ma\
t' with elementary divisors 'eldiv'.\ntrue\ngap> inv := InverseRatMat(mat);; t\
ime;                      \n840\ngap> ExponentSquareIntMatFullRank(mat, 101, 3\
); # same as without the `3'\n115200\n", count := [ 1, 3, 3, 8 ], name := "PCD\
ATA", root := ~ ) ], count := [ 1, 3, 3, 8 ], mathmode := "Text", 
                                                          name := "Example", 
                                                          next := 13545, 
                                                          root := ~, 
                                                          start := 13265, 
                                                          stop := 13544 ), 
                                                      rec( content := "\n", 
                                                          count := 
                                                            [ 1, 3, 3, 9 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ) ], 
                                                  count := [ 1, 3, 3, 3 ], 
                                                  mathmode := "Text", 
                                                  name := "Description", 
                                                  next := 13560, root := ~, 
                                                  start := 12184, 
                                                  stop := 13559 ) ], 
                                          count := [ 1, 3, 3, 1 ], 
                                          mathmode := "Text", 
                                          name := "ManSection", 
                                          next := 13574, root := ~, 
                                          start := 12083, stop := 13573 ), 
                                      rec( content := "\n\n", 
                                          count := [ 1, 3, 3, 9 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ) ], 
                                  count := [ 1, 3, 0, 1 ], mathmode := "Text",
                                  name := "Section", next := 13586, 
                                  root := ~, start := 10014, stop := 13585 ), 
                              rec( content := "\n\n", count := [ 1, 3, 3, 9 ],
                                  mathmode := "Text", name := "PCDATA", 
                                  root := ~ ), 
                              rec( 
                                  content := " ###############################\
############################# ", count := [ 1, 3, 3, 9 ], mathmode := "Text", 
                                  name := "XMLCOMMENT", root := ~ ), 
                              rec( content := "\n\n", count := [ 1, 3, 3, 9 ],
                                  mathmode := "Text", name := "PCDATA", 
                                  root := ~ ), 
                              rec( 
                                  attributes := rec( Label := "Sect-ElDivPad" 
                                     ), 
                                  content := 
                                    [ 
                                      rec( content := "\n", 
                                          count := [ 1, 4, 0, 1 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  content := "All Elementary Di\
visors Using p-adic Method", count := [ 1, 4, 0, 2 ], mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 4, 0, 2 ],
                                          mathmode := "Text", 
                                          name := "Heading", next := 13753, 
                                          root := ~, start := 13691, 
                                          stop := 13752 ), 
                                      rec( 
                                          content := "\n\nIn the  following tw\
o functions we  put things together. In particular\nwe  handle the prime parts\
  of the elementary divisors efficiently for\nprimes appearing with low  power\
s  in the highest  elementary  divisor\nrespectively determinant divisor.\n\n"
                                            , count := [ 1, 4, 0, 3 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  attributes := 
                                                    rec( Arg := " A ", 
                                                      Name := "ElementaryDivis\
orsSquareIntMatFullRank" ), content := 0, count := [ 1, 4, 1, 2 ], 
                                                  mathmode := "Text", 
                                                  name := "Func", 
                                                  next := 14087, root := ~, 
                                                  start := 14016, 
                                                  stop := 14086 ), 
                                              rec( attributes := rec(  ), 
                                                  content := 
                                                    [ 
                                                      rec( 
                                                          content := "\nThis  f\
unction returns  a list  of nonzero  elementary divisors of  an\ninteger matri\
x ", count := [ 1, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                         ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "A\
", count := [ 1, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 4, 1, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 14198, 
                                                          root := ~, 
                                                          start := 14190, 
                                                          stop := 14197 ), 
                                                      rec( content := ".", 
                                                          count := 
                                                            [ 1, 4, 1, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 4, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 14203, 
                                                          root := ~, 
                                                          start := 14199, 
                                                          stop := 14202 ), 
                                                      rec( 
                                                          content := "\n  \nHe\
re we  start with computing the biggest  elementary divisor via ", 
                                                          count := 
                                                            [ 1, 4, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec( 
                                                              Func := "Exponen\
tSquareIntMatFullRank" ), content := 0, count := [ 1, 4, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "Ref", 
                                                          next := 14315, 
                                                          root := ~, 
                                                          start := 14273, 
                                                          stop := 14314 ), 
                                                      rec( 
                                                          content := ".   If  \
 it runs   into a problem\nbecause  ", count := [ 1, 4, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "A\
", count := [ 1, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 4, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 14366, 
                                                          root := ~, 
                                                          start := 14358, 
                                                          stop := 14365 ), 
                                                      rec( 
                                                          content := " is  sin\
gular modulo  a choosen  prime (it  starts by\ndefault with 251) then the prim\
e is automatically replaced by the next\none.", count := [ 1, 4, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 4, 1, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 14499, 
                                                          root := ~, 
                                                          start := 14495, 
                                                          stop := 14498 ), 
                                                      rec( 
                                                          content := "\n\nThe \
rest  is done using ", count := [ 1, 4, 1, 5 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec( 
                                                              Func := "Element\
aryDivisorsPPartRkExp" ), content := 0, count := [ 1, 4, 1, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "Ref", 
                                                          next := 14567, 
                                                          root := ~, 
                                                          start := 14525, 
                                                          stop := 14566 ), 
                                                      rec( 
                                                          content := " and\n",
                                                          count := 
                                                            [ 1, 4, 1, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec( 
                                                              Func := "RankMod\
" ), content := 0, count := [ 1, 4, 1, 5 ], mathmode := "Text", 
                                                          name := "Ref", 
                                                          next := 14593, 
                                                          root := ~, 
                                                          start := 14572, 
                                                          stop := 14592 ), 
                                                      rec( content := ".", 
                                                          count := 
                                                            [ 1, 4, 1, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 4, 1, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 14598, 
                                                          root := ~, 
                                                          start := 14594, 
                                                          stop := 14597 ), 
                                                      rec( 
                                                          content := "\n\nThe \
 function   fails if  the  biggest  elementary divisor  cannot be\ncompletely \
factored and the non-factored part is  not a divisor of the\nbiggest elementar\
y divisor only.", count := [ 1, 4, 1, 6 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 4, 1, 7 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 14777, 
                                                          root := ~, 
                                                          start := 14773, 
                                                          stop := 14776 ), 
                                                      rec( 
                                                          content := "\n\nNote\
 that this function may  for many matrices  not be the best choice\nfor comput\
ing all elementary divisors.  You may first try the standard\n", 
                                                          count := 
                                                            [ 1, 4, 1, 7 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "G\
AP", count := [ 1, 4, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 4, 1, 7 ], 
                                                          mathmode := "Text", 
                                                          name := "Package", 
                                                          next := 14943, 
                                                          root := ~, 
                                                          start := 14921, 
                                                          stop := 14942 ), 
                                                      rec( 
                                                          content := " library\
 routines  for Smith normal form instead\nof         this     function.     Ne\
vertheless     remember       ", count := [ 1, 4, 1, 7 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec( 
                                                              Func := "Element\
aryDivisorsSquareIntMatFullRank" ), content := 0, count := [ 1, 4, 1, 7 ], 
                                                          mathmode := "Text", 
                                                          name := "Ref", 
                                                          next := 15110, 
                                                          root := ~, 
                                                          start := 15058, 
                                                          stop := 15109 ), 
                                                      rec( 
                                                          content := "   for h\
ard  and    big\nexamples.  It is particularly good when the largest elementar\
y divisor\nis a very small factor of the determinant.\n\n", 
                                                          count := 
                                                            [ 1, 4, 1, 7 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "\\
ngap> Collected(ElementaryDivisorsSquareIntMatFullRank(mat));      \n[ [ 1, 49\
 ], [ 3, 99 ], [ 6, 7 ], [ 30, 9 ], [ 60, 9 ], [ 120, 2 ], \n  [ 360, 10 ], [ \
720, 22 ], [ 3600, 12 ], [ 14400, 14 ], \n  [ 28800, 7 ], [ 115200, 2 ] ]\ngap\
> time;\n860\ngap> last2 = Collected(DiagonalOfMat(NormalFormIntMat(mat, 1).no\
rmal));\ntrue\ngap> time;\n5170\n", count := [ 1, 4, 1, 8 ], name := "PCDATA",
                                                                  root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 4, 1, 8 ], 
                                                          mathmode := "Text", 
                                                          name := "Example", 
                                                          next := 15603, 
                                                          root := ~, 
                                                          start := 15249, 
                                                          stop := 15602 ), 
                                                      rec( content := "\n", 
                                                          count := 
                                                            [ 1, 4, 1, 9 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ) ], 
                                                  count := [ 1, 4, 1, 3 ], 
                                                  mathmode := "Text", 
                                                  name := "Description", 
                                                  next := 15618, root := ~, 
                                                  start := 14090, 
                                                  stop := 15617 ) ], 
                                          count := [ 1, 4, 1, 1 ], 
                                          mathmode := "Text", 
                                          name := "ManSection", 
                                          next := 15632, root := ~, 
                                          start := 14003, stop := 15631 ), 
                                      rec( content := "\n\n", 
                                          count := [ 1, 4, 1, 9 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( 
                                          content := " #######################\
##################################### ", count := [ 1, 4, 1, 9 ], 
                                          mathmode := "Text", 
                                          name := "XMLCOMMENT", root := ~ ), 
                                      rec( content := "\n\n", 
                                          count := [ 1, 4, 1, 9 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  attributes := 
                                                    rec( 
                                                      Arg := " A, det[, rk] ",
                                                      Name := "ElementaryDivis\
orsIntMatDeterminant" ), content := 0, count := [ 1, 4, 2, 2 ], 
                                                  mathmode := "Text", 
                                                  name := "Func", 
                                                  next := 15797, root := ~, 
                                                  start := 15718, 
                                                  stop := 15796 ), 
                                              rec( attributes := rec(  ), 
                                                  content := 
                                                    [ 
                                                      rec( 
                                                          content := "\nThis fu\
nction  returns a list  of  nonzero elementary divisors  of  an\ninteger matri\
x ", count := [ 1, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                         ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "A\
", count := [ 1, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 4, 2, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 15907, 
                                                          root := ~, 
                                                          start := 15899, 
                                                          stop := 15906 ), 
                                                      rec( content := ".", 
                                                          count := 
                                                            [ 1, 4, 2, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 4, 2, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 15912, 
                                                          root := ~, 
                                                          start := 15908, 
                                                          stop := 15911 ), 
                                                      rec( 
                                                          content := "\n\nHere\
 ", count := [ 1, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ )
                                                        , 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "d\
et", count := [ 1, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 4, 2, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 15929, 
                                                          root := ~, 
                                                          start := 15919, 
                                                          stop := 15928 ), 
                                                      rec( 
                                                          content := " must be\
 an integer which is  a multiple of the biggest\ndeterminant divisor of ", 
                                                          count := 
                                                            [ 1, 4, 2, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "A\
", count := [ 1, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 4, 2, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 16016, 
                                                          root := ~, 
                                                          start := 16008, 
                                                          stop := 16015 ), 
                                                      rec( 
                                                          content := ".  If   \
the matrix does not have   full\nrank then its rank ", count := [ 1, 4, 2, 4 ]
                                                            , 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "r\
k", count := [ 1, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 4, 2, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 16084, 
                                                          root := ~, 
                                                          start := 16075, 
                                                          stop := 16083 ), 
                                                      rec( 
                                                          content := " must be\
 given, too.", count := [ 1, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 4, 2, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 16108, 
                                                          root := ~, 
                                                          start := 16104, 
                                                          stop := 16107 ), 
                                                      rec( 
                                                          content := "\n\nThe \
   argument    ", count := [ 1, 4, 2, 5 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "d\
et", count := [ 1, 4, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 4, 2, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 16139, 
                                                          root := ~, 
                                                          start := 16129, 
                                                          stop := 16138 ), 
                                                      rec( 
                                                          content := "    can \
be  given      in   the  form  of\n", count := [ 1, 4, 2, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "C\
ollected(FactorsInt(", count := [ 1, 4, 2, 5 ], mathmode := "Text", 
                                                                  name := "PCD\
ATA", root := ~ ), rec( attributes := rec(  ), content := [ rec( content := "d\
et", count := [ 1, 4, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                         ) ], 
                                                                  count := 
                                                                    [ 1, 4, 
                                                                      2, 5 ], 
                                                                  mathmode := \
"Text", name := "A", next := 16215, root := ~, start := 16205, stop := 16214 )
                                                                , 
                                                              rec( 
                                                                  content := "\
))", count := [ 1, 4, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 4, 2, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "C", 
                                                          next := 16221, 
                                                          root := ~, 
                                                          start := 16181, 
                                                          stop := 16220 ), 
                                                      rec( content := ".", 
                                                          count := 
                                                            [ 1, 4, 2, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 4, 2, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 16226, 
                                                          root := ~, 
                                                          start := 16222, 
                                                          stop := 16225 ), 
                                                      rec( 
                                                          content := "\n\nThis\
 function handles  prime divisors of ", count := [ 1, 4, 2, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "d\
et", count := [ 1, 4, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 4, 2, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 16279, 
                                                          root := ~, 
                                                          start := 16269, 
                                                          stop := 16278 ), 
                                                      rec( 
                                                          content := " with  m\
ultiplicity\nsmaller than  4  specially, for the other  prime  divisors ", 
                                                          count := 
                                                            [ 1, 4, 2, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "p\
", count := [ 1, 4, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 4, 2, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 16366, 
                                                          root := ~, 
                                                          start := 16358, 
                                                          stop := 16365 ), 
                                                      rec( 
                                                          content := " it\ndel\
egates to   ", count := [ 1, 4, 2, 6 ], mathmode := "Text", name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec( 
                                                              Func := "Element\
aryDivisorsPPartRkExp" ), content := 0, count := [ 1, 4, 2, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "Ref", 
                                                          next := 16429, 
                                                          root := ~, 
                                                          start := 16385, 
                                                          stop := 16428 ), 
                                                      rec( 
                                                          content := " where  \
the\n", count := [ 1, 4, 2, 6 ], mathmode := "Text", name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "e\
xp", count := [ 1, 4, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 4, 2, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 16451, 
                                                          root := ~, 
                                                          start := 16441, 
                                                          stop := 16450 ), 
                                                      rec( 
                                                          content := " argumen\
t is the multiplicity of the ", count := [ 1, 4, 2, 6 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "p\
", count := [ 1, 4, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 4, 2, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 16496, 
                                                          root := ~, 
                                                          start := 16488, 
                                                          stop := 16495 ), 
                                                      rec( content := " in ", 
                                                          count := 
                                                            [ 1, 4, 2, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "d\
et", count := [ 1, 4, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 4, 2, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 16510, 
                                                          root := ~, 
                                                          start := 16500, 
                                                          stop := 16509 ), 
                                                      rec( 
                                                          content := ".\n(Note\
 that  this is not very good  when  ", count := [ 1, 4, 2, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "p\
", count := [ 1, 4, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 4, 2, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 16561, 
                                                          root := ~, 
                                                          start := 16553, 
                                                          stop := 16560 ), 
                                                      rec( 
                                                          content := " has act\
ually  a much\nsmaller multiplicity in the largest elementary divisor.)", 
                                                          count := 
                                                            [ 1, 4, 2, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 4, 2, 7 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 16643, 
                                                          root := ~, 
                                                          start := 16639, 
                                                          stop := 16642 ), 
                                                      rec( content := "\n\n", 
                                                          count := 
                                                            [ 1, 4, 2, 7 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "\\
ngap> ReadPackage(\"edim\",  \"tst/mat\");\nReading 242x242 integer matrix 'ma\
t' with elementary divisors 'eldiv'.\ntrue\ngap> # not so good:\ngap> Elementa\
ryDivisorsIntMatDeterminant(mat,Product(eldiv)) = \n> Concatenation([1..49]*0+\
1, eldiv); time;\ntrue\n5490\n", count := [ 1, 4, 2, 8 ], name := "PCDATA", 
                                                                  root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 4, 2, 8 ], 
                                                          mathmode := "Text", 
                                                          name := "Example", 
                                                          next := 16916, 
                                                          root := ~, 
                                                          start := 16645, 
                                                          stop := 16915 ), 
                                                      rec( content := "\n", 
                                                          count := 
                                                            [ 1, 4, 2, 9 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ) ], 
                                                  count := [ 1, 4, 2, 3 ], 
                                                  mathmode := "Text", 
                                                  name := "Description", 
                                                  next := 16931, root := ~, 
                                                  start := 15799, 
                                                  stop := 16930 ) ], 
                                          count := [ 1, 4, 2, 1 ], 
                                          mathmode := "Text", 
                                          name := "ManSection", 
                                          next := 16945, root := ~, 
                                          start := 15705, stop := 16944 ), 
                                      rec( content := "\n\n", 
                                          count := [ 1, 4, 2, 9 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ) ], 
                                  count := [ 1, 4, 0, 1 ], mathmode := "Text",
                                  name := "Section", next := 16957, 
                                  root := ~, start := 13659, stop := 16956 ), 
                              rec( content := "\n\n", count := [ 1, 4, 2, 9 ],
                                  mathmode := "Text", name := "PCDATA", 
                                  root := ~ ), 
                              rec( 
                                  content := " ###############################\
############################# ", count := [ 1, 4, 2, 9 ], mathmode := "Text", 
                                  name := "XMLCOMMENT", root := ~ ), 
                              rec( content := "\n\n", count := [ 1, 4, 2, 9 ],
                                  mathmode := "Text", name := "PCDATA", 
                                  root := ~ ), 
                              rec( 
                                  attributes := 
                                    rec( Label := "Sect-NFIntMatLLL" ), 
                                  content := 
                                    [ 
                                      rec( content := "\n", 
                                          count := [ 1, 5, 0, 1 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  content := "Gcd and Normal Fo\
rms Using LLL", count := [ 1, 5, 0, 2 ], mathmode := "Text", name := "PCDATA",
                                                  root := ~ ) ], 
                                          count := [ 1, 5, 0, 2 ], 
                                          mathmode := "Text", 
                                          name := "Heading", next := 17114, 
                                          root := ~, start := 17065, 
                                          stop := 17113 ), 
                                      rec( content := "\n\nThe ", 
                                          count := [ 1, 5, 0, 3 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "EDIM", 
                                                  count := [ 1, 5, 0, 3 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 5, 0, 3 ],
                                          mathmode := "Text", 
                                          name := "Package", next := 17143, 
                                          root := ~, start := 17120, 
                                          stop := 17142 ), 
                                      rec( 
                                          content := "-mini package also conta\
ins implementations\nof an extended  Gcd-algorithm  for integers  and  a Hermi\
te and  Smith\nnormal form algorithm  for integer matrices using  LLL-techique\
s. They\nare well described in the paper ", count := [ 1, 5, 0, 3 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec( Key := "HMM98" )
                                            , content := 0, 
                                          count := [ 1, 5, 0, 3 ], 
                                          mathmode := "Text", name := "Cite", 
                                          next := 17380, root := ~, 
                                          start := 17361, stop := 17379 ), 
                                      rec( 
                                          content := " by Havas, Majewski\nand\
 Matthews.", count := [ 1, 5, 0, 3 ], mathmode := "Text", name := "PCDATA", 
                                          root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 0, 
                                          count := [ 1, 5, 0, 4 ], 
                                          mathmode := "Text", name := "P", 
                                          next := 17417, root := ~, 
                                          start := 17413, stop := 17416 ), 
                                      rec( 
                                          content := "\n\nThey are particularl\
y  useful if one  wants  to have the normal  forms\ntogether with transforming\
 matrices.  These transforming matrices have\nspectacularly nice   (i.e.,  ", 
                                          count := [ 1, 5, 0, 4 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "small", 
                                                  count := [ 1, 5, 0, 4 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 5, 0, 4 ],
                                          mathmode := "Text", name := "Q", 
                                          next := 17602, root := ~, 
                                          start := 17590, stop := 17601 ), 
                                      rec( 
                                          content := ") entries    in cases of\
   input\nmatrices which  are non-square  or not   of full rank   (otherwise t\
he\ntransformation to the Hermite normal form is unique).", 
                                          count := [ 1, 5, 0, 4 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 0, 
                                          count := [ 1, 5, 0, 5 ], 
                                          mathmode := "Text", name := "P", 
                                          next := 17763, root := ~, 
                                          start := 17759, stop := 17762 ), 
                                      rec( content := "\n\nIn detail:", 
                                          count := [ 1, 5, 0, 5 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 0, 
                                          count := [ 1, 5, 0, 6 ], 
                                          mathmode := "Text", name := "P", 
                                          next := 17779, root := ~, 
                                          start := 17775, stop := 17778 ), 
                                      rec( content := "\n\n", 
                                          count := [ 1, 5, 0, 6 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  attributes := 
                                                    rec( 
                                                      Arg := "n1,  n2, ...", 
                                                      Name := "GcdexIntLLL" ),
                                                  content := 0, 
                                                  count := [ 1, 5, 1, 2 ], 
                                                  mathmode := "Text", 
                                                  name := "Func", 
                                                  next := 17847, root := ~, 
                                                  start := 17794, 
                                                  stop := 17846 ), 
                                              rec( attributes := rec(  ), 
                                                  content := 
                                                    [ 
                                                      rec( 
                                                          content := "  \nThis \
function returns for integers ", count := [ 1, 5, 1, 3 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  attributes :=\
 rec(  ), content := [ rec( content := "n1", count := [ 1, 5, 1, 3 ], mathmode\
 := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 5, 1, 3 ], 
                                                                  mathmode := \
"Text", name := "A", next := 17913, root := ~, start := 17904, stop := 17912 )
                                                                , 
                                                              rec( 
                                                                  content := "\
, ", count := [ 1, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ), 
                                                              rec( 
                                                                  attributes :\
= rec(  ), content := [ rec( content := "n2", count := [ 1, 5, 1, 3 ], mathmod\
e := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 5, 1, 3 ], 
                                                                  mathmode := \
"Text", name := "A", next := 17924, root := ~, start := 17915, stop := 17923 )
                                                                , 
                                                              rec( 
                                                                  content := "\
, \\ldots", count := [ 1, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 5, 1, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 17936, 
                                                          root := ~, 
                                                          start := 17901, 
                                                          stop := 17935 ), 
                                                      rec( 
                                                          content := " a\nlist\
  ", count := [ 1, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                         ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "[\
g,  [c_1,  c_2,  \\ldots]]", count := [ 1, 5, 1, 3 ], mathmode := "Text", 
                                                                  name := "PCD\
ATA", root := ~ ) ], count := [ 1, 5, 1, 3 ], mathmode := "Text", 
                                                          name := "M", 
                                                          next := 17978, 
                                                          root := ~, 
                                                          start := 17945, 
                                                          stop := 17977 ), 
                                                      rec( 
                                                          content := ", where \
 ", count := [ 1, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ )
                                                        , 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "g\
  =  c_1", count := [ 1, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ),
                                                              rec( 
                                                                  attributes :\
= rec(  ), content := [ rec( content := "n1", count := [ 1, 5, 1, 3 ], mathmod\
e := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 5, 1, 3 ], 
                                                                  mathmode := \
"Text", name := "A", next := 18008, root := ~, start := 17999, stop := 18007 )
                                                                , 
                                                              rec( 
                                                                  content := "\
  +\nc_2", count := [ 1, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ),
                                                              rec( 
                                                                  attributes :\
= rec(  ), content := [ rec( content := "n2", count := [ 1, 5, 1, 3 ], mathmod\
e := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 5, 1, 3 ], 
                                                                  mathmode := \
"Text", name := "A", next := 18024, root := ~, start := 18015, stop := 18023 )
                                                                , 
                                                              rec( 
                                                                  content := "\
  +  \\ldots", count := [ 1, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 5, 1, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 18039, 
                                                          root := ~, 
                                                          start := 17987, 
                                                          stop := 18038 ), 
                                                      rec( 
                                                          content := "  is  th\
e  greatest  common  divisor  of  the\n", count := [ 1, 5, 1, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "n\
i", count := [ 1, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 5, 1, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 18094, 
                                                          root := ~, 
                                                          start := 18085, 
                                                          stop := 18093 ), 
                                                      rec( 
                                                          content := ". Here a\
ll the ", count := [ 1, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "c\
_i", count := [ 1, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 5, 1, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 18119, 
                                                          root := ~, 
                                                          start := 18109, 
                                                          stop := 18118 ), 
                                                      rec( 
                                                          content := " are usu\
ally very small.\n\n", count := [ 1, 5, 1, 3 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "\\
ngap> GcdexIntLLL( 517, 244, -304, -872, -286, 854, 866, 224, -765, -38);\n[ 1\
, [ 0, 0, 0, 0, 1, 0, 1, 1, 1, 1 ] ]\n", count := [ 1, 5, 1, 4 ], name := "PCD\
ATA", root := ~ ) ], count := [ 1, 5, 1, 4 ], mathmode := "Text", 
                                                          name := "Example", 
                                                          next := 18278, 
                                                          root := ~, 
                                                          start := 18145, 
                                                          stop := 18277 ), 
                                                      rec( content := "\n", 
                                                          count := 
                                                            [ 1, 5, 1, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ) ], 
                                                  count := [ 1, 5, 1, 3 ], 
                                                  mathmode := "Text", 
                                                  name := "Description", 
                                                  next := 18293, root := ~, 
                                                  start := 17850, 
                                                  stop := 18292 ) ], 
                                          count := [ 1, 5, 1, 1 ], 
                                          mathmode := "Text", 
                                          name := "ManSection", 
                                          next := 18307, root := ~, 
                                          start := 17781, stop := 18306 ), 
                                      rec( content := "\n\n", 
                                          count := [ 1, 5, 1, 5 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( 
                                          content := " #######################\
##################################### ", count := [ 1, 5, 1, 5 ], 
                                          mathmode := "Text", 
                                          name := "XMLCOMMENT", root := ~ ), 
                                      rec( content := "\n\n", 
                                          count := [ 1, 5, 1, 5 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  attributes := 
                                                    rec( Arg := "A", 
                                                      Name := "HermiteIntMatLL\
L" ), content := 0, count := [ 1, 5, 2, 2 ], mathmode := "Text", 
                                                  name := "Func", 
                                                  next := 18440, root := ~, 
                                                  start := 18393, 
                                                  stop := 18439 ), 
                                              rec( attributes := rec(  ), 
                                                  content := 
                                                    [ 
                                                      rec( 
                                                          content := "  This  r\
eturns  the Hermite normal   form  of an integer\nmatrix  ", 
                                                          count := 
                                                            [ 1, 5, 2, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "A\
", count := [ 1, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 5, 2, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 18529, 
                                                          root := ~, 
                                                          start := 18521, 
                                                          stop := 18528 ), 
                                                      rec( 
                                                          content := " and use\
s the  LLL-algorithm to avoid entry explosion.\n", count := [ 1, 5, 2, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ) ], 
                                                  count := [ 1, 5, 2, 3 ], 
                                                  mathmode := "Text", 
                                                  name := "Description", 
                                                  next := 18598, root := ~, 
                                                  start := 18442, 
                                                  stop := 18597 ) ], 
                                          count := [ 1, 5, 2, 1 ], 
                                          mathmode := "Text", 
                                          name := "ManSection", 
                                          next := 18612, root := ~, 
                                          start := 18380, stop := 18611 ), 
                                      rec( content := "\n\n", 
                                          count := [ 1, 5, 2, 3 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( 
                                          content := " #######################\
##################################### ", count := [ 1, 5, 2, 3 ], 
                                          mathmode := "Text", 
                                          name := "XMLCOMMENT", root := ~ ), 
                                      rec( content := "\n\n", 
                                          count := [ 1, 5, 2, 3 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  attributes := 
                                                    rec( Arg := " A ", 
                                                      Name := "HermiteIntMatLL\
LTrans" ), content := 0, count := [ 1, 5, 3, 2 ], mathmode := "Text", 
                                                  name := "Func", 
                                                  next := 18752, root := ~, 
                                                  start := 18698, 
                                                  stop := 18751 ), 
                                              rec( attributes := rec(  ), 
                                                  content := 
                                                    [ 
                                                      rec( 
                                                          content := " \nThis f\
unction returns a pair of matrices ", count := [ 1, 5, 3, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "[\
H, L]", count := [ 1, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 5, 3, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 18824, 
                                                          root := ~, 
                                                          start := 18811, 
                                                          stop := 18823 ), 
                                                      rec( 
                                                          content := "  where \
", count := [ 1, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ),
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "H\
 = L\n", count := [ 1, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ),
                                                              rec( 
                                                                  attributes :\
= rec(  ), content := [ rec( content := "A", count := [ 1, 5, 3, 3 ], mathmode\
 := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 5, 3, 3 ], 
                                                                  mathmode := \
"Text", name := "A", next := 18849, root := ~, start := 18841, stop := 18848 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 5, 3, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 18853, 
                                                          root := ~, 
                                                          start := 18832, 
                                                          stop := 18852 ), 
                                                      rec( 
                                                          content := " is the \
Hermite normal form of an integer matrix ", count := [ 1, 5, 3, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "A\
", count := [ 1, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 5, 3, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 18910, 
                                                          root := ~, 
                                                          start := 18902, 
                                                          stop := 18909 ), 
                                                      rec( 
                                                          content := ".\nThe t\
ransforming matrix ", count := [ 1, 5, 3, 3 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "L\
", count := [ 1, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 5, 3, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 18944, 
                                                          root := ~, 
                                                          start := 18936, 
                                                          stop := 18943 ), 
                                                      rec( 
                                                          content := " can hav\
e surprisingly small entries.\n\n", count := [ 1, 5, 3, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "\\
ngap> ReadPackage(\"edim\",  \"tst/mat2\");\nReading 34x34 integer matrix 'mat\
2' with elementary divisors 'eldiv2'.\ntrue\ngap> tr := HermiteIntMatLLLTrans(\
mat2);; Maximum(List(Flat(tr[2]), AbsInt));\n606\ngap> tr[2]*mat2 = tr[1];    \
                                            \ntrue\n", count := [ 1, 5, 3, 4 ]
                                                                    , 
                                                                  name := "PCD\
ATA", root := ~ ) ], count := [ 1, 5, 3, 4 ], mathmode := "Text", 
                                                          name := "Example", 
                                                          next := 19277, 
                                                          root := ~, 
                                                          start := 18983, 
                                                          stop := 19276 ), 
                                                      rec( content := "\n", 
                                                          count := 
                                                            [ 1, 5, 3, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ) ], 
                                                  count := [ 1, 5, 3, 3 ], 
                                                  mathmode := "Text", 
                                                  name := "Description", 
                                                  next := 19292, root := ~, 
                                                  start := 18755, 
                                                  stop := 19291 ) ], 
                                          count := [ 1, 5, 3, 1 ], 
                                          mathmode := "Text", 
                                          name := "ManSection", 
                                          next := 19306, root := ~, 
                                          start := 18685, stop := 19305 ), 
                                      rec( content := "\n\n", 
                                          count := [ 1, 5, 3, 5 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( 
                                          content := " #######################\
##################################### ", count := [ 1, 5, 3, 5 ], 
                                          mathmode := "Text", 
                                          name := "XMLCOMMENT", root := ~ ), 
                                      rec( content := "\n\n", 
                                          count := [ 1, 5, 3, 5 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  attributes := 
                                                    rec( Arg := " A ", 
                                                      Name := "SmithIntMatLLL"\
 ), content := 0, count := [ 1, 5, 4, 2 ], mathmode := "Text", 
                                                  name := "Func", 
                                                  next := 19439, root := ~, 
                                                  start := 19392, 
                                                  stop := 19438 ), 
                                              rec( attributes := rec(  ), 
                                                  content := 
                                                    [ 
                                                      rec( 
                                                          content := " \nThis  \
function  returns the  Smith  normal  form of  an integer matrix\n", 
                                                          count := 
                                                            [ 1, 5, 4, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "A\
", count := [ 1, 5, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 5, 4, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 19536, 
                                                          root := ~, 
                                                          start := 19528, 
                                                          stop := 19535 ), 
                                                      rec( 
                                                          content := " using t\
he LLL-algorithm to avoid entry explosion.\n", count := [ 1, 5, 4, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ) ], 
                                                  count := [ 1, 5, 4, 3 ], 
                                                  mathmode := "Text", 
                                                  name := "Description", 
                                                  next := 19601, root := ~, 
                                                  start := 19442, 
                                                  stop := 19600 ) ], 
                                          count := [ 1, 5, 4, 1 ], 
                                          mathmode := "Text", 
                                          name := "ManSection", 
                                          next := 19615, root := ~, 
                                          start := 19379, stop := 19614 ), 
                                      rec( content := "\n\n", 
                                          count := [ 1, 5, 4, 3 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( 
                                          content := " #######################\
##################################### ", count := [ 1, 5, 4, 3 ], 
                                          mathmode := "Text", 
                                          name := "XMLCOMMENT", root := ~ ), 
                                      rec( content := "\n\n", 
                                          count := [ 1, 5, 4, 3 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  attributes := 
                                                    rec( Arg := " A  ", 
                                                      Name := "SmithIntMatLLLT\
rans" ), content := 0, count := [ 1, 5, 5, 2 ], mathmode := "Text", 
                                                  name := "Func", 
                                                  next := 19754, root := ~, 
                                                  start := 19701, 
                                                  stop := 19753 ), 
                                              rec( attributes := rec(  ), 
                                                  content := 
                                                    [ 
                                                      rec( 
                                                          content := "\nThis  f\
unction returns ", count := [ 1, 5, 5, 3 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "[\
S, L, R]", count := [ 1, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 5, 5, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 19809, 
                                                          root := ~, 
                                                          start := 19793, 
                                                          stop := 19808 ), 
                                                      rec( 
                                                          content := "  where \
", count := [ 1, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ),
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "S\
 = L ", count := [ 1, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ),
                                                              rec( 
                                                                  attributes :\
= rec(  ), content := [ rec( content := "A", count := [ 1, 5, 5, 3 ], mathmode\
 := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 5, 5, 3 ], 
                                                                  mathmode := \
"Text", name := "A", next := 19834, root := ~, start := 19826, stop := 19833 )
                                                                , 
                                                              rec( 
                                                                  content := "\
 R", count := [ 1, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 5, 5, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 19840, 
                                                          root := ~, 
                                                          start := 19817, 
                                                          stop := 19839 ), 
                                                      rec( 
                                                          content := "\nis the\
 Smith normal form of an integer matrix ", count := [ 1, 5, 5, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "A\
", count := [ 1, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 5, 5, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 19895, 
                                                          root := ~, 
                                                          start := 19887, 
                                                          stop := 19894 ), 
                                                      rec( content := ".", 
                                                          count := 
                                                            [ 1, 5, 5, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 5, 5, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 19900, 
                                                          root := ~, 
                                                          start := 19896, 
                                                          stop := 19899 ), 
                                                      rec( 
                                                          content := "\n\nWe a\
pply  the algorithm for Hermite  normal form several times  to get\nthe   Smit\
h  normal     form, that  is   not    in  the    paper ", count := 
                                                            [ 1, 5, 5, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec( 
                                                              Key := "HMM98" )
                                                            , content := 0, 
                                                          count := 
                                                            [ 1, 5, 5, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "Cite", 
                                                          next := 20057, 
                                                          root := ~, 
                                                          start := 20038, 
                                                          stop := 20056 ), 
                                                      rec( 
                                                          content := ".  The t\
ransforming matrices need not  be as  nice as for\nthe Hermite normal form.", 
                                                          count := 
                                                            [ 1, 5, 5, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 5, 5, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 20143, 
                                                          root := ~, 
                                                          start := 20139, 
                                                          stop := 20142 ), 
                                                      rec( content := "\n\n", 
                                                          count := 
                                                            [ 1, 5, 5, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "\\
ngap> ReadPackage(\"edim\",  \"tst/mat2\");\nReading 34x34 integer matrix 'mat\
2' with elementary divisors 'eldiv2'.\ntrue\ngap> tr := SmithIntMatLLLTrans(ma\
t2);;\ngap> tr[2] * mat2 * tr[3] = tr[1];    \ntrue\n", count := [ 1, 5, 5, 6 
                                                                     ], 
                                                                  name := "PCD\
ATA", root := ~ ) ], count := [ 1, 5, 5, 6 ], mathmode := "Text", 
                                                          name := "Example", 
                                                          next := 20363, 
                                                          root := ~, 
                                                          start := 20145, 
                                                          stop := 20362 ), 
                                                      rec( content := "\n", 
                                                          count := 
                                                            [ 1, 5, 5, 7 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ) ], 
                                                  count := [ 1, 5, 5, 3 ], 
                                                  mathmode := "Text", 
                                                  name := "Description", 
                                                  next := 20378, root := ~, 
                                                  start := 19756, 
                                                  stop := 20377 ) ], 
                                          count := [ 1, 5, 5, 1 ], 
                                          mathmode := "Text", 
                                          name := "ManSection", 
                                          next := 20392, root := ~, 
                                          start := 19688, stop := 20391 ), 
                                      rec( content := "\n\n", 
                                          count := [ 1, 5, 5, 7 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ) ], 
                                  count := [ 1, 5, 0, 1 ], mathmode := "Text",
                                  name := "Section", next := 20404, 
                                  root := ~, start := 17030, stop := 20403 ), 
                              rec( content := "\n\n\n", 
                                  count := [ 1, 5, 5, 7 ], mathmode := "Text",
                                  name := "PCDATA", root := ~ ), 
                              rec( 
                                  content := " ###############################\
############################# ", count := [ 1, 5, 5, 7 ], mathmode := "Text", 
                                  name := "XMLCOMMENT", root := ~ ), 
                              rec( content := "\n\n", count := [ 1, 5, 5, 7 ],
                                  mathmode := "Text", name := "PCDATA", 
                                  root := ~ ), 
                              rec( attributes := rec( Label := "Sect-Util" ), 
                                  content := 
                                    [ 
                                      rec( content := "\n\n", 
                                          count := [ 1, 6, 0, 1 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  content := "Utility Functions\
 from the ", count := [ 1, 6, 0, 2 ], mathmode := "Text", name := "PCDATA", 
                                                  root := ~ ), 
                                              rec( attributes := rec(  ), 
                                                  content := 
                                                    [ 
                                                      rec( content := "EDIM", 
                                                          count := 
                                                            [ 1, 6, 0, 2 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ) ], 
                                                  count := [ 1, 6, 0, 2 ], 
                                                  mathmode := "Text", 
                                                  name := "Package", 
                                                  next := 20566, root := ~, 
                                                  start := 20543, 
                                                  stop := 20565 ), 
                                              rec( content := "-package", 
                                                  count := [ 1, 6, 0, 2 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 6, 0, 2 ],
                                          mathmode := "Text", 
                                          name := "Heading", next := 20584, 
                                          root := ~, start := 20507, 
                                          stop := 20583 ), 
                                      rec( content := "\n\n", 
                                          count := [ 1, 6, 0, 3 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  attributes := 
                                                    rec( Arg := "n, k, l, x", 
                                                      Name := "RatNumberFromMo\
dular" ), content := 0, count := [ 1, 6, 1, 2 ], mathmode := "Text", 
                                                  name := "Func", 
                                                  next := 20659, root := ~, 
                                                  start := 20599, 
                                                  stop := 20658 ), 
                                              rec( attributes := rec(  ), 
                                                  content := 
                                                    [ 
                                                      rec( 
                                                          content := "\nThis fu\
nction returns ", count := [ 1, 6, 1, 3 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "r\
/s =  ", count := [ 1, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ),
                                                              rec( 
                                                                  attributes :\
= rec(  ), content := [ rec( content := "x", count := [ 1, 6, 1, 3 ], mathmode\
 := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 6, 1, 3 ], 
                                                                  mathmode := \
"Text", name := "A", next := 20715, root := ~, start := 20707, stop := 20714 )
                                                                , 
                                                              rec( 
                                                                  content := "\
 \\pmod{", count := [ 1, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ),
                                                              rec( 
                                                                  attributes :\
= rec(  ), content := [ rec( content := "n", count := [ 1, 6, 1, 3 ], mathmode\
 := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 6, 1, 3 ], 
                                                                  mathmode := \
"Text", name := "A", next := 20730, root := ~, start := 20722, stop := 20729 )
                                                                , 
                                                              rec( 
                                                                  content := "\
}", count := [ 1, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 6, 1, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 20735, 
                                                          root := ~, 
                                                          start := 20697, 
                                                          stop := 20734 ), 
                                                      rec( 
                                                          content := ", if\nit\
 exists. More precisely:", count := [ 1, 6, 1, 3 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 6, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 20770, 
                                                          root := ~, 
                                                          start := 20766, 
                                                          stop := 20769 ), 
                                                      rec( content := "\n\n", 
                                                          count := 
                                                            [ 1, 6, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "n\
", count := [ 1, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 6, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 20780, 
                                                          root := ~, 
                                                          start := 20772, 
                                                          stop := 20779 ), 
                                                      rec( content := ",   ", 
                                                          count := 
                                                            [ 1, 6, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "k\
", count := [ 1, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 6, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 20792, 
                                                          root := ~, 
                                                          start := 20784, 
                                                          stop := 20791 ), 
                                                      rec( content := ",  ", 
                                                          count := 
                                                            [ 1, 6, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "l\
", count := [ 1, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 6, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 20803, 
                                                          root := ~, 
                                                          start := 20795, 
                                                          stop := 20802 ), 
                                                      rec( 
                                                          content := "   must \
 be   positive  integers   with\n", count := [ 1, 6, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "2\
", count := [ 1, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ),
                                                              rec( 
                                                                  attributes :\
= rec(  ), content := [ rec( content := "k", count := [ 1, 6, 1, 4 ], mathmode\
 := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 6, 1, 4 ], 
                                                                  mathmode := \
"Text", name := "A", next := 20855, root := ~, start := 20847, stop := 20854 )
                                                                , 
                                                              rec( 
                                                                  attributes :\
= rec(  ), content := [ rec( content := "l", count := [ 1, 6, 1, 4 ], mathmode\
 := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 6, 1, 4 ], 
                                                                  mathmode := \
"Text", name := "A", next := 20863, root := ~, start := 20855, stop := 20862 )
                                                                , 
                                                              rec( 
                                                                  content := "\
 \\leq  ", count := [ 1, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ),
                                                              rec( 
                                                                  attributes :\
= rec(  ), content := [ rec( content := "n", count := [ 1, 6, 1, 4 ], mathmode\
 := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 6, 1, 4 ], 
                                                                  mathmode := \
"Text", name := "A", next := 20878, root := ~, start := 20870, stop := 20877 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 6, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 20882, 
                                                          root := ~, 
                                                          start := 20843, 
                                                          stop := 20881 ), 
                                                      rec( 
                                                          content := " and  ",
                                                          count := 
                                                            [ 1, 6, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "x\
", count := [ 1, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 6, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 20896, 
                                                          root := ~, 
                                                          start := 20888, 
                                                          stop := 20895 ), 
                                                      rec( 
                                                          content := " an  int\
eger with\n", count := [ 1, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "-\
", count := [ 1, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ),
                                                              rec( 
                                                                  attributes :\
= rec(  ), content := [ rec( content := "n", count := [ 1, 6, 1, 4 ], mathmode\
 := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 6, 1, 4 ], 
                                                                  mathmode := \
"Text", name := "A", next := 20926, root := ~, start := 20918, stop := 20925 )
                                                                , 
                                                              rec( 
                                                                  content := "\
/2  <  ", count := [ 1, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ),
                                                              rec( 
                                                                  attributes :\
= rec(  ), content := [ rec( content := "x", count := [ 1, 6, 1, 4 ], mathmode\
 := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 6, 1, 4 ], 
                                                                  mathmode := \
"Text", name := "A", next := 20944, root := ~, start := 20936, stop := 20943 )
                                                                , 
                                                              rec( 
                                                                  content := "\
  \\leq   ", count := [ 1, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ),
                                                              rec( 
                                                                  attributes :\
= rec(  ), content := [ rec( content := "n", count := [ 1, 6, 1, 4 ], mathmode\
 := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 6, 1, 4 ], 
                                                                  mathmode := \
"Text", name := "A", next := 20961, root := ~, start := 20953, stop := 20960 )
                                                                , 
                                                              rec( 
                                                                  content := "\
/2", count := [ 1, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 6, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 20967, 
                                                          root := ~, 
                                                          start := 20914, 
                                                          stop := 20966 ), 
                                                      rec( 
                                                          content := ".  If  i\
t  exists\nthis  function returns  a rational  number ", count := 
                                                            [ 1, 6, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "r\
/s", count := [ 1, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 6, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 21038, 
                                                          root := ~, 
                                                          start := 21028, 
                                                          stop := 21037 ), 
                                                      rec( 
                                                          content := "  with "
                                                            , 
                                                          count := 
                                                            [ 1, 6, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "0\
  <\ns  <  ", count := [ 1, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ),
                                                              rec( 
                                                                  attributes :\
= rec(  ), content := [ rec( content := "l", count := [ 1, 6, 1, 4 ], mathmode\
 := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 6, 1, 4 ], 
                                                                  mathmode := \
"Text", name := "A", next := 21073, root := ~, start := 21065, stop := 21072 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 6, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 21077, 
                                                          root := ~, 
                                                          start := 21045, 
                                                          stop := 21076 ), 
                                                      rec( content := ",  ", 
                                                          count := 
                                                            [ 1, 6, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "\\
\gcd(s,  ", count := [ 1, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ),
                                                              rec( 
                                                                  attributes :\
= rec(  ), content := [ rec( content := "n", count := [ 1, 6, 1, 4 ], mathmode\
 := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 6, 1, 4 ], 
                                                                  mathmode := \
"Text", name := "A", next := 21100, root := ~, start := 21092, stop := 21099 )
                                                                , 
                                                              rec( 
                                                                  content := "\
)  =  1", count := [ 1, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 6, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 21111, 
                                                          root := ~, 
                                                          start := 21080, 
                                                          stop := 21110 ), 
                                                      rec( content := ",  ", 
                                                          count := 
                                                            [ 1, 6, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "-\
", count := [ 1, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ),
                                                              rec( 
                                                                  attributes :\
= rec(  ), content := [ rec( content := "k", count := [ 1, 6, 1, 4 ], mathmode\
 := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 6, 1, 4 ], 
                                                                  mathmode := \
"Text", name := "A", next := 21126, root := ~, start := 21118, stop := 21125 )
                                                                , 
                                                              rec( 
                                                                  content := "\
\n<  r  < ", count := [ 1, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ),
                                                              rec( 
                                                                  attributes :\
= rec(  ), content := [ rec( content := "k", count := [ 1, 6, 1, 4 ], mathmode\
 := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 6, 1, 4 ], 
                                                                  mathmode := \
"Text", name := "A", next := 21149, root := ~, start := 21141, stop := 21148 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 6, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 21153, 
                                                          root := ~, 
                                                          start := 21114, 
                                                          stop := 21152 ), 
                                                      rec( 
                                                          content := "  and  "
                                                            , 
                                                          count := 
                                                            [ 1, 6, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "r\
/s", count := [ 1, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 6, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 21170, 
                                                          root := ~, 
                                                          start := 21160, 
                                                          stop := 21169 ), 
                                                      rec( 
                                                          content := "  congru\
ent to  ", count := [ 1, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  attributes :=\
 rec(  ), content := [ rec( content := "x", count := [ 1, 6, 1, 4 ], mathmode \
:= "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 6, 1, 4 ], 
                                                                  mathmode := \
"Text", name := "A", next := 21197, root := ~, start := 21189, stop := 21196 )
                                                                , 
                                                              rec( 
                                                                  content := "\
\n\\pmod{", count := [ 1, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ),
                                                              rec( 
                                                                  attributes :\
= rec(  ), content := [ rec( content := "n", count := [ 1, 6, 1, 4 ], mathmode\
 := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 6, 1, 4 ], 
                                                                  mathmode := \
"Text", name := "A", next := 21212, root := ~, start := 21204, stop := 21211 )
                                                                , 
                                                              rec( 
                                                                  content := "\
}", count := [ 1, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 6, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 21217, 
                                                          root := ~, 
                                                          start := 21186, 
                                                          stop := 21216 ), 
                                                      rec( 
                                                          content := " (i.e., \
", count := [ 1, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ),
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  attributes :=\
 rec(  ), content := [ rec( content := "n", count := [ 1, 6, 1, 4 ], mathmode \
:= "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 6, 1, 4 ], 
                                                                  mathmode := \
"Text", name := "A", next := 21236, root := ~, start := 21228, stop := 21235 )
                                                                , 
                                                              rec( 
                                                                  content := "\
 \\mid  r - s ", count := [ 1, 6, 1, 4 ], mathmode := "Text", name := "PCDATA"
                                                                    , 
                                                                  root := ~ ),
                                                              rec( 
                                                                  attributes :\
= rec(  ), content := [ rec( content := "x", count := [ 1, 6, 1, 4 ], mathmode\
 := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 6, 1, 4 ], 
                                                                  mathmode := \
"Text", name := "A", next := 21257, root := ~, start := 21249, stop := 21256 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 6, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 21261, 
                                                          root := ~, 
                                                          start := 21225, 
                                                          stop := 21260 ), 
                                                      rec( 
                                                          content := "). Such\\
nan ", count := [ 1, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "r\
/s", count := [ 1, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 6, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 21282, 
                                                          root := ~, 
                                                          start := 21272, 
                                                          stop := 21281 ), 
                                                      rec( 
                                                          content := "  is uni\
que. The  function returns ", count := [ 1, 6, 1, 4 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "f\
ail", count := [ 1, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 6, 1, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "K", 
                                                          next := 21328, 
                                                          root := ~, 
                                                          start := 21317, 
                                                          stop := 21327 ), 
                                                      rec( 
                                                          content := " if  suc\
h a\nnumber does not exist.", count := [ 1, 6, 1, 4 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 6, 1, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 21366, 
                                                          root := ~, 
                                                          start := 21362, 
                                                          stop := 21365 ), 
                                                      rec( content := "\n", 
                                                          count := 
                                                            [ 1, 6, 1, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ) ], 
                                                  count := [ 1, 6, 1, 3 ], 
                                                  mathmode := "Text", 
                                                  name := "Description", 
                                                  next := 21381, root := ~, 
                                                  start := 20661, 
                                                  stop := 21380 ) ], 
                                          count := [ 1, 6, 1, 1 ], 
                                          mathmode := "Text", 
                                          name := "ManSection", 
                                          next := 21395, root := ~, 
                                          start := 20586, stop := 21394 ), 
                                      rec( content := "\n\n", 
                                          count := [ 1, 6, 1, 5 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( 
                                          content := " #######################\
##################################### ", count := [ 1, 6, 1, 5 ], 
                                          mathmode := "Text", 
                                          name := "XMLCOMMENT", root := ~ ), 
                                      rec( content := "\n\n", 
                                          count := [ 1, 6, 1, 5 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  attributes := 
                                                    rec( Arg := "A, p", 
                                                      Name := "InverseIntMatMo\
d" ), content := 0, count := [ 1, 6, 2, 2 ], mathmode := "Text", 
                                                  name := "Func", 
                                                  next := 21531, root := ~, 
                                                  start := 21481, 
                                                  stop := 21530 ), 
                                              rec( attributes := rec(  ), 
                                                  content := 
                                                    [ 
                                                      rec( 
                                                          content := "\nThis fu\
nction  returns  an inverse  matrix modulo  a prime ", count := [ 1, 6, 2, 3 ]
                                                            , 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "p\
", count := [ 1, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 6, 2, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 21615, 
                                                          root := ~, 
                                                          start := 21607, 
                                                          stop := 21614 ), 
                                                      rec( content := " or\n",
                                                          count := 
                                                            [ 1, 6, 2, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "f\
ail", count := [ 1, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 6, 2, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "K", 
                                                          next := 21630, 
                                                          root := ~, 
                                                          start := 21619, 
                                                          stop := 21629 ), 
                                                      rec( 
                                                          content := ". More p\
recisely:", count := [ 1, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 6, 2, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 21651, 
                                                          root := ~, 
                                                          start := 21647, 
                                                          stop := 21650 ), 
                                                      rec( content := "\n\n", 
                                                          count := 
                                                            [ 1, 6, 2, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "A\
", count := [ 1, 6, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 6, 2, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 21661, 
                                                          root := ~, 
                                                          start := 21653, 
                                                          stop := 21660 ), 
                                                      rec( 
                                                          content := "  must  \
be an  integer matrix and   ", count := [ 1, 6, 2, 4 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "p\
", count := [ 1, 6, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 6, 2, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 21705, 
                                                          root := ~, 
                                                          start := 21697, 
                                                          stop := 21704 ), 
                                                      rec( 
                                                          content := " a prime\
 such that\n", count := [ 1, 6, 2, 4 ], mathmode := "Text", name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "A\
", count := [ 1, 6, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 6, 2, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 21732, 
                                                          root := ~, 
                                                          start := 21724, 
                                                          stop := 21731 ), 
                                                      rec( 
                                                          content := "  is inv\
ertible modulo   ", count := [ 1, 6, 2, 4 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "p\
", count := [ 1, 6, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 6, 2, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 21765, 
                                                          root := ~, 
                                                          start := 21757, 
                                                          stop := 21764 ), 
                                                      rec( 
                                                          content := ".   This\
 function  returns an\ninteger matrix ", count := [ 1, 6, 2, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "i\
nv", count := [ 1, 6, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 6, 2, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 21820, 
                                                          root := ~, 
                                                          start := 21810, 
                                                          stop := 21819 ), 
                                                      rec( 
                                                          content := " with   \
entries in the range  ", count := [ 1, 6, 2, 4 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "]\
-", count := [ 1, 6, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ )
                                                                , 
                                                              rec( 
                                                                  attributes :\
= rec(  ), content := [ rec( content := "p", count := [ 1, 6, 2, 4 ], mathmode\
 := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 6, 2, 4 ], 
                                                                  mathmode := \
"Text", name := "A", next := 21863, root := ~, start := 21855, stop := 21862 )
                                                                , 
                                                              rec( 
                                                                  content := "\
/2\n\\ldots ", count := [ 1, 6, 2, 4 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ),
                                                              rec( 
                                                                  attributes :\
= rec(  ), content := [ rec( content := "p", count := [ 1, 6, 2, 4 ], mathmode\
 := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 6, 2, 4 ], 
                                                                  mathmode := \
"Text", name := "A", next := 21881, root := ~, start := 21873, stop := 21880 )
                                                                , 
                                                              rec( 
                                                                  content := "\
/2]", count := [ 1, 6, 2, 4 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 6, 2, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "M", 
                                                          next := 21888, 
                                                          root := ~, 
                                                          start := 21850, 
                                                          stop := 21887 ), 
                                                      rec( 
                                                          content := " such th\
at  ", count := [ 1, 6, 2, 4 ], mathmode := "Text", name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "i\
nv", count := [ 1, 6, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 6, 2, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 21910, 
                                                          root := ~, 
                                                          start := 21900, 
                                                          stop := 21909 ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "A\
", count := [ 1, 6, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 6, 2, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 21918, 
                                                          root := ~, 
                                                          start := 21910, 
                                                          stop := 21917 ), 
                                                      rec( 
                                                          content := " reduced\
  modulo p\nis the identity matrix.", count := [ 1, 6, 2, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 6, 2, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 21964, 
                                                          root := ~, 
                                                          start := 21960, 
                                                          stop := 21963 ), 
                                                      rec( 
                                                          content := "\n\nIt r\
eturns ", count := [ 1, 6, 2, 5 ], mathmode := "Text", name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "f\
ail", count := [ 1, 6, 2, 5 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 6, 2, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "K", 
                                                          next := 21988, 
                                                          root := ~, 
                                                          start := 21977, 
                                                          stop := 21987 ), 
                                                      rec( 
                                                          content := "  if the\
 inverse modulo ", count := [ 1, 6, 2, 5 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "p\
", count := [ 1, 6, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 6, 2, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 22020, 
                                                          root := ~, 
                                                          start := 22012, 
                                                          stop := 22019 ), 
                                                      rec( 
                                                          content := " does no\
t exist.\n\nThis function is particularly fast for primes smaller 256.\n", 
                                                          count := 
                                                            [ 1, 6, 2, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ) ], 
                                                  count := [ 1, 6, 2, 3 ], 
                                                  mathmode := "Text", 
                                                  name := "Description", 
                                                  next := 22111, root := ~, 
                                                  start := 21534, 
                                                  stop := 22110 ) ], 
                                          count := [ 1, 6, 2, 1 ], 
                                          mathmode := "Text", 
                                          name := "ManSection", 
                                          next := 22125, root := ~, 
                                          start := 21468, stop := 22124 ), 
                                      rec( content := "\n\n", 
                                          count := [ 1, 6, 2, 5 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( 
                                          content := " #######################\
##################################### ", count := [ 1, 6, 2, 5 ], 
                                          mathmode := "Text", 
                                          name := "XMLCOMMENT", root := ~ ), 
                                      rec( content := "\n\n", 
                                          count := [ 1, 6, 2, 5 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  attributes := 
                                                    rec( Arg := "A", 
                                                      Name := "HadamardBoundIn\
tMat" ), content := 0, count := [ 1, 6, 3, 2 ], mathmode := "Text", 
                                                  name := "Func", 
                                                  next := 22261, root := ~, 
                                                  start := 22211, 
                                                  stop := 22260 ), 
                                              rec( attributes := rec(  ), 
                                                  content := 
                                                    [ 
                                                      rec( 
                                                          content := "\nThe Had\
amard bound for a square integer matrix ", count := [ 1, 6, 3, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "A\
", count := [ 1, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 6, 3, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 22333, 
                                                          root := ~, 
                                                          start := 22325, 
                                                          stop := 22332 ), 
                                                      rec( 
                                                          content := " is the \
product\nof Euclidean norms of the nonzero rows (or columns) of ", 
                                                          count := 
                                                            [ 1, 6, 3, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "A\
", count := [ 1, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 6, 3, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 22412, 
                                                          root := ~, 
                                                          start := 22404, 
                                                          stop := 22411 ), 
                                                      rec( 
                                                          content := ". It is\\
nan upper bound for the absolute value of the determinant of ", 
                                                          count := 
                                                            [ 1, 6, 3, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "A\
", count := [ 1, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 6, 3, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 22488, 
                                                          root := ~, 
                                                          start := 22480, 
                                                          stop := 22487 ), 
                                                      rec( content := ".\n", 
                                                          count := 
                                                            [ 1, 6, 3, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ) ], 
                                                  count := [ 1, 6, 3, 3 ], 
                                                  mathmode := "Text", 
                                                  name := "Description", 
                                                  next := 22504, root := ~, 
                                                  start := 22264, 
                                                  stop := 22503 ) ], 
                                          count := [ 1, 6, 3, 1 ], 
                                          mathmode := "Text", 
                                          name := "ManSection", 
                                          next := 22518, root := ~, 
                                          start := 22198, stop := 22517 ), 
                                      rec( content := "\n\n", 
                                          count := [ 1, 6, 3, 3 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( 
                                          content := " #######################\
##################################### ", count := [ 1, 6, 3, 3 ], 
                                          mathmode := "Text", 
                                          name := "XMLCOMMENT", root := ~ ), 
                                      rec( content := "\n\n", 
                                          count := [ 1, 6, 3, 3 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  attributes := 
                                                    rec( Arg := "n[, nr]", 
                                                      Name := "CheapFactorsInt\
" ), content := 0, count := [ 1, 6, 4, 2 ], mathmode := "Text", 
                                                  name := "Func", 
                                                  next := 22656, root := ~, 
                                                  start := 22604, 
                                                  stop := 22655 ), 
                                              rec( attributes := rec(  ), 
                                                  content := 
                                                    [ 
                                                      rec( 
                                                          content := "\nThis  f\
unction  returns a  list  of factors  of   an integer ", count := 
                                                            [ 1, 6, 4, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "n\
", count := [ 1, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 6, 4, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 22742, 
                                                          root := ~, 
                                                          start := 22734, 
                                                          stop := 22741 ), 
                                                      rec( 
                                                          content := ",\ninclu\
ding  ", count := [ 1, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "s\
mall", count := [ 1, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 6, 4, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "Q", 
                                                          next := 22767, 
                                                          root := ~, 
                                                          start := 22755, 
                                                          stop := 22766 ), 
                                                      rec( 
                                                          content := " prime  \
factors -  here   the optional argument\n", count := [ 1, 6, 4, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "n\
r", count := [ 1, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 6, 4, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 22824, 
                                                          root := ~, 
                                                          start := 22815, 
                                                          stop := 22823 ), 
                                                      rec( 
                                                          content := " is the \
  number of iterations  for `FactorsRho'  (default is\n2000).", 
                                                          count := 
                                                            [ 1, 6, 4, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 6, 4, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 22896, 
                                                          root := ~, 
                                                          start := 22892, 
                                                          stop := 22895 ), 
                                                      rec( 
                                                          content := "\n\nThis\
  is  only  a  slight  modification of  the  library  function  ", 
                                                          count := 
                                                            [ 1, 6, 4, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec( 
                                                              BookName := "ref\
", Func := "FactorsInt" ), content := 0, count := [ 1, 6, 4, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "Ref", 
                                                          next := 23005, 
                                                          root := ~, 
                                                          start := 22966, 
                                                          stop := 23004 ), 
                                                      rec( 
                                                          content := "  which \
avoids  an error  message when\nthe number is not completely factored.", 
                                                          count := 
                                                            [ 1, 6, 4, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 6, 4, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 23086, 
                                                          root := ~, 
                                                          start := 23082, 
                                                          stop := 23085 ), 
                                                      rec( content := " \n", 
                                                          count := 
                                                            [ 1, 6, 4, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ) ], 
                                                  count := [ 1, 6, 4, 3 ], 
                                                  mathmode := "Text", 
                                                  name := "Description", 
                                                  next := 23102, root := ~, 
                                                  start := 22659, 
                                                  stop := 23101 ) ], 
                                          count := [ 1, 6, 4, 1 ], 
                                          mathmode := "Text", 
                                          name := "ManSection", 
                                          next := 23117, root := ~, 
                                          start := 22591, stop := 23116 ), 
                                      rec( content := "\n\n", 
                                          count := [ 1, 6, 4, 5 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( 
                                          content := " #######################\
##################################### ", count := [ 1, 6, 4, 5 ], 
                                          mathmode := "Text", 
                                          name := "XMLCOMMENT", root := ~ ), 
                                      rec( content := "\n\n", 
                                          count := [ 1, 6, 4, 5 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  attributes := 
                                                    rec( Arg := "A, p", 
                                                      Name := "RankMod" ), 
                                                  content := 0, 
                                                  count := [ 1, 6, 5, 2 ], 
                                                  mathmode := "Text", 
                                                  name := "Func", 
                                                  next := 23244, root := ~, 
                                                  start := 23203, 
                                                  stop := 23243 ), 
                                              rec( attributes := rec(  ), 
                                                  content := 
                                                    [ 
                                                      rec( 
                                                          content := "\nThis fu\
nction returns the rank  of  an integer matrix ", count := [ 1, 6, 5, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "A\
", count := [ 1, 6, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 6, 5, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 23322, 
                                                          root := ~, 
                                                          start := 23314, 
                                                          stop := 23321 ), 
                                                      rec( 
                                                          content := "  modulo\
\n", count := [ 1, 6, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                         ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "p\
", count := [ 1, 6, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 6, 5, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 23339, 
                                                          root := ~, 
                                                          start := 23331, 
                                                          stop := 23338 ), 
                                                      rec( 
                                                          content := ". Here "
                                                            , 
                                                          count := 
                                                            [ 1, 6, 5, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "p\
", count := [ 1, 6, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 6, 5, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 23354, 
                                                          root := ~, 
                                                          start := 23346, 
                                                          stop := 23353 ), 
                                                      rec( 
                                                          content := " must no\
t necessarily be  a prime. If it is not\nand  this function   returns an  inte\
ger, then   this is   the rank of\n", count := [ 1, 6, 5, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "A\
", count := [ 1, 6, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 6, 5, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 23481, 
                                                          root := ~, 
                                                          start := 23473, 
                                                          stop := 23480 ), 
                                                      rec( 
                                                          content := " for all\
 prime divisors of ", count := [ 1, 6, 5, 3 ], mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "p\
", count := [ 1, 6, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 6, 5, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 23516, 
                                                          root := ~, 
                                                          start := 23508, 
                                                          stop := 23515 ), 
                                                      rec( content := ".", 
                                                          count := 
                                                            [ 1, 6, 5, 3 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 6, 5, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 23521, 
                                                          root := ~, 
                                                          start := 23517, 
                                                          stop := 23520 ), 
                                                      rec( 
                                                          content := "\n\nIf  \
during the   computation  a  factorisation  of   ", count := [ 1, 6, 5, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "p\
", count := [ 1, 6, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 6, 5, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 23584, 
                                                          root := ~, 
                                                          start := 23576, 
                                                          stop := 23583 ), 
                                                      rec( 
                                                          content := " is foun\
d\n(because some pivot entry has nontrivial  greatest common divisor with\n", 
                                                          count := 
                                                            [ 1, 6, 5, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "p\
", count := [ 1, 6, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 6, 5, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 23673, 
                                                          root := ~, 
                                                          start := 23665, 
                                                          stop := 23672 ), 
                                                      rec( 
                                                          content := ") then t\
he function     is recursively applied to   the  found\nfactors ", 
                                                          count := 
                                                            [ 1, 6, 5, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "f\
_i", count := [ 1, 6, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                                 ) ], 
                                                          count := 
                                                            [ 1, 6, 5, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "C", 
                                                          next := 23754, 
                                                          root := ~, 
                                                          start := 23744, 
                                                          stop := 23753 ), 
                                                      rec( content := " of ", 
                                                          count := 
                                                            [ 1, 6, 5, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "p\
", count := [ 1, 6, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 6, 5, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 23766, 
                                                          root := ~, 
                                                          start := 23758, 
                                                          stop := 23765 ), 
                                                      rec( 
                                                          content := ".  The r\
esult is then given in the form\n", count := [ 1, 6, 5, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "[\
[f_1, rk_1], [f_2, rk_2], ...]", count := [ 1, 6, 5, 4 ], mathmode := "Text", 
                                                                  name := "PCD\
ATA", root := ~ ) ], count := [ 1, 6, 5, 4 ], mathmode := "Text", 
                                                          name := "C", 
                                                          next := 23844, 
                                                          root := ~, 
                                                          start := 23806, 
                                                          stop := 23843 ), 
                                                      rec( content := ".", 
                                                          count := 
                                                            [ 1, 6, 5, 4 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 6, 5, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 23849, 
                                                          root := ~, 
                                                          start := 23845, 
                                                          stop := 23848 ), 
                                                      rec( 
                                                          content := "\n\nThe \
idea to make  this function useful  for non primes  was to use  it\nwith large\
 factors of the biggest elementary divisor of ", count := [ 1, 6, 5, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "A\
", count := [ 1, 6, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) 
                                                             ], 
                                                          count := 
                                                            [ 1, 6, 5, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "A", 
                                                          next := 23986, 
                                                          root := ~, 
                                                          start := 23978, 
                                                          stop := 23985 ), 
                                                      rec( 
                                                          content := " whose\n\
prime factorization cannot be found easily.", count := [ 1, 6, 5, 5 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 0, 
                                                          count := 
                                                            [ 1, 6, 5, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "P", 
                                                          next := 24040, 
                                                          root := ~, 
                                                          start := 24036, 
                                                          stop := 24039 ), 
                                                      rec( content := "\n\n", 
                                                          count := 
                                                            [ 1, 6, 5, 6 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "\\
ngap> ReadPackage(\"edim\",  \"tst/mat\");\nReading 242x242 integer matrix 'ma\
t' with elementary divisors 'eldiv'.\ntrue\ngap> RankMod(mat, 5);\n155\ngap> R\
ankMod(mat, (2*79*4001));\n[ [ 2, 148 ], [ 79, 242 ], [ 4001, 242 ] ]\n", 
                                                                  count := 
                                                                    [ 1, 6, 
                                                                      5, 7 ], 
                                                                  name := "PCD\
ATA", root := ~ ) ], count := [ 1, 6, 5, 7 ], mathmode := "Text", 
                                                          name := "Example", 
                                                          next := 24277, 
                                                          root := ~, 
                                                          start := 24042, 
                                                          stop := 24276 ), 
                                                      rec( content := "\n", 
                                                          count := 
                                                            [ 1, 6, 5, 8 ], 
                                                          mathmode := "Text", 
                                                          name := "PCDATA", 
                                                          root := ~ ) ], 
                                                  count := [ 1, 6, 5, 3 ], 
                                                  mathmode := "Text", 
                                                  name := "Description", 
                                                  next := 24292, root := ~, 
                                                  start := 23246, 
                                                  stop := 24291 ) ], 
                                          count := [ 1, 6, 5, 1 ], 
                                          mathmode := "Text", 
                                          name := "ManSection", 
                                          next := 24306, root := ~, 
                                          start := 23190, stop := 24305 ), 
                                      rec( content := "\n\n", 
                                          count := [ 1, 6, 5, 8 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ) ], 
                                  count := [ 1, 6, 0, 1 ], mathmode := "Text",
                                  name := "Section", next := 24318, 
                                  root := ~, start := 20478, stop := 24317 ), 
                              rec( content := "\n\n", count := [ 1, 6, 5, 8 ],
                                  mathmode := "Text", name := "PCDATA", 
                                  root := ~ ), 
                              rec( 
                                  content := " ###############################\
############################# ", count := [ 1, 6, 5, 8 ], mathmode := "Text", 
                                  name := "XMLCOMMENT", root := ~ ), 
                              rec( content := "\n\n", count := [ 1, 6, 5, 8 ],
                                  mathmode := "Text", name := "PCDATA", 
                                  root := ~ ), 
                              rec( 
                                  attributes := 
                                    rec( Label := "Sect-InvRatMatAlg" ), 
                                  content := 
                                    [ 
                                      rec( content := "\n", 
                                          count := [ 1, 7, 0, 1 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  content := "InverseRatMat - t\
he Algorithm", count := [ 1, 7, 0, 2 ], mathmode := "Text", name := "PCDATA", 
                                                  root := ~ ) ], 
                                          count := [ 1, 7, 0, 2 ], 
                                          mathmode := "Text", 
                                          name := "Heading", next := 24475, 
                                          root := ~, start := 24427, 
                                          stop := 24474 ), 
                                      rec( 
                                          content := "\n\n\nThe  idea   is to \
 recover a rational    matrix from  an ", count := [ 1, 7, 0, 3 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "l", 
                                                  count := [ 1, 7, 0, 3 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 3 ],
                                          mathmode := "Text", name := "M", 
                                          next := 24543, root := ~, 
                                          start := 24535, stop := 24542 ), 
                                      rec( 
                                          content := "-adic\napproximation for\
 some prime ", count := [ 1, 7, 0, 3 ], mathmode := "Text", name := "PCDATA", 
                                          root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "l", 
                                                  count := [ 1, 7, 0, 3 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 3 ],
                                          mathmode := "Text", name := "M", 
                                          next := 24586, root := ~, 
                                          start := 24578, stop := 24585 ), 
                                      rec( 
                                          content := ".   This description cam\
e out  of\ndiscussions with Jürgen Müller.  I  thank John Cannon for pointin\
g out\nthat the basic idea already appeared in the paper ", 
                                          count := [ 1, 7, 0, 3 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec( Key := "D82" ), 
                                          content := 0, 
                                          count := [ 1, 7, 0, 3 ], 
                                          mathmode := "Text", name := "Cite", 
                                          next := 24760, root := ~, 
                                          start := 24743, stop := 24759 ), 
                                      rec( content := " of\nDixon.", 
                                          count := [ 1, 7, 0, 3 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 0, 
                                          count := [ 1, 7, 0, 4 ], 
                                          mathmode := "Text", name := "P", 
                                          next := 24774, root := ~, 
                                          start := 24770, stop := 24773 ), 
                                      rec( content := "\n\nLet ", 
                                          count := [ 1, 7, 0, 4 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "A", 
                                                  count := [ 1, 7, 0, 4 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 4 ],
                                          mathmode := "Text", name := "M", 
                                          next := 24788, root := ~, 
                                          start := 24780, stop := 24787 ), 
                                      rec( 
                                          content := " be an  invertible matri\
x over  the rational  numbers.  By\nmultiplying with a constant we may assume \
that its entries are in fact\nintegers.", count := [ 1, 7, 0, 4 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 0, 
                                          count := [ 1, 7, 0, 5 ], 
                                          mathmode := "Text", name := "P", 
                                          next := 24931, root := ~, 
                                          start := 24927, stop := 24930 ), 
                                      rec( 
                                          content := "\n\n(1) We first  descri\
be how to find  an  ", count := [ 1, 7, 0, 5 ], mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "l", 
                                                  count := [ 1, 7, 0, 5 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 5 ],
                                          mathmode := "Text", name := "M", 
                                          next := 24981, root := ~, 
                                          start := 24973, stop := 24980 ), 
                                      rec( 
                                          content := "-adic approximation of\n\
", count := [ 1, 7, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ),
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "A^{-1}", 
                                                  count := [ 1, 7, 0, 5 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 5 ],
                                          mathmode := "Text", name := "M", 
                                          next := 25017, root := ~, 
                                          start := 25004, stop := 25016 ), 
                                      rec( content := ".  Find a prime ", 
                                          count := [ 1, 7, 0, 5 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "l", 
                                                  count := [ 1, 7, 0, 5 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 5 ],
                                          mathmode := "Text", name := "M", 
                                          next := 25041, root := ~, 
                                          start := 25033, stop := 25040 ), 
                                      rec( content := " such that ", 
                                          count := [ 1, 7, 0, 5 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "A", 
                                                  count := [ 1, 7, 0, 5 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 5 ],
                                          mathmode := "Text", name := "M", 
                                          next := 25060, root := ~, 
                                          start := 25052, stop := 25059 ), 
                                      rec( 
                                          content := " is invertible\nmodulo "
                                            , count := [ 1, 7, 0, 5 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "l", 
                                                  count := [ 1, 7, 0, 5 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 5 ],
                                          mathmode := "Text", name := "M", 
                                          next := 25090, root := ~, 
                                          start := 25082, stop := 25089 ), 
                                      rec( content := " and let ", 
                                          count := [ 1, 7, 0, 5 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "B", 
                                                  count := [ 1, 7, 0, 5 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 5 ],
                                          mathmode := "Text", name := "M", 
                                          next := 25107, root := ~, 
                                          start := 25099, stop := 25106 ), 
                                      rec( 
                                          content := " be the integer matrix w\
ith entries in\nthe   range   ", count := [ 1, 7, 0, 5 ], mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  content := "\\left]-l/2,l/2\\\
right]", count := [ 1, 7, 0, 5 ], mathmode := "Text", name := "PCDATA", 
                                                  root := ~ ) ], 
                                          count := [ 1, 7, 0, 5 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 25188, root := ~, 
                                          start := 25160, stop := 25187 ), 
                                      rec( content := "  such  that   ", 
                                          count := [ 1, 7, 0, 5 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "BA", 
                                                  count := [ 1, 7, 0, 5 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 5 ],
                                          mathmode := "Text", name := "M", 
                                          next := 25212, root := ~, 
                                          start := 25203, stop := 25211 ), 
                                      rec( 
                                          content := "  is\ncongruent  to  the\
 identity matrix   modulo  ", count := [ 1, 7, 0, 5 ], mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "l", 
                                                  count := [ 1, 7, 0, 5 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 5 ],
                                          mathmode := "Text", name := "M", 
                                          next := 25270, root := ~, 
                                          start := 25262, stop := 25269 ), 
                                      rec( 
                                          content := ".  (This  can  be\ncompu\
ted fast by usual Gauß elimination.)", count := [ 1, 7, 0, 5 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 0, 
                                          count := [ 1, 7, 0, 6 ], 
                                          mathmode := "Text", name := "P", 
                                          next := 25334, root := ~, 
                                          start := 25330, stop := 25333 ), 
                                      rec( content := "\n\nNow let  ", 
                                          count := [ 1, 7, 0, 6 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "v \\in  ℤ^r", 
                                                  count := [ 1, 7, 0, 6 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 6 ],
                                          mathmode := "Text", name := "M", 
                                          next := 25365, root := ~, 
                                          start := 25345, stop := 25364 ), 
                                      rec( 
                                          content := " be  a row  vector. Defi\
ne  two sequences\n", count := [ 1, 7, 0, 6 ], mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "v_i", 
                                                  count := [ 1, 7, 0, 6 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 6 ],
                                          mathmode := "Text", name := "M", 
                                          next := 25417, root := ~, 
                                          start := 25407, stop := 25416 ), 
                                      rec( content := " and ", 
                                          count := [ 1, 7, 0, 6 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "x_i", 
                                                  count := [ 1, 7, 0, 6 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 6 ],
                                          mathmode := "Text", name := "M", 
                                          next := 25432, root := ~, 
                                          start := 25422, stop := 25431 ), 
                                      rec( content := "  of row vectors in  ",
                                          count := [ 1, 7, 0, 6 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "ℤ^r", 
                                                  count := [ 1, 7, 0, 6 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 6 ],
                                          mathmode := "Text", name := "M", 
                                          next := 25466, root := ~, 
                                          start := 25453, stop := 25465 ), 
                                      rec( content := " by: ", 
                                          count := [ 1, 7, 0, 6 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  content := "x_0\n:= 0  \\in �\
��^r", count := [ 1, 7, 0, 6 ], mathmode := "Text", name := "PCDATA", 
                                                  root := ~ ) ], 
                                          count := [ 1, 7, 0, 6 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 25498, root := ~, 
                                          start := 25471, stop := 25497 ), 
                                      rec( content := ",  ", 
                                          count := [ 1, 7, 0, 6 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "v_0 :=  -v", 
                                                  count := [ 1, 7, 0, 6 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 6 ],
                                          mathmode := "Text", name := "M", 
                                          next := 25518, root := ~, 
                                          start := 25501, stop := 25517 ), 
                                      rec( content := " and for  ", 
                                          count := [ 1, 7, 0, 6 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "i >  0", 
                                                  count := [ 1, 7, 0, 6 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 6 ],
                                          mathmode := "Text", name := "M", 
                                          next := 25544, root := ~, 
                                          start := 25528, stop := 25543 ), 
                                      rec( content := " set\n", 
                                          count := [ 1, 7, 0, 6 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "x_i", 
                                                  count := [ 1, 7, 0, 6 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 6 ],
                                          mathmode := "Text", name := "M", 
                                          next := 25559, root := ~, 
                                          start := 25549, stop := 25558 ), 
                                      rec( 
                                          content := "  to  the  vector  congr\
uent  to  ", count := [ 1, 7, 0, 6 ], mathmode := "Text", name := "PCDATA", 
                                          root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "-v_{i-1}  B", 
                                                  count := [ 1, 7, 0, 6 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 6 ],
                                          mathmode := "Text", name := "M", 
                                          next := 25611, root := ~, 
                                          start := 25593, stop := 25610 ), 
                                      rec( content := "  modulo\n", 
                                          count := [ 1, 7, 0, 6 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "l", 
                                                  count := [ 1, 7, 0, 6 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 6 ],
                                          mathmode := "Text", name := "M", 
                                          next := 25628, root := ~, 
                                          start := 25620, stop := 25627 ), 
                                      rec( 
                                          content := " having  entries in  the\
 range  ", count := [ 1, 7, 0, 6 ], mathmode := "Text", name := "PCDATA", 
                                          root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  content := "\\left]-l/2, l/2\\
\right]", count := [ 1, 7, 0, 6 ], mathmode := "Text", name := "PCDATA", 
                                                  root := ~ ) ], 
                                          count := [ 1, 7, 0, 6 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 25689, root := ~, 
                                          start := 25660, stop := 25688 ), 
                                      rec( 
                                          content := ".\nThen all entries  of \
", count := [ 1, 7, 0, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ),
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  content := "x_i A + v_{i-1}",
                                                  count := [ 1, 7, 0, 6 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 6 ],
                                          mathmode := "Text", name := "M", 
                                          next := 25734, root := ~, 
                                          start := 25712, stop := 25733 ), 
                                      rec( content := " are  divisible by ", 
                                          count := [ 1, 7, 0, 6 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "l", 
                                                  count := [ 1, 7, 0, 6 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 6 ],
                                          mathmode := "Text", name := "M", 
                                          next := 25761, root := ~, 
                                          start := 25753, stop := 25760 ), 
                                      rec( content := "\nand we set ", 
                                          count := [ 1, 7, 0, 6 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  content := "v_i := (1/l) \\cd\
ot (x_i A + v_{i-1})", count := [ 1, 7, 0, 6 ], mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 6 ],
                                          mathmode := "Text", name := "M", 
                                          next := 25816, root := ~, 
                                          start := 25773, stop := 25815 ), 
                                      rec( content := ".", 
                                          count := [ 1, 7, 0, 6 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 0, 
                                          count := [ 1, 7, 0, 7 ], 
                                          mathmode := "Text", name := "P", 
                                          next := 25821, root := ~, 
                                          start := 25817, stop := 25820 ), 
                                      rec( 
                                          content := "\n\nInduction  shows tha\
t for ", count := [ 1, 7, 0, 7 ], mathmode := "Text", name := "PCDATA", 
                                          root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  content := "y_i  := \\sum_{k=\
1}^{i} l^{k-1} x_k", count := [ 1, 7, 0, 7 ], mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 7 ],
                                          mathmode := "Text", name := "M", 
                                          next := 25890, root := ~, 
                                          start := 25849, stop := 25889 ), 
                                      rec( content := " we\nhave ", 
                                          count := [ 1, 7, 0, 7 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  content := "y_i A  = v +  l^i\
 v_i", count := [ 1, 7, 0, 7 ], mathmode := "Text", name := "PCDATA", 
                                                  root := ~ ) ], 
                                          count := [ 1, 7, 0, 7 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 25927, root := ~, 
                                          start := 25899, stop := 25926 ), 
                                      rec( content := " for all  ", 
                                          count := [ 1, 7, 0, 7 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "i \\geq 0", 
                                                  count := [ 1, 7, 0, 7 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 7 ],
                                          mathmode := "Text", name := "M", 
                                          next := 25952, root := ~, 
                                          start := 25937, stop := 25951 ), 
                                      rec( 
                                          content := ".  Hence the\nsequence  \
", count := [ 1, 7, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ),
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "y_i", 
                                                  count := [ 1, 7, 0, 7 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 7 ],
                                          mathmode := "Text", name := "M", 
                                          next := 25985, root := ~, 
                                          start := 25975, stop := 25984 ), 
                                      rec( content := ",    ", 
                                          count := [ 1, 7, 0, 7 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "i \\geq    0", 
                                                  count := [ 1, 7, 0, 7 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 7 ],
                                          mathmode := "Text", name := "M", 
                                          next := 26008, root := ~, 
                                          start := 25990, stop := 26007 ), 
                                      rec( content := ",  gives  an  ", 
                                          count := [ 1, 7, 0, 7 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "l", 
                                                  count := [ 1, 7, 0, 7 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 7 ],
                                          mathmode := "Text", name := "M", 
                                          next := 26030, root := ~, 
                                          start := 26022, stop := 26029 ), 
                                      rec( 
                                          content := "-adic\napproximation to \
 the vector ", count := [ 1, 7, 0, 7 ], mathmode := "Text", name := "PCDATA", 
                                          root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "y \\in ℚ^r", 
                                                  count := [ 1, 7, 0, 7 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 7 ],
                                          mathmode := "Text", name := "M", 
                                          next := 26084, root := ~, 
                                          start := 26065, stop := 26083 ), 
                                      rec( content := "  with ", 
                                          count := [ 1, 7, 0, 7 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "y  A =\nv", 
                                                  count := [ 1, 7, 0, 7 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 7 ],
                                          mathmode := "Text", name := "M", 
                                          next := 26106, root := ~, 
                                          start := 26091, stop := 26105 ), 
                                      rec( content := ".", 
                                          count := [ 1, 7, 0, 7 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 0, 
                                          count := [ 1, 7, 0, 8 ], 
                                          mathmode := "Text", name := "P", 
                                          next := 26111, root := ~, 
                                          start := 26107, stop := 26110 ), 
                                      rec( 
                                          content := "\n\n(2) The second point\
 is  to show how  we can  get the vector  ", count := [ 1, 7, 0, 8 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "y", 
                                                  count := [ 1, 7, 0, 8 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 8 ],
                                          mathmode := "Text", name := "M", 
                                          next := 26183, root := ~, 
                                          start := 26175, stop := 26182 ), 
                                      rec( 
                                          content := "\nfrom a  sufficiently  \
good  approximation ", count := [ 1, 7, 0, 8 ], mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "y_i", 
                                                  count := [ 1, 7, 0, 8 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 8 ],
                                          mathmode := "Text", name := "M", 
                                          next := 26236, root := ~, 
                                          start := 26226, stop := 26235 ), 
                                      rec( 
                                          content := ".  Note   that the\nsequ\
ence of ", count := [ 1, 7, 0, 8 ], mathmode := "Text", name := "PCDATA", 
                                          root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "y_i", 
                                                  count := [ 1, 7, 0, 8 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 8 ],
                                          mathmode := "Text", name := "M", 
                                          next := 26277, root := ~, 
                                          start := 26267, stop := 26276 ), 
                                      rec( 
                                          content := " becomes  constant for "
                                            , count := [ 1, 7, 0, 8 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "i  \\geq i_0", 
                                                  count := [ 1, 7, 0, 8 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 8 ],
                                          mathmode := "Text", name := "M", 
                                          next := 26318, root := ~, 
                                          start := 26300, stop := 26317 ), 
                                      rec( 
                                          content := " if all\nentries   of  "
                                            , count := [ 1, 7, 0, 8 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "y", 
                                                  count := [ 1, 7, 0, 8 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 8 ],
                                          mathmode := "Text", name := "M", 
                                          next := 26348, root := ~, 
                                          start := 26340, stop := 26347 ), 
                                      rec( 
                                          content := " are  integers  of  abso\
lute  value smaller than\n", count := [ 1, 7, 0, 8 ], mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "l^{i_0} / 2", 
                                                  count := [ 1, 7, 0, 8 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 8 ],
                                          mathmode := "Text", name := "M", 
                                          next := 26415, root := ~, 
                                          start := 26397, stop := 26414 ), 
                                      rec( 
                                          content := " because of our choice o\
f representatives of residue\nclasses    modulo  ", count := [ 1, 7, 0, 8 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "l", 
                                                  count := [ 1, 7, 0, 8 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 8 ],
                                          mathmode := "Text", name := "M", 
                                          next := 26495, root := ~, 
                                          start := 26487, stop := 26494 ), 
                                      rec( 
                                          content := "    in      the interval\
     ", count := [ 1, 7, 0, 8 ], mathmode := "Text", name := "PCDATA", 
                                          root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  content := "\\left]-l/2,\nl/2\
\\right]", count := [ 1, 7, 0, 8 ], mathmode := "Text", name := "PCDATA", 
                                                  root := ~ ) ], 
                                          count := [ 1, 7, 0, 8 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 26553, root := ~, 
                                          start := 26524, stop := 26552 ), 
                                      rec( content := ".", 
                                          count := [ 1, 7, 0, 8 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 0, 
                                          count := [ 1, 7, 0, 9 ], 
                                          mathmode := "Text", name := "P", 
                                          next := 26558, root := ~, 
                                          start := 26554, stop := 26557 ), 
                                      rec( 
                                          content := "\n\nMore generally consi\
der ", count := [ 1, 7, 0, 9 ], mathmode := "Text", name := "PCDATA", 
                                          root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "a / b \\in ℚ",
                                                  count := [ 1, 7, 0, 9 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 9 ],
                                          mathmode := "Text", name := "M", 
                                          next := 26605, root := ~, 
                                          start := 26584, stop := 26604 ), 
                                      rec( content := " with ", 
                                          count := [ 1, 7, 0, 9 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "b > 0", 
                                                  count := [ 1, 7, 0, 9 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 9 ],
                                          mathmode := "Text", name := "M", 
                                          next := 26626, root := ~, 
                                          start := 26611, stop := 26625 ), 
                                      rec( content := " and\n", 
                                          count := [ 1, 7, 0, 9 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "a, b", 
                                                  count := [ 1, 7, 0, 9 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 9 ],
                                          mathmode := "Text", name := "M", 
                                          next := 26642, root := ~, 
                                          start := 26631, stop := 26641 ), 
                                      rec( 
                                          content := " coprime. Then there is \
for each ", count := [ 1, 7, 0, 9 ], mathmode := "Text", name := "PCDATA", 
                                          root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "n \\in ℕ", 
                                                  count := [ 1, 7, 0, 9 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 9 ],
                                          mathmode := "Text", name := "M", 
                                          next := 26692, root := ~, 
                                          start := 26675, stop := 26691 ), 
                                      rec( content := " which is\ncoprime to "
                                            , count := [ 1, 7, 0, 9 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "b", 
                                                  count := [ 1, 7, 0, 9 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 9 ],
                                          mathmode := "Text", name := "M", 
                                          next := 26721, root := ~, 
                                          start := 26713, stop := 26720 ), 
                                      rec( content := " a  unique ", 
                                          count := [ 1, 7, 0, 9 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "c \\in ℤ", 
                                                  count := [ 1, 7, 0, 9 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 9 ],
                                          mathmode := "Text", name := "M", 
                                          next := 26749, root := ~, 
                                          start := 26732, stop := 26748 ), 
                                      rec( content := " with ", 
                                          count := [ 1, 7, 0, 9 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  content := "-n  / 2 < c\n\\le\
q n / 2", count := [ 1, 7, 0, 9 ], mathmode := "Text", name := "PCDATA", 
                                                  root := ~ ) ], 
                                          count := [ 1, 7, 0, 9 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 26787, root := ~, 
                                          start := 26755, stop := 26786 ), 
                                      rec( content := "  and ", 
                                          count := [ 1, 7, 0, 9 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  content := "a \\equiv c b \\\
pmod{n}", count := [ 1, 7, 0, 9 ], mathmode := "Text", name := "PCDATA", 
                                                  root := ~ ) ], 
                                          count := [ 1, 7, 0, 9 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 26821, root := ~, 
                                          start := 26793, stop := 26820 ), 
                                      rec( content := ". This ", 
                                          count := [ 1, 7, 0, 9 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "c", 
                                                  count := [ 1, 7, 0, 9 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 9 ],
                                          mathmode := "Text", name := "M", 
                                          next := 26836, root := ~, 
                                          start := 26828, stop := 26835 ), 
                                      rec( 
                                          content := " can be\ncomputed via th\
e extended Euclidean  algorithm applied to ", count := [ 1, 7, 0, 9 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "b", 
                                                  count := [ 1, 7, 0, 9 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 9 ],
                                          mathmode := "Text", name := "M", 
                                          next := 26910, root := ~, 
                                          start := 26902, stop := 26909 ), 
                                      rec( content := " and\n", 
                                          count := [ 1, 7, 0, 9 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "n", 
                                                  count := [ 1, 7, 0, 9 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 0, 9 ],
                                          mathmode := "Text", name := "M", 
                                          next := 26923, root := ~, 
                                          start := 26915, stop := 26922 ), 
                                      rec( content := ".", 
                                          count := [ 1, 7, 0, 9 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 0, 
                                          count := [ 1, 7, 0, 10 ], 
                                          mathmode := "Text", name := "P", 
                                          next := 26928, root := ~, 
                                          start := 26924, stop := 26927 ), 
                                      rec( content := "\n\nNow let ", 
                                          count := [ 1, 7, 0, 10 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  content := "n,  \\alpha, \\be\
ta \\in ℕ", count := [ 1, 7, 0, 10 ], mathmode := "Text", name := "PCDATA", 
                                                  root := ~ ) ], 
                                          count := [ 1, 7, 0, 10 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 26971, root := ~, 
                                          start := 26938, stop := 26970 ), 
                                      rec( content := " with ", 
                                          count := [ 1, 7, 0, 10 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  content := "2  \\alpha \\beta\
 \\leq\nn", count := [ 1, 7, 0, 10 ], mathmode := "Text", name := "PCDATA", 
                                                  root := ~ ) ], 
                                          count := [ 1, 7, 0, 10 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 27006, root := ~, 
                                          start := 26977, stop := 27005 ), 
                                      rec( content := ". Then the map ", 
                                          count := [ 1, 7, 0, 10 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  content := "\\{a/b \\in ℚ \\
\mid -\\alpha \\leq a \\leq \\alpha,\n1 \\leq b < \\beta  \\} \\rightarrow \\l\
eft]-n/2, n/2\\right]", count := [ 1, 7, 0, 10 ], mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], 
                                          count := [ 1, 7, 0, 10 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 27134, root := ~, 
                                          start := 27021, stop := 27133 ), 
                                      rec( content := ", ", 
                                          count := [ 1, 7, 0, 10 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  content := "a/b\n\\mapsto c",
                                                  count := [ 1, 7, 0, 10 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], 
                                          count := [ 1, 7, 0, 10 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 27156, root := ~, 
                                          start := 27136, stop := 27155 ), 
                                      rec( 
                                          content := " (defined  as above) is \
injective  (since for ", count := [ 1, 7, 0, 10 ], mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "a/b", 
                                                  count := [ 1, 7, 0, 10 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], 
                                          count := [ 1, 7, 0, 10 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 27212, root := ~, 
                                          start := 27202, stop := 27211 ), 
                                      rec( content := ",\n", 
                                          count := [ 1, 7, 0, 10 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "a'/b'", 
                                                  count := [ 1, 7, 0, 10 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], 
                                          count := [ 1, 7, 0, 10 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 27226, root := ~, 
                                          start := 27214, stop := 27225 ), 
                                      rec( 
                                          content := "  in the  above  set we \
 have  ", count := [ 1, 7, 0, 10 ], mathmode := "Text", name := "PCDATA", 
                                          root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  content := "a  b' -  a'  b \\\
equiv  0\n\\pmod{n}", count := [ 1, 7, 0, 10 ], mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], 
                                          count := [ 1, 7, 0, 10 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 27297, root := ~, 
                                          start := 27257, stop := 27296 ), 
                                      rec( content := " if and only if ", 
                                          count := [ 1, 7, 0, 10 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  content := "a b' - a' b = 0",
                                                  count := [ 1, 7, 0, 10 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], 
                                          count := [ 1, 7, 0, 10 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 27335, root := ~, 
                                          start := 27313, stop := 27334 ), 
                                      rec( content := ").", 
                                          count := [ 1, 7, 0, 10 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 0, 
                                          count := [ 1, 7, 0, 11 ], 
                                          mathmode := "Text", name := "P", 
                                          next := 27341, root := ~, 
                                          start := 27337, stop := 27340 ), 
                                      rec( 
                                          content := "\n\nIn practice we  can \
use for any  ", count := [ 1, 7, 0, 11 ], mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  content := "c \\in \\left]-n/\
2, n/2\\right]", count := [ 1, 7, 0, 11 ], mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], 
                                          count := [ 1, 7, 0, 11 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 27411, root := ~, 
                                          start := 27376, stop := 27410 ), 
                                      rec( 
                                          content := " a\ncertain extended Euc\
lidean  algorithm applied to ", count := [ 1, 7, 0, 11 ], mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "n", 
                                                  count := [ 1, 7, 0, 11 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], 
                                          count := [ 1, 7, 0, 11 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 27471, root := ~, 
                                          start := 27463, stop := 27470 ), 
                                      rec( content := " and ", 
                                          count := [ 1, 7, 0, 11 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "c", 
                                                  count := [ 1, 7, 0, 11 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], 
                                          count := [ 1, 7, 0, 11 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 27484, root := ~, 
                                          start := 27476, stop := 27483 ), 
                                      rec( content := "\nto decide if ", 
                                          count := [ 1, 7, 0, 11 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "c", 
                                                  count := [ 1, 7, 0, 11 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], 
                                          count := [ 1, 7, 0, 11 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 27506, root := ~, 
                                          start := 27498, stop := 27505 ), 
                                      rec( 
                                          content := " is in the image of the \
above map and to find the\ncorresponding ", count := [ 1, 7, 0, 11 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "a/b", 
                                                  count := [ 1, 7, 0, 11 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], 
                                          count := [ 1, 7, 0, 11 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 27580, root := ~, 
                                          start := 27570, stop := 27579 ), 
                                      rec( content := " if it exists.", 
                                          count := [ 1, 7, 0, 11 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 0, 
                                          count := [ 1, 7, 0, 12 ], 
                                          mathmode := "Text", name := "P", 
                                          next := 27598, root := ~, 
                                          start := 27594, stop := 27597 ), 
                                      rec( 
                                          content := "\n\n(3) To put things to\
gether we apply (2)  to the entries of the vectors\n", 
                                          count := [ 1, 7, 0, 12 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "y_i", 
                                                  count := [ 1, 7, 0, 12 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], 
                                          count := [ 1, 7, 0, 12 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 27681, root := ~, 
                                          start := 27671, stop := 27680 ), 
                                      rec( 
                                          content := " constructed  in (1), ch\
oosing ", count := [ 1, 7, 0, 12 ], mathmode := "Text", name := "PCDATA", 
                                          root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "n   = l^i", 
                                                  count := [ 1, 7, 0, 12 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], 
                                          count := [ 1, 7, 0, 12 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 27728, root := ~, 
                                          start := 27712, stop := 27727 ), 
                                      rec( content := ", ", 
                                          count := [ 1, 7, 0, 12 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  content := "\\alpha =\n\\sqrt\
{n}/2", count := [ 1, 7, 0, 12 ], mathmode := "Text", name := "PCDATA", 
                                                  root := ~ ) ], 
                                          count := [ 1, 7, 0, 12 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 27756, root := ~, 
                                          start := 27730, stop := 27755 ), 
                                      rec( content := " and ", 
                                          count := [ 1, 7, 0, 12 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  content := "\\beta = \\sqrt{n\
}", count := [ 1, 7, 0, 12 ], mathmode := "Text", name := "PCDATA", root := ~ 
                                                 ) ], 
                                          count := [ 1, 7, 0, 12 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 27784, root := ~, 
                                          start := 27761, stop := 27783 ), 
                                      rec( 
                                          content := ". If  we have found this\
 way\na candidate  for  ", count := [ 1, 7, 0, 12 ], mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "y", 
                                                  count := [ 1, 7, 0, 12 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], 
                                          count := [ 1, 7, 0, 12 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 27839, root := ~, 
                                          start := 27831, stop := 27838 ), 
                                      rec( 
                                          content := " we can  easily  check i\
f it  is correct  by\ncomputing ", count := [ 1, 7, 0, 12 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "y A", 
                                                  count := [ 1, 7, 0, 12 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], 
                                          count := [ 1, 7, 0, 12 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 27904, root := ~, 
                                          start := 27894, stop := 27903 ), 
                                      rec( content := ".  If ", 
                                          count := [ 1, 7, 0, 12 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "\\mu", 
                                                  count := [ 1, 7, 0, 12 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], 
                                          count := [ 1, 7, 0, 12 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 27920, root := ~, 
                                          start := 27910, stop := 27919 ), 
                                      rec( 
                                          content := " is the  maximal absolut\
e value of\nall numerators and denominators of the entries of ", 
                                          count := [ 1, 7, 0, 12 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "y", 
                                                  count := [ 1, 7, 0, 12 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], 
                                          count := [ 1, 7, 0, 12 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 28013, root := ~, 
                                          start := 28005, stop := 28012 ), 
                                      rec( 
                                          content := " it is clear\nfrom (2) t\
hat we will find  ", count := [ 1, 7, 0, 12 ], mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "y", 
                                                  count := [ 1, 7, 0, 12 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], 
                                          count := [ 1, 7, 0, 12 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 28062, root := ~, 
                                          start := 28054, stop := 28061 ), 
                                      rec( content := " from ", 
                                          count := [ 1, 7, 0, 12 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "y_i", 
                                                  count := [ 1, 7, 0, 12 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], 
                                          count := [ 1, 7, 0, 12 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 28078, root := ~, 
                                          start := 28068, stop := 28077 ), 
                                      rec( content := "  if ", 
                                          count := [ 1, 7, 0, 12 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( 
                                                  content := "l^i > 2\n\\mu^2",
                                                  count := [ 1, 7, 0, 12 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], 
                                          count := [ 1, 7, 0, 12 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 28106, root := ~, 
                                          start := 28083, stop := 28105 ), 
                                      rec( content := ".", 
                                          count := [ 1, 7, 0, 12 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 0, 
                                          count := [ 1, 7, 0, 13 ], 
                                          mathmode := "Text", name := "P", 
                                          next := 28111, root := ~, 
                                          start := 28107, stop := 28110 ), 
                                      rec( content := "\n\n(4) If we take as "
                                            , count := [ 1, 7, 0, 13 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "v", 
                                                  count := [ 1, 7, 0, 13 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], 
                                          count := [ 1, 7, 0, 13 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 28139, root := ~, 
                                          start := 28131, stop := 28138 ), 
                                      rec( 
                                          content := "  in (1) to(3)  all stan\
dard unit vectors we\nclearly get  the   rows  of  ", 
                                          count := [ 1, 7, 0, 13 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "A^{-1}", 
                                                  count := [ 1, 7, 0, 13 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], 
                                          count := [ 1, 7, 0, 13 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 28226, root := ~, 
                                          start := 28213, stop := 28225 ), 
                                      rec( 
                                          content := ".    But   we  can  do  \
  it\nbetter.  Namely we  can take   as ", count := [ 1, 7, 0, 13 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "v", 
                                                  count := [ 1, 7, 0, 13 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], 
                                          count := [ 1, 7, 0, 13 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 28297, root := ~, 
                                          start := 28289, stop := 28296 ), 
                                      rec( 
                                          content := "  the standard unit  vec\
tors\nmultiplied   by the  least  common   multiple  ", 
                                          count := [ 1, 7, 0, 13 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "\\epsilon", 
                                                  count := [ 1, 7, 0, 13 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], 
                                          count := [ 1, 7, 0, 13 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 28388, root := ~, 
                                          start := 28373, stop := 28387 ), 
                                      rec( 
                                          content := " of  the\ndenominators  \
of the already   computed entries of ", count := [ 1, 7, 0, 13 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "A^{-1}", 
                                                  count := [ 1, 7, 0, 13 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], 
                                          count := [ 1, 7, 0, 13 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 28461, root := ~, 
                                          start := 28448, stop := 28460 ), 
                                      rec( 
                                          content := ".   In\nmany examples  t\
his ", count := [ 1, 7, 0, 13 ], mathmode := "Text", name := "PCDATA", 
                                          root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "\\epsilon", 
                                                  count := [ 1, 7, 0, 13 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], 
                                          count := [ 1, 7, 0, 13 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 28503, root := ~, 
                                          start := 28488, stop := 28502 ), 
                                      rec( content := "  actually equals ", 
                                          count := [ 1, 7, 0, 13 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "\\epsilon_r", 
                                                  count := [ 1, 7, 0, 13 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], 
                                          count := [ 1, 7, 0, 13 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 28538, root := ~, 
                                          start := 28521, stop := 28537 ), 
                                      rec( 
                                          content := "\nafter  the computation\
 of  the first or  first few rows.  Therefore we\nwill often find quickly the \
next row of  ", count := [ 1, 7, 0, 13 ], mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "A^{-1}", 
                                                  count := [ 1, 7, 0, 13 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], 
                                          count := [ 1, 7, 0, 13 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 28664, root := ~, 
                                          start := 28651, stop := 28663 ), 
                                      rec( 
                                          content := " already in (1),\nbecaus\
e we find a ", count := [ 1, 7, 0, 13 ], mathmode := "Text", name := "PCDATA",
                                          root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "v_i = 0", 
                                                  count := [ 1, 7, 0, 13 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], 
                                          count := [ 1, 7, 0, 13 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 28713, root := ~, 
                                          start := 28699, stop := 28712 ), 
                                      rec( 
                                          content := " such  that the sequence\
 of ", count := [ 1, 7, 0, 13 ], mathmode := "Text", name := "PCDATA", 
                                          root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "y_i", 
                                                  count := [ 1, 7, 0, 13 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], 
                                          count := [ 1, 7, 0, 13 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 28751, root := ~, 
                                          start := 28741, stop := 28750 ), 
                                      rec( content := "\nbecomes constant (", 
                                          count := [ 1, 7, 0, 13 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 
                                            [ 
                                              rec( content := "=y", 
                                                  count := [ 1, 7, 0, 13 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], 
                                          count := [ 1, 7, 0, 13 ], 
                                          mathmode := "Text", name := "M", 
                                          next := 28779, root := ~, 
                                          start := 28770, stop := 28778 ), 
                                      rec( content := ").", 
                                          count := [ 1, 7, 0, 13 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( attributes := rec(  ), 
                                          content := 0, 
                                          count := [ 1, 7, 0, 14 ], 
                                          mathmode := "Text", name := "P", 
                                          next := 28785, root := ~, 
                                          start := 28781, stop := 28784 ), 
                                      rec( content := "\n\n", 
                                          count := [ 1, 7, 0, 14 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ), 
                                      rec( 
                                          attributes := 
                                            rec( Label := "Ssect-rankintmat" )
                                            , 
                                          content := 
                                            [ 
                                              rec( content := "\n", 
                                                  count := [ 1, 7, 1, 1 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ), 
                                              rec( attributes := rec(  ), 
                                                  content := 
                                                    [ 
                                                      rec( 
                                                          content := "Rank of I\
nteger Matrix", count := [ 1, 7, 1, 2 ], mathmode := "Text", name := "PCDATA",
                                                          root := ~ ) ], 
                                                  count := [ 1, 7, 1, 2 ], 
                                                  mathmode := "Text", 
                                                  name := "Heading", 
                                                  next := 28866, root := ~, 
                                                  start := 28825, 
                                                  stop := 28865 ), 
                                              rec( 
                                                  content := "\n\nThe followin\
g strategy has shown to  be useful in proving that some very\nbig integer matr\
ix is not invertible.", count := [ 1, 7, 1, 3 ], mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ), 
                                              rec( attributes := rec(  ), 
                                                  content := 0, 
                                                  count := [ 1, 7, 1, 4 ], 
                                                  mathmode := "Text", 
                                                  name := "P", next := 28982, 
                                                  root := ~, start := 28978, 
                                                  stop := 28981 ), 
                                              rec( content := "\n\n", 
                                                  count := [ 1, 7, 1, 4 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ), 
                                              rec( attributes := rec(  ), 
                                                  content := 
                                                    [ 
                                                      rec( 
                                                          attributes := rec(  )
                                                            , 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "C\
heck  the   rank  modulo  some   small  primes,  say   with  ", count := 
                                                                    [ 1, 7, 
                                                                      1, 6 ], 
                                                                  mathmode := \
"Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "RankMo\
d" ), content := 0, count := [ 1, 7, 1, 6 ], mathmode := "Text", name := "Ref"
                                                                    , 
                                                                  next := 2908\
0, root := ~, start := 29059, stop := 29079 ), rec( content := ". ", 
                                                                  count := 
                                                                    [ 1, 7, 
                                                                      1, 6 ], 
                                                                  mathmode := \
"Text", name := "PCDATA", root := ~ ) ], count := [ 1, 7, 1, 6 ], 
                                                          name := "Item", 
                                                          next := 29089, 
                                                          root := ~, 
                                                          start := 28991, 
                                                          stop := 29088 ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "I\
f the  rank seems  less than  the number of  rows choose  a prime\n", 
                                                                  count := 
                                                                    [ 1, 7, 
                                                                      1, 8 ], 
                                                                  mathmode := \
"Text", name := "PCDATA", root := ~ ), rec( attributes := rec(  ), 
                                                                  content := 
                                                                    [ rec( 
                                                                          cont\
ent := "p", count := [ 1, 7, 1, 8 ], mathmode := "Text", name := "PCDATA", 
                                                                          root\
 := ~ ) ], count := [ 1, 7, 1, 8 ], mathmode := "Text", name := "M", 
                                                                  next := 2917\
1, root := ~, start := 29163, stop := 29170 ), rec( content := ", a  collectio\
n of  lines which  is linearly  independent modulo\n", count := [ 1, 7, 1, 8 ]
                                                                    , 
                                                                  mathmode := \
"Text", name := "PCDATA", root := ~ ), rec( attributes := rec(  ), 
                                                                  content := 
                                                                    [ rec( 
                                                                          cont\
ent := "p", count := [ 1, 7, 1, 8 ], mathmode := "Text", name := "PCDATA", 
                                                                          root\
 := ~ ) ], count := [ 1, 7, 1, 8 ], mathmode := "Text", name := "M", 
                                                                  next := 2924\
4, root := ~, start := 29236, stop := 29243 ), rec( content := ", and another \
line linearly  dependend on these. Guess that this\nlast line is  also linearl\
y dependend on the chosen  collection over the\nrational numbers (maybe check \
modulo several small primes).", count := [ 1, 7, 1, 8 ], mathmode := "Text", 
                                                                  name := "PCD\
ATA", root := ~ ) ], count := [ 1, 7, 1, 8 ], name := "Item", next := 29448, 
                                                          root := ~, 
                                                          start := 29090, 
                                                          stop := 29447 ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "F\
ind columns of  the collection of lines which  give an invertible\nmatrix modu\
lo some prime.", count := [ 1, 7, 1, 10 ], mathmode := "Text", name := "PCDATA\
", root := ~ ) ], count := [ 1, 7, 1, 10 ], name := "Item", next := 29554, 
                                                          root := ~, 
                                                          start := 29449, 
                                                          stop := 29553 ), 
                                                      rec( 
                                                          attributes := 
                                                            rec(  ), 
                                                          content := 
                                                            [ 
                                                              rec( 
                                                                  content := "T\
hen use ", count := [ 1, 7, 1, 12 ], mathmode := "Text", name := "PCDATA", 
                                                                  root := ~ ),
                                                              rec( 
                                                                  attributes :\
= rec( Func := "RationalSolutionIntMat" ), content := 0, count := [ 1, 7, 1, 
                                                                      12 ], 
                                                                  mathmode := \
"Text", name := "Ref", next := 29607, root := ~, start := 29570, stop := 29606\
 ), rec( content := " with the invertible\nsubmatrix and  corresponding entrie\
s  of the  linearly dependend  row to\nprove this.", count := [ 1, 7, 1, 12 ],
                                                                  mathmode := \
"Text", name := "PCDATA", root := ~ ) ], count := [ 1, 7, 1, 12 ], 
                                                          name := "Item", 
                                                          next := 29719, 
                                                          root := ~, 
                                                          start := 29555, 
                                                          stop := 29718 ) ], 
                                                  count := [ 1, 7, 1, 5 ], 
                                                  mathmode := "Text", 
                                                  name := "List", 
                                                  next := 29727, root := ~, 
                                                  start := 28984, 
                                                  stop := 29726 ), 
                                              rec( 
                                                  content := "\n\nGuessing  th\
e rank  of a  matrix from  the rank  modulo several  primes,\nchosing a  maxim\
al set  of lines which  are linearly  independent modulo\nsome  primes, and  u\
sing ", count := [ 1, 7, 1, 14 ], mathmode := "Text", name := "PCDATA", 
                                                  root := ~ ), 
                                              rec( 
                                                  attributes := 
                                                    rec( 
                                                      Func := "RationalSolutio\
nIntMat" ), content := 0, count := [ 1, 7, 1, 14 ], mathmode := "Text", 
                                                  name := "Ref", 
                                                  next := 29937, root := ~, 
                                                  start := 29900, 
                                                  stop := 29936 ), 
                                              rec( 
                                                  content := " with  the\nrema\
ining lines,  one may  also find  the exact rank  of a  huge integer\nmatrix. \
", count := [ 1, 7, 1, 14 ], mathmode := "Text", name := "PCDATA", root := ~ )
                                                , 
                                              rec( attributes := rec(  ), 
                                                  content := 0, 
                                                  count := [ 1, 7, 1, 15 ], 
                                                  mathmode := "Text", 
                                                  name := "P", next := 30033, 
                                                  root := ~, start := 30029, 
                                                  stop := 30032 ), 
                                              rec( content := "\n\n", 
                                                  count := [ 1, 7, 1, 15 ], 
                                                  mathmode := "Text", 
                                                  name := "PCDATA", root := ~ 
                                                 ) ], count := [ 1, 7, 1, 1 ],
                                          mathmode := "Text", 
                                          name := "Subsection", 
                                          next := 30048, root := ~, 
                                          start := 28787, stop := 30047 ), 
                                      rec( content := "\n\n", 
                                          count := [ 1, 7, 1, 15 ], 
                                          mathmode := "Text", 
                                          name := "PCDATA", root := ~ ) ], 
                                  count := [ 1, 7, 0, 1 ], mathmode := "Text",
                                  name := "Section", next := 30060, 
                                  root := ~, start := 24391, stop := 30059 ), 
                              rec( content := "\n\n", 
                                  count := [ 1, 7, 1, 15 ], 
                                  mathmode := "Text", name := "PCDATA", 
                                  root := ~ ) ], count := [ 1, 0, 0, 1 ], 
                          mathmode := "Text", name := "Chapter", 
                          next := 30072, root := ~, start := 1605, 
                          stop := 30071 ), 
                      rec( content := "\n\n\n", count := [ 1, 7, 1, 15 ], 
                          mathmode := "Text", name := "PCDATA", root := ~ ) ],
                  count := [ 0, 0, 2, 1 ], mathmode := "Text", 
                  name := "Body", next := 30082, root := ~, start := 1391, 
                  stop := 30081 ), 
              rec( content := "  We want a bibliography  ", 
                  count := [ 1, 7, 1, 15 ], mathmode := "Text", 
                  name := "XMLCOMMENT", root := ~ ), 
              rec( attributes := rec( Databases := "edim" ), content := 0, 
                  count := [ "Bib", 0, 0, 1 ], mathmode := "Text", 
                  name := "Bibliography", next := 30150, root := ~, 
                  start := 30118, stop := 30149 ), 
              rec( content := "  And finally an index as well   ", 
                  count := [ "Bib", 0, 0, 1 ], mathmode := "Text", 
                  name := "XMLCOMMENT", root := ~ ), 
              rec( attributes := rec(  ), content := 0, 
                  count := [ "Ind", 0, 0, 1 ], mathmode := "Text", 
                  name := "TheIndex", next := 30204, root := ~, 
                  start := 30193, stop := 30203 ) ], count := [ 0, 0, 0, 1 ], 
          name := "Book", next := 30213, root := ~, start := 234, 
          stop := 30212 ) ], count := [ 0, 0, 0, 1 ], indent := "", 
  indextext := "<strong class=\"pkg\">EDIM</strong>  <a href=\"chap1_mj.html#X\
7AA826067CC8C395\">1.</a>  <br />\n<code class=\"func\">CheapFactorsInt</code>\
  <a href=\"chap1_mj.html#X7BAB977C7EB05067\">1.6-4</a>  <br />\n<code class=\
\"func\">ElementaryDivisorsIntMatDeterminant</code>  <a href=\"chap1_mj.html#X\
821E30477A5DCE68\">1.4-2</a>  <br />\n<code class=\"func\">ElementaryDivisorsP\
PartHavasSterling</code>  <a href=\"chap1_mj.html#X82EC4724865F4DF9\">1.2-2</a\
>  <br />\n<code class=\"func\">ElementaryDivisorsPPartRk</code>  <a href=\"ch\
ap1_mj.html#X813B0D73868CD751\">1.2-1</a>  <br />\n<code class=\"func\">Elemen\
taryDivisorsPPartRkExp</code>  <a href=\"chap1_mj.html#X813B0D73868CD751\">1.2\
-1</a>  <br />\n<code class=\"func\">ElementaryDivisorsPPartRkExpSmall</code> \
 <a href=\"chap1_mj.html#X813B0D73868CD751\">1.2-1</a>  <br />\n<code class=\"\
func\">ElementaryDivisorsPPartRkI</code>  <a href=\"chap1_mj.html#X813B0D73868\
CD751\">1.2-1</a>  <br />\n<code class=\"func\">ElementaryDivisorsPPartRkII</c\
ode>  <a href=\"chap1_mj.html#X813B0D73868CD751\">1.2-1</a>  <br />\n<code cla\
ss=\"func\">ElementaryDivisorsSquareIntMatFullRank</code>  <a href=\"chap1_mj.\
html#X7B6A3B8486872B0C\">1.4-1</a>  <br />\n<code class=\"func\">ExponentSquar\
eIntMatFullRank</code>  <a href=\"chap1_mj.html#X86DA61D978D2D889\">1.3-3</a> \
 <br />\n<code class=\"func\">GcdexIntLLL</code>  <a href=\"chap1_mj.html#X799\
B1A5285D00859\">1.5-1</a>  <br />\n<code class=\"func\">HadamardBoundIntMat</c\
ode>  <a href=\"chap1_mj.html#X7CDF3D2081207080\">1.6-3</a>  <br />\n<code cla\
ss=\"func\">HermiteIntMatLLL</code>  <a href=\"chap1_mj.html#X7C6AE6777B72F9D2\
\">1.5-2</a>  <br />\n<code class=\"func\">HermiteIntMatLLLTrans</code>  <a hr\
ef=\"chap1_mj.html#X862962B7878A284F\">1.5-3</a>  <br />\n<code class=\"func\"\
>InfoEDIM</code>  <a href=\"chap1_mj.html#X7AF83FEC7CD0311C\">1.1-1</a>  <br /\
>\n<code class=\"func\">InverseIntMatMod</code>  <a href=\"chap1_mj.html#X7851\
8E1D81435762\">1.6-2</a>  <br />\n<code class=\"func\">InverseRatMat</code>  <\
a href=\"chap1_mj.html#X7A9656D47C4D2D16\">1.3-1</a>  <br />\nLicense  <a href\
=\"chap0_mj.html#X81488B807F2A1CF1\">.-1</a>  <br />\n<code class=\"func\">Ran\
kMod</code>  <a href=\"chap1_mj.html#X8312EDA78209B4EA\">1.6-5</a>  <br />\n<c\
ode class=\"func\">RationalSolutionIntMat</code>  <a href=\"chap1_mj.html#X830\
2B31E86B3AFDB\">1.3-2</a>  <br />\n<code class=\"func\">RatNumberFromModular</\
code>  <a href=\"chap1_mj.html#X7EC844F97C469C03\">1.6-1</a>  <br />\n<code cl\
ass=\"func\">SmithIntMatLLL</code>  <a href=\"chap1_mj.html#X8626F15179C09798\
\">1.5-4</a>  <br />\n<code class=\"func\">SmithIntMatLLLTrans</code>  <a href\
=\"chap1_mj.html#X86094B1B7C87EBF6\">1.5-5</a>  <br />\n", 
  input := "<?xml version=\"1.0\" encoding=\"UTF-8\"?>\n\n<!DOCTYPE Book SYSTE\
M \"gapdoc.dtd\">\n\n<!--\n\n  manual.xml          EDIM mini-package documenta\
tion        Frank Lübeck\n\n  Copyright (C) 2000-2018, Lehrstuhl D für Mathe\
matik, RWTH-Aachen\n\n-->\n\n<Book Name=\"edim\">\n\n<TitlePage>\n    <Title><\
Package>EDIM</Package></Title>\n    <Subtitle>Elementary Divisors and Integer \
Matrices</Subtitle>\n    <Version>\n      Version 1.3.3\n\n    </Version>\n   \
 <Author>Frank Lübeck\n<Address>\nLehrstuhl D für Mathematik\nRWTH Aachen\nP\
ontdriesch 14/16\n52062 Aachen\nGermany\n</Address>\n    <Email>Frank.Luebeck@\
Math.RWTH-Aachen.De</Email>\n    <Homepage>http://www.math.rwth-aachen.de/&til\
de;Frank.Luebeck</Homepage>\n    </Author>\n    <Date>January 2018</Date>\n   \
 <Copyright> <Index>License</Index>\n              &copyright; 2000-2018 by Fr\
ank Lübeck <P/>\n              <Package>EDIM</Package>  \n              is fr\
ee software; you can redistribute it\n              and/or modify it under the\
 terms of the <URL Text=\"GNU\n              General Public License\">http://w\
ww.fsf.org/licenses/gpl.html</URL>              as published by the Free Softw\
are Foundation;\n              either version 2 of the License, or (at your op\
tion) any\n              later version.\n    </Copyright>\n\n</TitlePage>\n\n<\
!--  TableOfContents  . . . . . .  generate a table of contents   -->\n<TableO\
fContents/>\n\n<!--  The only chapter of the book  --> \n<Body>\n<!-- \n\n  ma\
in.xml            EDIM package documentation        Frank Lübeck\n\n         \
Copyright (C) 1999-2004, Lehrstuhl D für Mathematik, RWTH-Aachen\n\nThis is t\
he main chapter of the documentation.\n\n-->\n\n<Chapter Label=\"Chap-EDIM\">\
\n<Heading>The <Package>EDIM</Package>-Package</Heading>\n<Index><Package>EDIM\
</Package></Index>\n\n<Emph>(Elementary   Divisors   and   Integer    Matrices\
,  by    Frank\nLübeck)</Emph><P/>\n\nThis  chapter describes  the  functions\
 defined  in  the &GAP;4  package\n<Package>EDIM</Package>.  The main  functio\
ns implement  variants of  an\nalgorithm for computing for a given prime <M>p<\
/M> the <M>p</M>-parts of\nthe  elementary divisors  of  an integer  matrix.  \
These algorithms  use\na  <M>p</M>-adic  method  and  are  described by  the  \
author  in  <Cite\nKey=\"L98\"/> (see <Ref Func=\"ElementaryDivisorsPPartRk\"/\
>).<P/>\n\nThese functions were already applied to  integer matrices of dimens\
ion\ngreater  than  <M>11000</M> (which had  many   non-trivial  elementary\nd\
ivisors which were products of small primes).<P/>\n\nFurthermore  there are  f\
unctions  for finding  the biggest elementary\ndivisor of an invertible intege\
r matrix and  the inverse of a rational\ninvertible matrix  (see <Ref Func=\"E\
xponentSquareIntMatFullRank\"/> and\n<Ref  Func=\"InverseRatMat\"/>).   These \
  algorithms use  <M>p</M>-adic\napproximations, explained in <Ref Sect=\"Sect\
-InvRatMatAlg\"/>.<P/>\n\nFinally we distribute  implementations  of some othe\
r   algorithms for\nfinding elementary  divisors or normal  forms  of integer \
 matrices: A\n<M>p</M>-modular  algorithm  by    Havas  and   Sterling from   \
 <Cite\nKey=\"HS79\"/>  (see <Ref Func=\"ElementaryDivisorsPPartHavasSterling\
\"/>)\nand LLL-based  algorithms for extended   greatest  common divisors  of\
\nintegers (see <Ref Func=\"GcdexIntLLL\"/>) and  for Hermite normal forms\nof\
 integer matrices with  (very nice) transforming matrices  (see <Ref\nFunc=\"H\
ermiteIntMatLLL\"/>).<P/>\n\nBy   default  the   <Package>EDIM</Package>   is \
 automatically   loaded\nby   &GAP;   when   it   is  installed.   If   the   \
automatic   loading\nis   disabled  in   your  installation   you  must   load\
  the   package\nwith   <C>RequirePackage(\"edim\");</C>   before  its   funct\
ions   become\navailable.<P/>\n\nPlease,            send             me       \
     an            e-mail\n(<Email>Frank.Luebeck@Math.RWTH-Aachen.De</Email>) \
 if  you  have  any\nquestions,  remarks, suggestions,  etc. concerning  this \
mini-package.\nAlso, I would like to hear about applications of this package.<\
P/>\n\nFrank Lübeck \n\n<Section Label=\"Sect-Install\">\n<Heading>Installati\
on of the <Package>EDIM</Package> package</Heading>\n\nTo install this package\
 first unpack it inside some GAP root directory\ninto the subdirectory <F>pkg/\
edim</F> (see \n<Ref Sect=\"Installing a GAP Package\" BookName=\"Reference\"/\
>). Then\nthe <Package>EDIM</Package> package can already be loaded and used.\
\nBut we strongly recommend to compile a kernel function as well during\ninsta\
llation, otherwise \nthe function    <Ref Func=\"ElementaryDivisorsPPartRkExpS\
mall\"/> \nwill not be available.<P/>\n\nTo install the kernel function  go to\
  the  directory  <F>pkg/edim</F> to\nwhich the package was extracted and call\
<P/>\n\n<C>/bin/sh ./configure [path] [CONFIGNAME=...]</C><P/>\n\nwhere <C>pat\
h</C>  is a path  to  the main <Package>GAP</Package> root\ndirectory (if not \
given, the default <F>../..</F> is assumed). \nIf you have installed several G\
AP kernels you can compile the corresponding\n<Package>EDIM</Package> kernel f\
unction by specifying the <C>CONFIGNAME</C>\nthat was used to configure that k\
ernel.<P/>\n\nAfterwards call <C>make</C>  to  compile  a binary file. <P/>\n\
\nIf you have installed several GAP kernels repeat these two steps for each \n\
of them, using the various values of <C>CONFIGNAME</C>.\n\nYou  can   run  a t\
est of    the installation  by  typing <C>make test</C>.\n\n<ManSection> \n\n<\
InfoClass Name=\"InfoEDIM\"/> \n\n<Description>    \nThis is  an <K>Info</K> c\
lass for the <Package>EDIM</Package>-package.\nBy <C>SetInfoLevel(InfoEDIM, 1)\
;</C> you can switch on the printing of\nsome information     during      the \
   computations     of    certain\n<Package>EDIM</Package>-functions.  \n</Des\
cription>\n</ManSection>\n\n</Section>\n\n<!-- ###############################\
############################# -->\n<Section Label=\"Sect-PPElDiv\">\n<Heading>\
<M>p</M>-Parts of Elementary Divisors</Heading>\n\nHere we explain the main fu\
nctions of the package.\n\n<ManSection>\n<Func Name=\"ElementaryDivisorsPPartR\
k\"\n      Arg=\"A, p[, rk ]\"/>\n<Func Name=\"ElementaryDivisorsPPartRkI\"\n \
     Arg=\"A, p, rk\"/>\n<Func Name=\"ElementaryDivisorsPPartRkII\"\n      Arg\
=\"A, p, rk\"/>\n<Func Name=\"ElementaryDivisorsPPartRkExp\"\n      Arg=\"A, p\
, rk, exp\"/>\n<Func Name=\"ElementaryDivisorsPPartRkExpSmall\"\n      Arg=\"A\
, p, rk, exp, il\"/>\n\n<Description>  These functions  return  a list  <M>[m_\
1, m_2,  \\ldots,\nm_r]</M> where <M>m_i</M> is the number of nonzero elementa\
ry divisors\nof <A>A</A>  divisible by  <M><A>p</A>^i</M> (see  <Ref BookName=\
\"ref\"\nFunc=\"ElementaryDivisorsMat\"  /> for  a definition  of the  element\
ary\ndivisors).<P/>\n\nThe    algorithms   for   these  functions  are     des\
cribed in <Cite\nKey=\"L98\"/>.<P/>\n\n<A>A</A> must be a matrix with integer \
entries,  <A>p</A> a prime, and\n<A>rk</A> the  rank of  <A>A</A> (as rational\
  matrix).  In  the first\nversion of the command <A>rk</A> is computed, if it\
 is not given. <P/>\n\nThe first   version of the  command delegates  its  job\
  to the fourth\nversion by trying growing values for <A>exp</A>, see below. <\
P/>\n\nThe second  and third versions implement  the main algorithm described\
\nin   <Cite     Key=\"L98\"/>    and   a    variation.      Here     <Ref\nFu\
nc=\"ElementaryDivisorsPPartRkII\"/> has a bit  more overhead, but can\nbe   a\
dvantageous    because  the  intermediate    entries  during the\ncomputation \
can be much smaller.<P/>\n\nIn the fourth form <A>exp</A> must be an  upper bo\
und for the  highest\npower of <A>p</A>  appearing  in an  elementary divisor \
 of  <A>A</A>.\nThis  information  allows    reduction   of  matrix   entries \
  modulo\n<M><A>p</A>^{<A>exp</A>}</M> during the computation. <P/>\n\nIf <A>e\
xp</A> is too  small or the given <A>rk</A>  is not correct the\nfunction retu\
rns `fail'. <P/>\n\nAs long as <M><A>p</A>^{<A>exp</A>}</M>  is smaller than <\
M>2^{28}</M>\nand <M><A>p</A>^{<A>exp</A> +  2}</M> is smaller than <M>2^{31}<\
/M> we\nuse  internally a kernel  function which can also  be used directly in\
\nthe  fifth form  of the command.   There <A>il</A>  can be <M>0</M> or\n<M>1\
</M> where in the  second case some  information is printed during\nthe comput\
ation.<P/>\n\nThis last form  of the  function was  already  succesfully appli\
ed  to\ndense matrices of rank up to <M>11000</M>.<P/>\n\nNote that you have t\
o compile a  file (see <Ref Sect=\"Sect-Install\"/>)\nwhile  installing this  \
package, if   you  want  to  have this  kernel\nfunction available.<P/> \n\n<E\
xample>\ngap> ReadPackage(\"edim\",  \"tst/mat\");\nReading 242x242 integer ma\
trix 'mat' with elementary divisors 'eldiv'.\ntrue\ngap> ElementaryDivisorsPPa\
rtRkI(mat, 2, 242); time; # mat has full rank\n[ 94, 78, 69, 57, 23, 23, 9, 2,\
 2, 0 ]\n490\ngap> ElementaryDivisorsPPartRkExpSmall(mat, 2, 242, 10, 0); time\
;\n[ 94, 78, 69, 57, 23, 23, 9, 2, 2, 0 ]\n10\n</Example>\n</Description>\n</M\
anSection>\n\n<!-- ###########################################################\
# -->\n<ManSection>\n<Func Name=\"ElementaryDivisorsPPartHavasSterling\"\n    \
  Arg=\"A, p, d\"\n      Comm=\"another modular algorithm\" />\n\n<Description\
> \nFor an integer  matrix  <A>A</A> and a  prime <A>p</A>   this function\nre\
turns a list <M>[m_1, m_2,  \\ldots, m_r]</M> where <M>m_i</M> is the\nnumber \
  of nonzero  elementary   divisors of   <A>A</A>  divisible by\n<M><A>p</A>^i\
</M>. <P/>\n\nAn upper bound <A>d</A> for the highest power of <A>p</A> appear\
ing in\nan  elementary  divisor of  <A>A</A> must be   given. Smaller <A>d</A>\
\nimprove the performance of the algorithm considerably.<P/>\n\nThis is an imp\
lementation of  the modular algorithm described in <Cite\nKey=\"HS79\"/>. <P/>\
\n\nWe added a slight improvement: we divide the considered submatrices by\nth\
e <A>p</A>-part  of the greatest common divisor  of all entries (and\nlower th\
e  <A>d</A> appropriately).  This  reduces  the size  of   the\nentries and of\
ten shortens the pivot search.<P/>\n\n<Example>\ngap> ReadPackage(\"edim\",  \
\"tst/mat\");\nReading 242x242 integer matrix 'mat' with elementary divisors '\
eldiv'.\ntrue\ngap> ElementaryDivisorsPPartHavasSterling(mat, 2, 10); time;\n[\
 94, 78, 69, 57, 23, 23, 9, 2, 2 ]\n1260\n</Example>\n</Description>\n</ManSec\
tion>\n</Section>\n\n<!-- ####################################################\
######## -->\n\n<Section Label=\"Sect-InvRatMat\">\n<Heading>Inverse of Ration\
al Matrices</Heading>\n\n<ManSection>\n<Func Name=\"InverseRatMat\"\n      Arg\
=\"A[, p ]\"\n      Comm=\"inverse of a rational matrix\"/>\n\n<Description>  \
\nThis  function returns  the inverse  of  an invertible matrix over the\nrati\
onal numbers.<P/>\n\nIt first  computes the  inverse  modulo some  prime <A>p<\
/A>, computes\nfrom this  a  <A>p</A>-adic approximation  to  the inverse and \
finally\nconstructs  the     rational    entries from      their  <A>p</A>-adi\
c\napproximations. See  section  <Ref Sect=\"Sect-InvRatMatAlg\"/>  for more\n\
details.<P/>\n\nThis    seems    to    be   better    than    <Package>GAP</Pa\
ckage>'s\nstandard    Gauß   algorithm    (<C>A^-1</C>)   already    for   sm\
all\nmatrices.   (Try,  e.g.,   <C>RandomMat(20,20,[-10000..10000])</C>  or\n<\
C>RandomMat(100,100)</C>.)<P/>\n\nThe  optional argument <A>p</A> should  be  \
a prime such that <A>A</A>\nmodulo  <A>p</A> is  invertible  (default is  <M><\
A>p</A>=251</M>). If\n<A>A</A>   is   not  invertible   modulo   <A>p</A>  the\
n <A>p</A>  is\nautomatically replaced by the next prime.<P/>\n</Description>\
\n</ManSection>\n\n<!-- ######################################################\
###### -->\n<ManSection>\n<Func Name=\"RationalSolutionIntMat\"\n      Arg=\"A\
, v[, p[, invA ] ]\"\n      Comm=\"rational solution of linear system over int\
egers\"/>\n\n<Description>  \nThis  function returns the solution <M>x</M> of \
the system of linear\nequations <M>x <A>A</A> = <A>v</A></M>.<P/>\n\nHere, <A>\
A</A> must be a matrix with integer entries which is invertible\nover the rati\
onals and <A>v</A> must be a vector with integer entries of\nthe appropriate l\
ength.<P/>\n\nThe  optional arguments  are a  prime <A>p</A>  such that  <M><A\
>A</A>\n\\pmod{p}</M> is invertible  (if not given, <M>p =  251</M> is assumed\
)\nand the inverse <A>invA</A> of <M><A>A</A> \\pmod{p}</M>.<P/>\n\nThe soluti\
on is computed via <M>p</M>-adic approximation as explained in \n<Ref Sect=\"S\
ect-InvRatMatAlg\"/>.<P/>\n</Description>\n</ManSection>\n\n<!-- #############\
############################################### -->\n\n<ManSection>\n<Func Nam\
e=\"ExponentSquareIntMatFullRank\"\n      Arg=\" A[, p[, nr ] ]\"\n      Comm=\
\"\"/> \n\n<Description>\nThis function  returns the biggest   elementary  div\
isor of  a  square\ninteger matrix <A>A</A> of full rank.<P/>\n\nFor  such a  \
matrix <A>A</A>   the   least  common  multiple of    the\ndenominators of all\
 entries of the inverse matrix <M><A>A</A>^{-1}</M>\nis exactly the biggest el\
ementary divisor of <A>A</A>.<P/>\n  \nThis function  is  implemented  by   a \
 slight modification   of  <Ref\nFunc=\"InverseRatMat\"/>.   The  third   argu\
ment <A>nr</A>   tells  the\nfunction  to return the least common   multiple o\
f the first <A>nr</A>\nrows  of the rational  inverse matrix  only.  Very  oft\
en the function\nwill    already    return   the  biggest    elementary    div\
isor with\n<M><A>nr</A>=2</M> or <M>3</M>  (and the command without this argum\
ent\nwould spend most time in checking, that this is correct).<P/>\n\nThe opti\
onal argument  <A>p</A> should be  a prime such  that <A>A</A>\nmodulo <A>p</A\
> is  invertible  (default  is  <M><A>p</A>=251</M>). If\n<A>A</A>     is not \
 invertible   modulo   <A>p</A>  then <A>p</A>  is\nautomatically replaced by \
the next prime.<P/>\n\n<Example>\ngap> ReadPackage(\"edim\",  \"tst/mat\");\nR\
eading 242x242 integer matrix 'mat' with elementary divisors 'eldiv'.\ntrue\ng\
ap> inv := InverseRatMat(mat);; time;                      \n840\ngap> Exponen\
tSquareIntMatFullRank(mat, 101, 3); # same as without the `3'\n115200\n</Examp\
le>\n</Description>\n</ManSection>\n\n</Section>\n\n<!-- #####################\
####################################### -->\n\n<Section Label=\"Sect-ElDivPad\
\">\n<Heading>All Elementary Divisors Using p-adic Method</Heading>\n\nIn the \
 following two functions we  put things together. In particular\nwe  handle th\
e prime parts  of the elementary divisors efficiently for\nprimes appearing wi\
th low  powers  in the highest  elementary  divisor\nrespectively determinant \
divisor.\n\n<ManSection>\n<Func Name=\"ElementaryDivisorsSquareIntMatFullRank\
\"\n      Arg=\" A \"  /> \n\n<Description>\nThis  function returns  a list  o\
f nonzero  elementary divisors of  an\ninteger matrix <A>A</A>.<P/>\n  \nHere \
we  start with computing the biggest  elementary divisor via <Ref\nFunc=\"Expo\
nentSquareIntMatFullRank\"/>.   If   it runs   into a problem\nbecause  <A>A</\
A> is  singular modulo  a choosen  prime (it  starts by\ndefault with 251) the\
n the prime is automatically replaced by the next\none.<P/>\n\nThe rest  is do\
ne using <Ref Func=\"ElementaryDivisorsPPartRkExp\"/> and\n<Ref Func=\"RankMod\
\"/>.<P/>\n\nThe  function   fails if  the  biggest  elementary divisor  canno\
t be\ncompletely factored and the non-factored part is  not a divisor of the\n\
biggest elementary divisor only.<P/>\n\nNote that this function may  for many \
matrices  not be the best choice\nfor computing all elementary divisors.  You \
may first try the standard\n<Package>GAP</Package> library routines  for Smith\
 normal form instead\nof         this     function.     Nevertheless     remem\
ber       <Ref\nFunc=\"ElementaryDivisorsSquareIntMatFullRank\"/>   for hard  \
and    big\nexamples.  It is particularly good when the largest elementary div\
isor\nis a very small factor of the determinant.\n\n<Example>\ngap> Collected(\
ElementaryDivisorsSquareIntMatFullRank(mat));      \n[ [ 1, 49 ], [ 3, 99 ], [\
 6, 7 ], [ 30, 9 ], [ 60, 9 ], [ 120, 2 ], \n  [ 360, 10 ], [ 720, 22 ], [ 360\
0, 12 ], [ 14400, 14 ], \n  [ 28800, 7 ], [ 115200, 2 ] ]\ngap> time;\n860\nga\
p> last2 = Collected(DiagonalOfMat(NormalFormIntMat(mat, 1).normal));\ntrue\ng\
ap> time;\n5170\n</Example>\n</Description>\n</ManSection>\n\n<!-- ###########\
################################################# -->\n\n<ManSection>\n<Func N\
ame=\"ElementaryDivisorsIntMatDeterminant\"\n      Arg=\" A, det[, rk] \"  />\
\n\n<Description>\nThis function  returns a list  of  nonzero elementary divis\
ors  of  an\ninteger matrix <A>A</A>.<P/>\n\nHere <A>det</A> must be an intege\
r which is  a multiple of the biggest\ndeterminant divisor of <A>A</A>.  If   \
the matrix does not have   full\nrank then its rank <A>rk</A> must be given, t\
oo.<P/>\n\nThe    argument    <A>det</A>    can be  given      in   the  form \
 of\n<C>Collected(FactorsInt(<A>det</A>))</C>.<P/>\n\nThis function handles  p\
rime divisors of <A>det</A> with  multiplicity\nsmaller than  4  specially, fo\
r the other  prime  divisors <M>p</M> it\ndelegates to   <Ref   Func=\"Element\
aryDivisorsPPartRkExp\"/> where  the\n<A>exp</A> argument is the multiplicity \
of the <M>p</M> in <A>det</A>.\n(Note that  this is not very good  when  <M>p<\
/M> has actually  a much\nsmaller multiplicity in the largest elementary divis\
or.)<P/>\n\n<Example>\ngap> ReadPackage(\"edim\",  \"tst/mat\");\nReading 242x\
242 integer matrix 'mat' with elementary divisors 'eldiv'.\ntrue\ngap> # not s\
o good:\ngap> ElementaryDivisorsIntMatDeterminant(mat,Product(eldiv)) = \n> Co\
ncatenation([1..49]*0+1, eldiv); time;\ntrue\n5490\n</Example>\n</Description>\
\n</ManSection>\n\n</Section>\n\n<!-- ########################################\
#################### -->\n\n<Section Label=\"Sect-NFIntMatLLL\">\n<Heading>Gcd\
 and Normal Forms Using LLL</Heading>\n\nThe <Package>EDIM</Package>-mini pack\
age also contains implementations\nof an extended  Gcd-algorithm  for integers\
  and  a Hermite and  Smith\nnormal form algorithm  for integer matrices using\
  LLL-techiques. They\nare well described in the paper <Cite Key=\"HMM98\"/> b\
y Havas, Majewski\nand Matthews.<P/>\n\nThey are particularly  useful if one  \
wants  to have the normal  forms\ntogether with transforming matrices.  These \
transforming matrices have\nspectacularly nice   (i.e.,  <Q>small</Q>) entries\
    in cases of   input\nmatrices which  are non-square  or not   of full rank\
   (otherwise the\ntransformation to the Hermite normal form is unique).<P/>\n\
\nIn detail:<P/>\n\n<ManSection>\n<Func Name=\"GcdexIntLLL\"\n      Arg=\"n1, \
 n2, ...\"  /> \n\n<Description>  \nThis function returns for integers <M><A>n\
1</A>, <A>n2</A>, \\ldots</M> a\nlist  <M>[g,  [c_1,  c_2,  \\ldots]]</M>, whe\
re  <M>g  =  c_1<A>n1</A>  +\nc_2<A>n2</A>  +  \\ldots</M>  is  the  greatest \
 common  divisor  of  the\n<A>ni</A>. Here all the <M>c_i</M> are usually very\
 small.\n\n<Example>\ngap> GcdexIntLLL( 517, 244, -304, -872, -286, 854, 866, \
224, -765, -38);\n[ 1, [ 0, 0, 0, 0, 1, 0, 1, 1, 1, 1 ] ]\n</Example>\n</Descr\
iption>\n</ManSection>\n\n<!-- ###############################################\
############# -->\n\n<ManSection>\n<Func Name=\"HermiteIntMatLLL\"\n      Arg=\
\"A\"  />\n\n<Description>  This  returns  the Hermite normal   form  of an in\
teger\nmatrix  <A>A</A> and uses the  LLL-algorithm to avoid entry explosion.\
\n</Description>\n</ManSection>\n\n<!-- ######################################\
###################### -->\n\n<ManSection>\n<Func Name=\"HermiteIntMatLLLTrans\
\"\n      Arg=\" A \"  /> \n\n<Description> \nThis function returns a pair of \
matrices <M>[H, L]</M>  where <M>H = L\n<A>A</A></M> is the Hermite normal for\
m of an integer matrix <A>A</A>.\nThe transforming matrix <M>L</M> can have su\
rprisingly small entries.\n\n<Example>\ngap> ReadPackage(\"edim\",  \"tst/mat2\
\");\nReading 34x34 integer matrix 'mat2' with elementary divisors 'eldiv2'.\n\
true\ngap> tr := HermiteIntMatLLLTrans(mat2);; Maximum(List(Flat(tr[2]), AbsIn\
t));\n606\ngap> tr[2]*mat2 = tr[1];                                           \
     \ntrue\n</Example>\n</Description>\n</ManSection>\n\n<!-- ###############\
############################################# -->\n\n<ManSection>\n<Func Name=\
\"SmithIntMatLLL\"\n      Arg=\" A \"  /> \n\n<Description> \nThis  function  \
returns the  Smith  normal  form of  an integer matrix\n<A>A</A> using the LLL\
-algorithm to avoid entry explosion.\n</Description>\n</ManSection>\n\n<!-- ##\
########################################################## -->\n\n<ManSection>\
\n<Func Name=\"SmithIntMatLLLTrans\"\n      Arg=\" A  \"  />\n\n<Description>\
\nThis  function returns <M>[S, L, R]</M>  where <M>S = L <A>A</A> R</M>\nis t\
he Smith normal form of an integer matrix <A>A</A>.<P/>\n\nWe apply  the algor\
ithm for Hermite  normal form several times  to get\nthe   Smith  normal     f\
orm, that  is   not    in  the    paper <Cite\nKey=\"HMM98\"/>.  The transform\
ing matrices need not  be as  nice as for\nthe Hermite normal form.<P/>\n\n<Ex\
ample>\ngap> ReadPackage(\"edim\",  \"tst/mat2\");\nReading 34x34 integer matr\
ix 'mat2' with elementary divisors 'eldiv2'.\ntrue\ngap> tr := SmithIntMatLLLT\
rans(mat2);;\ngap> tr[2] * mat2 * tr[3] = tr[1];    \ntrue\n</Example>\n</Desc\
ription>\n</ManSection>\n\n</Section>\n\n\n<!-- ##############################\
############################## -->\n\n<Section Label=\"Sect-Util\">\n\n<Headin\
g>Utility Functions from the <Package>EDIM</Package>-package</Heading>\n\n<Man\
Section>\n<Func Name=\"RatNumberFromModular\"\n      Arg=\"n, k, l, x\"  />\n\
\n<Description>\nThis function returns <M>r/s =  <A>x</A> \\pmod{<A>n</A>}</M>\
, if\nit exists. More precisely:<P/>\n\n<A>n</A>,   <A>k</A>,  <A>l</A>   must\
  be   positive  integers   with\n<M>2<A>k</A><A>l</A> \\leq  <A>n</A></M> and\
  <A>x</A> an  integer with\n<M>-<A>n</A>/2  &lt;  <A>x</A>  \\leq   <A>n</A>/\
2</M>.  If  it  exists\nthis  function returns  a rational  number <M>r/s</M> \
 with <M>0  &lt;\ns  &lt;  <A>l</A></M>,  <M>\\gcd(s,  <A>n</A>)  =  1</M>,  <\
M>-<A>k</A>\n&lt;  r  &lt; <A>k</A></M>  and  <M>r/s</M>  congruent to  <M><A>\
x</A>\n\\pmod{<A>n</A>}</M> (i.e., <M><A>n</A> \\mid  r - s <A>x</A></M>). Suc\
h\nan <M>r/s</M>  is unique. The  function returns <K>fail</K> if  such a\nnum\
ber does not exist.<P/>\n</Description>\n</ManSection>\n\n<!-- ###############\
############################################# -->\n\n<ManSection>\n<Func Name=\
\"InverseIntMatMod\"\n      Arg=\"A, p\"  /> \n\n<Description>\nThis function \
 returns  an inverse  matrix modulo  a prime <A>p</A> or\n<K>fail</K>. More pr\
ecisely:<P/>\n\n<A>A</A>  must  be an  integer matrix and   <A>p</A> a prime s\
uch that\n<A>A</A>  is invertible modulo   <A>p</A>.   This function  returns \
an\ninteger matrix <A>inv</A> with   entries in the range  <M>]-<A>p</A>/2\n\\\
ldots <A>p</A>/2]</M> such that  <A>inv</A><A>A</A> reduced  modulo p\nis the \
identity matrix.<P/>\n\nIt returns <K>fail</K>  if the inverse modulo <A>p</A>\
 does not exist.\n\nThis function is particularly fast for primes smaller 256.\
\n</Description>\n</ManSection>\n\n<!-- ######################################\
###################### -->\n\n<ManSection>\n<Func Name=\"HadamardBoundIntMat\"\
\n      Arg=\"A\"  /> \n\n<Description>\nThe Hadamard bound for a square integ\
er matrix <A>A</A> is the product\nof Euclidean norms of the nonzero rows (or \
columns) of <A>A</A>. It is\nan upper bound for the absolute value of the dete\
rminant of <A>A</A>.\n</Description>\n</ManSection>\n\n<!-- ##################\
########################################## -->\n\n<ManSection>\n<Func Name=\"C\
heapFactorsInt\"\n      Arg=\"n[, nr]\"  /> \n\n<Description>\nThis  function \
 returns a  list  of factors  of   an integer <A>n</A>,\nincluding  <Q>small</\
Q> prime  factors -  here   the optional argument\n<A>nr</A> is the   number o\
f iterations  for `FactorsRho'  (default is\n2000).<P/>\n\nThis  is  only  a  \
slight  modification of  the  library  function  <Ref\nFunc=\"FactorsInt\" Boo\
kName=\"ref\"/>  which avoids  an error  message when\nthe number is not compl\
etely factored.<P/> \n</Description> \n</ManSection>\n\n<!-- #################\
########################################### -->\n\n<ManSection>\n<Func Name=\"\
RankMod\"\n      Arg=\"A, p\"  />\n\n<Description>\nThis function returns the \
rank  of  an integer matrix <A>A</A>  modulo\n<A>p</A>. Here <A>p</A> must not\
 necessarily be  a prime. If it is not\nand  this function   returns an  integ\
er, then   this is   the rank of\n<A>A</A> for all prime divisors of <A>p</A>.\
<P/>\n\nIf  during the   computation  a  factorisation  of   <A>p</A> is found\
\n(because some pivot entry has nontrivial  greatest common divisor with\n<A>p\
</A>) then the function     is recursively applied to   the  found\nfactors <C\
>f_i</C> of <A>p</A>.  The result is then given in the form\n<C>[[f_1, rk_1], \
[f_2, rk_2], ...]</C>.<P/>\n\nThe idea to make  this function useful  for non \
primes  was to use  it\nwith large factors of the biggest elementary divisor o\
f <A>A</A> whose\nprime factorization cannot be found easily.<P/>\n\n<Example>\
\ngap> ReadPackage(\"edim\",  \"tst/mat\");\nReading 242x242 integer matrix 'm\
at' with elementary divisors 'eldiv'.\ntrue\ngap> RankMod(mat, 5);\n155\ngap> \
RankMod(mat, (2*79*4001));\n[ [ 2, 148 ], [ 79, 242 ], [ 4001, 242 ] ]\n</Exam\
ple>\n</Description>\n</ManSection>\n\n</Section>\n\n<!-- ####################\
######################################## -->\n\n<Section Label=\"Sect-InvRatMa\
tAlg\">\n<Heading>InverseRatMat - the Algorithm</Heading>\n\n\nThe  idea   is \
to  recover a rational    matrix from  an <M>l</M>-adic\napproximation for som\
e prime <M>l</M>.   This description came out  of\ndiscussions with Jürgen M�\
�ller.  I  thank John Cannon for pointing out\nthat the basic idea already app\
eared in the paper <Cite Key=\"D82\"/> of\nDixon.<P/>\n\nLet <M>A</M> be an  i\
nvertible matrix over  the rational  numbers.  By\nmultiplying with a constant\
 we may assume that its entries are in fact\nintegers.<P/>\n\n(1) We first  de\
scribe how to find  an  <M>l</M>-adic approximation of\n<M>A^{-1}</M>.  Find a\
 prime <M>l</M> such that <M>A</M> is invertible\nmodulo <M>l</M> and let <M>B\
</M> be the integer matrix with entries in\nthe   range   <M>\\left]-l/2,l/2\\\
right]</M>  such  that   <M>BA</M>  is\ncongruent  to  the identity matrix   m\
odulo  <M>l</M>.  (This  can  be\ncomputed fast by usual Gauß elimination.)<P\
/>\n\nNow let  <M>v \\in  &ZZ;^r</M> be  a row  vector. Define  two sequences\
\n<M>v_i</M> and <M>x_i</M>  of row vectors in  <M>&ZZ;^r</M> by: <M>x_0\n:= 0\
  \\in &ZZ;^r</M>,  <M>v_0 :=  -v</M> and for  <M>i &gt;  0</M> set\n<M>x_i</M\
>  to  the  vector  congruent  to  <M>-v_{i-1}  B</M>  modulo\n<M>l</M> having\
  entries in  the range  <M>\\left]-l/2, l/2\\right]</M>.\nThen all entries  o\
f <M>x_i A + v_{i-1}</M> are  divisible by <M>l</M>\nand we set <M>v_i := (1/l\
) \\cdot (x_i A + v_{i-1})</M>.<P/>\n\nInduction  shows that for <M>y_i  := \\\
sum_{k=1}^{i} l^{k-1} x_k</M> we\nhave <M>y_i A  = v +  l^i v_i</M> for all  <\
M>i \\geq 0</M>.  Hence the\nsequence  <M>y_i</M>,    <M>i \\geq    0</M>,  gi\
ves  an  <M>l</M>-adic\napproximation to  the vector <M>y \\in &QQ;^r</M>  wit\
h <M>y  A =\nv</M>.<P/>\n\n(2) The second point is  to show how  we can  get t\
he vector  <M>y</M>\nfrom a  sufficiently  good  approximation <M>y_i</M>.  No\
te   that the\nsequence of <M>y_i</M> becomes  constant for <M>i  \\geq i_0</M\
> if all\nentries   of  <M>y</M> are  integers  of  absolute  value smaller th\
an\n<M>l^{i_0} / 2</M> because of our choice of representatives of residue\ncl\
asses    modulo  <M>l</M>    in      the interval     <M>\\left]-l/2,\nl/2\\ri\
ght]</M>.<P/>\n\nMore generally consider <M>a / b \\in &QQ;</M> with <M>b &gt;\
 0</M> and\n<M>a, b</M> coprime. Then there is for each <M>n \\in &NN;</M> whi\
ch is\ncoprime to <M>b</M> a  unique <M>c \\in &ZZ;</M> with <M>-n  / 2 &lt; c\
\n\\leq n / 2</M>  and <M>a \\equiv c b \\pmod{n}</M>. This <M>c</M> can be\nc\
omputed via the extended Euclidean  algorithm applied to <M>b</M> and\n<M>n</M\
>.<P/>\n\nNow let <M>n,  \\alpha, \\beta \\in &NN;</M> with <M>2  \\alpha \\be\
ta \\leq\nn</M>. Then the map <M>\\{a/b \\in &QQ; \\mid -\\alpha \\leq a \\leq\
 \\alpha,\n1 \\leq b &lt; \\beta  \\} \\rightarrow \\left]-n/2, n/2\\right]</M\
>, <M>a/b\n\\mapsto c</M> (defined  as above) is injective  (since for <M>a/b<\
/M>,\n<M>a'/b'</M>  in the  above  set we  have  <M>a  b' -  a'  b \\equiv  0\
\n\\pmod{n}</M> if and only if <M>a b' - a' b = 0</M>).<P/>\n\nIn practice we \
 can use for any  <M>c \\in \\left]-n/2, n/2\\right]</M> a\ncertain extended E\
uclidean  algorithm applied to <M>n</M> and <M>c</M>\nto decide if <M>c</M> is\
 in the image of the above map and to find the\ncorresponding <M>a/b</M> if it\
 exists.<P/>\n\n(3) To put things together we apply (2)  to the entries of the\
 vectors\n<M>y_i</M> constructed  in (1), choosing <M>n   = l^i</M>, <M>\\alph\
a =\n\\sqrt{n}/2</M> and <M>\\beta = \\sqrt{n}</M>. If  we have found this way\
\na candidate  for  <M>y</M> we can  easily  check if it  is correct  by\ncomp\
uting <M>y A</M>.  If <M>\\mu</M> is the  maximal absolute value of\nall numer\
ators and denominators of the entries of <M>y</M> it is clear\nfrom (2) that w\
e will find  <M>y</M> from <M>y_i</M>  if <M>l^i &gt; 2\n\\mu^2</M>.<P/>\n\n(4\
) If we take as <M>v</M>  in (1) to(3)  all standard unit vectors we\nclearly \
get  the   rows  of  <M>A^{-1}</M>.    But   we  can  do    it\nbetter.  Namel\
y we  can take   as <M>v</M>  the standard unit  vectors\nmultiplied   by the \
 least  common   multiple  <M>\\epsilon</M> of  the\ndenominators  of the alre\
ady   computed entries of <M>A^{-1}</M>.   In\nmany examples  this <M>\\epsilo\
n</M>  actually equals <M>\\epsilon_r</M>\nafter  the computation of  the firs\
t or  first few rows.  Therefore we\nwill often find quickly the next row of  \
<M>A^{-1}</M> already in (1),\nbecause we find a <M>v_i = 0</M> such  that the\
 sequence of <M>y_i</M>\nbecomes constant (<M>=y</M>).<P/>\n\n<Subsection Labe\
l=\"Ssect-rankintmat\">\n<Heading>Rank of Integer Matrix</Heading>\n\nThe foll\
owing strategy has shown to  be useful in proving that some very\nbig integer \
matrix is not invertible.<P/>\n\n<List>\n<Item>Check  the   rank  modulo  some\
   small  primes,  say   with  <Ref\nFunc=\"RankMod\"/>. </Item>\n<Item>If the\
  rank seems  less than  the number of  rows choose  a prime\n<M>p</M>, a  col\
lection of  lines which  is linearly  independent modulo\n<M>p</M>, and anothe\
r line linearly  dependend on these. Guess that this\nlast line is  also linea\
rly dependend on the chosen  collection over the\nrational numbers (maybe chec\
k modulo several small primes).</Item>\n<Item>Find columns of  the collection \
of lines which  give an invertible\nmatrix modulo some prime.</Item>\n<Item>Th\
en use <Ref  Func=\"RationalSolutionIntMat\"/> with the invertible\nsubmatrix \
and  corresponding entries  of the  linearly dependend  row to\nprove this.</I\
tem>\n</List>\n\nGuessing  the rank  of a  matrix from  the rank  modulo sever\
al  primes,\nchosing a  maximal set  of lines which  are linearly  independent\
 modulo\nsome  primes, and  using <Ref  Func=\"RationalSolutionIntMat\"/> with\
  the\nremaining lines,  one may  also find  the exact rank  of a  huge intege\
r\nmatrix. <P/>\n\n</Subsection>\n\n</Section>\n\n</Chapter>\n\n\n</Body>\n\n<\
!--  We want a bibliography  -->\n<Bibliography Databases=\"edim\"/>\n\n<!--  \
And finally an index as well   -->\n<TheIndex/>\n\n</Book>\n\n", 
  inputorigins := [ [ 1, "doc/edim.xml", 1 ], [ 40, "doc/edim.xml", 2 ], 
      [ 41, "doc/edim.xml", 3 ], [ 77, "doc/edim.xml", 4 ], 
      [ 78, "doc/edim.xml", 5 ], [ 83, "doc/edim.xml", 6 ], 
      [ 84, "doc/edim.xml", 7 ], [ 159, "doc/edim.xml", 8 ], 
      [ 160, "doc/edim.xml", 9 ], [ 228, "doc/edim.xml", 10 ], 
      [ 229, "doc/edim.xml", 11 ], [ 233, "doc/edim.xml", 12 ], 
      [ 234, "doc/edim.xml", 13 ], [ 253, "doc/edim.xml", 14 ], 
      [ 254, "doc/edim.xml", 15 ], [ 266, "doc/edim.xml", 16 ], 
      [ 309, "doc/edim.xml", 17 ], [ 375, "doc/edim.xml", 18 ], 
      [ 389, "doc/edim.xml", 19 ], [ 403, "doc/../VERSION", 1 ], 
      [ 409, "doc/edim.xml", 19 ], [ 410, "doc/edim.xml", 20 ], 
      [ 425, "doc/edim.xml", 21 ], [ 451, "doc/edim.xml", 22 ], 
      [ 461, "doc/edim.xml", 23 ], [ 489, "doc/edim.xml", 24 ], 
      [ 501, "doc/edim.xml", 25 ], [ 519, "doc/edim.xml", 26 ], 
      [ 532, "doc/edim.xml", 27 ], [ 540, "doc/edim.xml", 28 ], 
      [ 551, "doc/edim.xml", 29 ], [ 604, "doc/edim.xml", 30 ], 
      [ 681, "doc/edim.xml", 31 ], [ 695, "doc/edim.xml", 32 ], 
      [ 725, "doc/edim.xml", 33 ], [ 764, "doc/edim.xml", 34 ], 
      [ 822, "doc/edim.xml", 35 ], [ 862, "doc/edim.xml", 36 ], 
      [ 918, "doc/edim.xml", 37 ], [ 987, "doc/edim.xml", 38 ], 
      [ 1127, "doc/edim.xml", 39 ], [ 1198, "doc/edim.xml", 40 ], 
      [ 1227, "doc/edim.xml", 41 ], [ 1244, "doc/edim.xml", 42 ], 
      [ 1245, "doc/edim.xml", 43 ], [ 1258, "doc/edim.xml", 44 ], 
      [ 1259, "doc/edim.xml", 45 ], [ 1330, "doc/edim.xml", 46 ], 
      [ 1349, "doc/edim.xml", 47 ], [ 1350, "doc/edim.xml", 48 ], 
      [ 1391, "doc/edim.xml", 49 ], [ 1398, "doc/main.xml", 1 ], 
      [ 1404, "doc/main.xml", 2 ], [ 1405, "doc/main.xml", 3 ], 
      [ 1475, "doc/main.xml", 4 ], [ 1476, "doc/main.xml", 5 ], 
      [ 1551, "doc/main.xml", 6 ], [ 1552, "doc/main.xml", 7 ], 
      [ 1599, "doc/main.xml", 8 ], [ 1600, "doc/main.xml", 9 ], 
      [ 1604, "doc/main.xml", 10 ], [ 1605, "doc/main.xml", 11 ], 
      [ 1633, "doc/main.xml", 12 ], [ 1688, "doc/main.xml", 13 ], 
      [ 1727, "doc/main.xml", 14 ], [ 1728, "doc/main.xml", 15 ], 
      [ 1799, "doc/main.xml", 16 ], [ 1819, "doc/main.xml", 17 ], 
      [ 1820, "doc/main.xml", 18 ], [ 1893, "doc/main.xml", 19 ], 
      [ 1966, "doc/main.xml", 20 ], [ 2039, "doc/main.xml", 21 ], 
      [ 2112, "doc/main.xml", 22 ], [ 2185, "doc/main.xml", 23 ], 
      [ 2248, "doc/main.xml", 24 ], [ 2249, "doc/main.xml", 25 ], 
      [ 2320, "doc/main.xml", 26 ], [ 2391, "doc/main.xml", 27 ], 
      [ 2442, "doc/main.xml", 28 ], [ 2443, "doc/main.xml", 29 ], 
      [ 2514, "doc/main.xml", 30 ], [ 2585, "doc/main.xml", 31 ], 
      [ 2656, "doc/main.xml", 32 ], [ 2727, "doc/main.xml", 33 ], 
      [ 2793, "doc/main.xml", 34 ], [ 2794, "doc/main.xml", 35 ], 
      [ 2865, "doc/main.xml", 36 ], [ 2936, "doc/main.xml", 37 ], 
      [ 3007, "doc/main.xml", 38 ], [ 3078, "doc/main.xml", 39 ], 
      [ 3149, "doc/main.xml", 40 ], [ 3220, "doc/main.xml", 41 ], 
      [ 3291, "doc/main.xml", 42 ], [ 3323, "doc/main.xml", 43 ], 
      [ 3324, "doc/main.xml", 44 ], [ 3397, "doc/main.xml", 45 ], 
      [ 3470, "doc/main.xml", 46 ], [ 3543, "doc/main.xml", 47 ], 
      [ 3616, "doc/main.xml", 48 ], [ 3631, "doc/main.xml", 49 ], 
      [ 3632, "doc/main.xml", 50 ], [ 3703, "doc/main.xml", 51 ], 
      [ 3774, "doc/main.xml", 52 ], [ 3845, "doc/main.xml", 53 ], 
      [ 3912, "doc/main.xml", 54 ], [ 3913, "doc/main.xml", 55 ], 
      [ 3928, "doc/main.xml", 56 ], [ 3929, "doc/main.xml", 57 ], 
      [ 3960, "doc/main.xml", 58 ], [ 4031, "doc/main.xml", 59 ], 
      [ 4032, "doc/main.xml", 60 ], [ 4103, "doc/main.xml", 61 ], 
      [ 4147, "doc/main.xml", 62 ], [ 4214, "doc/main.xml", 63 ], 
      [ 4282, "doc/main.xml", 64 ], [ 4352, "doc/main.xml", 65 ], 
      [ 4377, "doc/main.xml", 66 ], [ 4442, "doc/main.xml", 67 ], 
      [ 4469, "doc/main.xml", 68 ], [ 4470, "doc/main.xml", 69 ], 
      [ 4544, "doc/main.xml", 70 ], [ 4589, "doc/main.xml", 71 ], 
      [ 4590, "doc/main.xml", 72 ], [ 4645, "doc/main.xml", 73 ], 
      [ 4646, "doc/main.xml", 74 ], [ 4717, "doc/main.xml", 75 ], 
      [ 4781, "doc/main.xml", 76 ], [ 4857, "doc/main.xml", 77 ], 
      [ 4933, "doc/main.xml", 78 ], [ 4977, "doc/main.xml", 79 ], 
      [ 4978, "doc/main.xml", 80 ], [ 5040, "doc/main.xml", 81 ], 
      [ 5041, "doc/main.xml", 82 ], [ 5116, "doc/main.xml", 83 ], 
      [ 5172, "doc/main.xml", 84 ], [ 5173, "doc/main.xml", 85 ], 
      [ 5249, "doc/main.xml", 86 ], [ 5250, "doc/main.xml", 87 ], 
      [ 5264, "doc/main.xml", 88 ], [ 5265, "doc/main.xml", 89 ], 
      [ 5295, "doc/main.xml", 90 ], [ 5296, "doc/main.xml", 91 ], 
      [ 5314, "doc/main.xml", 92 ], [ 5385, "doc/main.xml", 93 ], 
      [ 5456, "doc/main.xml", 94 ], [ 5527, "doc/main.xml", 95 ], 
      [ 5564, "doc/main.xml", 96 ], [ 5579, "doc/main.xml", 97 ], 
      [ 5593, "doc/main.xml", 98 ], [ 5594, "doc/main.xml", 99 ], 
      [ 5605, "doc/main.xml", 100 ], [ 5606, "doc/main.xml", 101 ], 
      [ 5676, "doc/main.xml", 102 ], [ 5707, "doc/main.xml", 103 ], 
      [ 5764, "doc/main.xml", 104 ], [ 5765, "doc/main.xml", 105 ], 
      [ 5816, "doc/main.xml", 106 ], [ 5817, "doc/main.xml", 107 ], 
      [ 5830, "doc/main.xml", 108 ], [ 5869, "doc/main.xml", 109 ], 
      [ 5895, "doc/main.xml", 110 ], [ 5935, "doc/main.xml", 111 ], 
      [ 5958, "doc/main.xml", 112 ], [ 5999, "doc/main.xml", 113 ], 
      [ 6022, "doc/main.xml", 114 ], [ 6064, "doc/main.xml", 115 ], 
      [ 6092, "doc/main.xml", 116 ], [ 6139, "doc/main.xml", 117 ], 
      [ 6171, "doc/main.xml", 118 ], [ 6172, "doc/main.xml", 119 ], 
      [ 6243, "doc/main.xml", 120 ], [ 6314, "doc/main.xml", 121 ], 
      [ 6385, "doc/main.xml", 122 ], [ 6456, "doc/main.xml", 123 ], 
      [ 6471, "doc/main.xml", 124 ], [ 6472, "doc/main.xml", 125 ], 
      [ 6543, "doc/main.xml", 126 ], [ 6560, "doc/main.xml", 127 ], 
      [ 6561, "doc/main.xml", 128 ], [ 6632, "doc/main.xml", 129 ], 
      [ 6703, "doc/main.xml", 130 ], [ 6774, "doc/main.xml", 131 ], 
      [ 6775, "doc/main.xml", 132 ], [ 6846, "doc/main.xml", 133 ], 
      [ 6911, "doc/main.xml", 134 ], [ 6912, "doc/main.xml", 135 ], 
      [ 6983, "doc/main.xml", 136 ], [ 7054, "doc/main.xml", 137 ], 
      [ 7125, "doc/main.xml", 138 ], [ 7196, "doc/main.xml", 139 ], 
      [ 7233, "doc/main.xml", 140 ], [ 7234, "doc/main.xml", 141 ], 
      [ 7305, "doc/main.xml", 142 ], [ 7376, "doc/main.xml", 143 ], 
      [ 7447, "doc/main.xml", 144 ], [ 7505, "doc/main.xml", 145 ], 
      [ 7506, "doc/main.xml", 146 ], [ 7577, "doc/main.xml", 147 ], 
      [ 7607, "doc/main.xml", 148 ], [ 7608, "doc/main.xml", 149 ], 
      [ 7679, "doc/main.xml", 150 ], [ 7750, "doc/main.xml", 151 ], 
      [ 7821, "doc/main.xml", 152 ], [ 7892, "doc/main.xml", 153 ], 
      [ 7963, "doc/main.xml", 154 ], [ 7984, "doc/main.xml", 155 ], 
      [ 7985, "doc/main.xml", 156 ], [ 8056, "doc/main.xml", 157 ], 
      [ 8103, "doc/main.xml", 158 ], [ 8104, "doc/main.xml", 159 ], 
      [ 8175, "doc/main.xml", 160 ], [ 8246, "doc/main.xml", 161 ], 
      [ 8271, "doc/main.xml", 162 ], [ 8272, "doc/main.xml", 163 ], 
      [ 8282, "doc/main.xml", 164 ], [ 8320, "doc/main.xml", 165 ], 
      [ 8391, "doc/main.xml", 166 ], [ 8396, "doc/main.xml", 167 ], 
      [ 8468, "doc/main.xml", 168 ], [ 8507, "doc/main.xml", 169 ], 
      [ 8511, "doc/main.xml", 170 ], [ 8577, "doc/main.xml", 171 ], 
      [ 8616, "doc/main.xml", 172 ], [ 8619, "doc/main.xml", 173 ], 
      [ 8630, "doc/main.xml", 174 ], [ 8645, "doc/main.xml", 175 ], 
      [ 8659, "doc/main.xml", 176 ], [ 8660, "doc/main.xml", 177 ], 
      [ 8730, "doc/main.xml", 178 ], [ 8743, "doc/main.xml", 179 ], 
      [ 8793, "doc/main.xml", 180 ], [ 8813, "doc/main.xml", 181 ], 
      [ 8855, "doc/main.xml", 182 ], [ 8856, "doc/main.xml", 183 ], 
      [ 8871, "doc/main.xml", 184 ], [ 8942, "doc/main.xml", 185 ], 
      [ 9013, "doc/main.xml", 186 ], [ 9084, "doc/main.xml", 187 ], 
      [ 9108, "doc/main.xml", 188 ], [ 9109, "doc/main.xml", 189 ], 
      [ 9180, "doc/main.xml", 190 ], [ 9251, "doc/main.xml", 191 ], 
      [ 9310, "doc/main.xml", 192 ], [ 9311, "doc/main.xml", 193 ], 
      [ 9382, "doc/main.xml", 194 ], [ 9401, "doc/main.xml", 195 ], 
      [ 9402, "doc/main.xml", 196 ], [ 9473, "doc/main.xml", 197 ], 
      [ 9544, "doc/main.xml", 198 ], [ 9615, "doc/main.xml", 199 ], 
      [ 9664, "doc/main.xml", 200 ], [ 9665, "doc/main.xml", 201 ], 
      [ 9675, "doc/main.xml", 202 ], [ 9713, "doc/main.xml", 203 ], 
      [ 9784, "doc/main.xml", 204 ], [ 9789, "doc/main.xml", 205 ], 
      [ 9850, "doc/main.xml", 206 ], [ 9886, "doc/main.xml", 207 ], 
      [ 9891, "doc/main.xml", 208 ], [ 9902, "doc/main.xml", 209 ], 
      [ 9917, "doc/main.xml", 210 ], [ 9931, "doc/main.xml", 211 ], 
      [ 9942, "doc/main.xml", 212 ], [ 9943, "doc/main.xml", 213 ], 
      [ 10013, "doc/main.xml", 214 ], [ 10014, "doc/main.xml", 215 ], 
      [ 10047, "doc/main.xml", 216 ], [ 10095, "doc/main.xml", 217 ], 
      [ 10096, "doc/main.xml", 218 ], [ 10109, "doc/main.xml", 219 ], 
      [ 10136, "doc/main.xml", 220 ], [ 10156, "doc/main.xml", 221 ], 
      [ 10200, "doc/main.xml", 222 ], [ 10201, "doc/main.xml", 223 ], 
      [ 10217, "doc/main.xml", 224 ], [ 10288, "doc/main.xml", 225 ], 
      [ 10310, "doc/main.xml", 226 ], [ 10311, "doc/main.xml", 227 ], 
      [ 10382, "doc/main.xml", 228 ], [ 10453, "doc/main.xml", 229 ], 
      [ 10524, "doc/main.xml", 230 ], [ 10596, "doc/main.xml", 231 ], 
      [ 10609, "doc/main.xml", 232 ], [ 10610, "doc/main.xml", 233 ], 
      [ 10681, "doc/main.xml", 234 ], [ 10753, "doc/main.xml", 235 ], 
      [ 10824, "doc/main.xml", 236 ], [ 10856, "doc/main.xml", 237 ], 
      [ 10857, "doc/main.xml", 238 ], [ 10928, "doc/main.xml", 239 ], 
      [ 10999, "doc/main.xml", 240 ], [ 11070, "doc/main.xml", 241 ], 
      [ 11116, "doc/main.xml", 242 ], [ 11131, "doc/main.xml", 243 ], 
      [ 11145, "doc/main.xml", 244 ], [ 11146, "doc/main.xml", 245 ], 
      [ 11216, "doc/main.xml", 246 ], [ 11229, "doc/main.xml", 247 ], 
      [ 11265, "doc/main.xml", 248 ], [ 11297, "doc/main.xml", 249 ], 
      [ 11361, "doc/main.xml", 250 ], [ 11362, "doc/main.xml", 251 ], 
      [ 11378, "doc/main.xml", 252 ], [ 11447, "doc/main.xml", 253 ], 
      [ 11491, "doc/main.xml", 254 ], [ 11492, "doc/main.xml", 255 ], 
      [ 11565, "doc/main.xml", 256 ], [ 11638, "doc/main.xml", 257 ], 
      [ 11666, "doc/main.xml", 258 ], [ 11667, "doc/main.xml", 259 ], 
      [ 11738, "doc/main.xml", 260 ], [ 11809, "doc/main.xml", 261 ], 
      [ 11870, "doc/main.xml", 262 ], [ 11871, "doc/main.xml", 263 ], 
      [ 11945, "doc/main.xml", 264 ], [ 11982, "doc/main.xml", 265 ], 
      [ 11997, "doc/main.xml", 266 ], [ 12011, "doc/main.xml", 267 ], 
      [ 12012, "doc/main.xml", 268 ], [ 12082, "doc/main.xml", 269 ], 
      [ 12083, "doc/main.xml", 270 ], [ 12096, "doc/main.xml", 271 ], 
      [ 12138, "doc/main.xml", 272 ], [ 12166, "doc/main.xml", 273 ], 
      [ 12183, "doc/main.xml", 274 ], [ 12184, "doc/main.xml", 275 ], 
      [ 12198, "doc/main.xml", 276 ], [ 12269, "doc/main.xml", 277 ], 
      [ 12311, "doc/main.xml", 278 ], [ 12312, "doc/main.xml", 279 ], 
      [ 12383, "doc/main.xml", 280 ], [ 12454, "doc/main.xml", 281 ], 
      [ 12513, "doc/main.xml", 282 ], [ 12516, "doc/main.xml", 283 ], 
      [ 12587, "doc/main.xml", 284 ], [ 12658, "doc/main.xml", 285 ], 
      [ 12729, "doc/main.xml", 286 ], [ 12800, "doc/main.xml", 287 ], 
      [ 12871, "doc/main.xml", 288 ], [ 12942, "doc/main.xml", 289 ], 
      [ 13004, "doc/main.xml", 290 ], [ 13005, "doc/main.xml", 291 ], 
      [ 13076, "doc/main.xml", 292 ], [ 13147, "doc/main.xml", 293 ], 
      [ 13218, "doc/main.xml", 294 ], [ 13264, "doc/main.xml", 295 ], 
      [ 13265, "doc/main.xml", 296 ], [ 13275, "doc/main.xml", 297 ], 
      [ 13313, "doc/main.xml", 298 ], [ 13384, "doc/main.xml", 299 ], 
      [ 13389, "doc/main.xml", 300 ], [ 13450, "doc/main.xml", 301 ], 
      [ 13454, "doc/main.xml", 302 ], [ 13528, "doc/main.xml", 303 ], 
      [ 13535, "doc/main.xml", 304 ], [ 13546, "doc/main.xml", 305 ], 
      [ 13561, "doc/main.xml", 306 ], [ 13575, "doc/main.xml", 307 ], 
      [ 13576, "doc/main.xml", 308 ], [ 13587, "doc/main.xml", 309 ], 
      [ 13588, "doc/main.xml", 310 ], [ 13658, "doc/main.xml", 311 ], 
      [ 13659, "doc/main.xml", 312 ], [ 13691, "doc/main.xml", 313 ], 
      [ 13754, "doc/main.xml", 314 ], [ 13755, "doc/main.xml", 315 ], 
      [ 13826, "doc/main.xml", 316 ], [ 13897, "doc/main.xml", 317 ], 
      [ 13968, "doc/main.xml", 318 ], [ 14002, "doc/main.xml", 319 ], 
      [ 14003, "doc/main.xml", 320 ], [ 14016, "doc/main.xml", 321 ], 
      [ 14068, "doc/main.xml", 322 ], [ 14089, "doc/main.xml", 323 ], 
      [ 14090, "doc/main.xml", 324 ], [ 14104, "doc/main.xml", 325 ], 
      [ 14175, "doc/main.xml", 326 ], [ 14204, "doc/main.xml", 327 ], 
      [ 14207, "doc/main.xml", 328 ], [ 14278, "doc/main.xml", 329 ], 
      [ 14349, "doc/main.xml", 330 ], [ 14420, "doc/main.xml", 331 ], 
      [ 14491, "doc/main.xml", 332 ], [ 14500, "doc/main.xml", 333 ], 
      [ 14501, "doc/main.xml", 334 ], [ 14572, "doc/main.xml", 335 ], 
      [ 14599, "doc/main.xml", 336 ], [ 14600, "doc/main.xml", 337 ], 
      [ 14670, "doc/main.xml", 338 ], [ 14741, "doc/main.xml", 339 ], 
      [ 14778, "doc/main.xml", 340 ], [ 14779, "doc/main.xml", 341 ], 
      [ 14850, "doc/main.xml", 342 ], [ 14921, "doc/main.xml", 343 ], 
      [ 14992, "doc/main.xml", 344 ], [ 15063, "doc/main.xml", 345 ], 
      [ 15134, "doc/main.xml", 346 ], [ 15205, "doc/main.xml", 347 ], 
      [ 15248, "doc/main.xml", 348 ], [ 15249, "doc/main.xml", 349 ], 
      [ 15259, "doc/main.xml", 350 ], [ 15326, "doc/main.xml", 351 ], 
      [ 15395, "doc/main.xml", 352 ], [ 15453, "doc/main.xml", 353 ], 
      [ 15485, "doc/main.xml", 354 ], [ 15496, "doc/main.xml", 355 ], 
      [ 15500, "doc/main.xml", 356 ], [ 15572, "doc/main.xml", 357 ], 
      [ 15577, "doc/main.xml", 358 ], [ 15588, "doc/main.xml", 359 ], 
      [ 15593, "doc/main.xml", 360 ], [ 15604, "doc/main.xml", 361 ], 
      [ 15619, "doc/main.xml", 362 ], [ 15633, "doc/main.xml", 363 ], 
      [ 15634, "doc/main.xml", 364 ], [ 15704, "doc/main.xml", 365 ], 
      [ 15705, "doc/main.xml", 366 ], [ 15718, "doc/main.xml", 367 ], 
      [ 15767, "doc/main.xml", 368 ], [ 15798, "doc/main.xml", 369 ], 
      [ 15799, "doc/main.xml", 370 ], [ 15813, "doc/main.xml", 371 ], 
      [ 15884, "doc/main.xml", 372 ], [ 15913, "doc/main.xml", 373 ], 
      [ 15914, "doc/main.xml", 374 ], [ 15985, "doc/main.xml", 375 ], 
      [ 16056, "doc/main.xml", 376 ], [ 16109, "doc/main.xml", 377 ], 
      [ 16110, "doc/main.xml", 378 ], [ 16181, "doc/main.xml", 379 ], 
      [ 16227, "doc/main.xml", 380 ], [ 16228, "doc/main.xml", 381 ], 
      [ 16299, "doc/main.xml", 382 ], [ 16370, "doc/main.xml", 383 ], 
      [ 16441, "doc/main.xml", 384 ], [ 16512, "doc/main.xml", 385 ], 
      [ 16583, "doc/main.xml", 386 ], [ 16644, "doc/main.xml", 387 ], 
      [ 16645, "doc/main.xml", 388 ], [ 16655, "doc/main.xml", 389 ], 
      [ 16693, "doc/main.xml", 390 ], [ 16764, "doc/main.xml", 391 ], 
      [ 16769, "doc/main.xml", 392 ], [ 16789, "doc/main.xml", 393 ], 
      [ 16853, "doc/main.xml", 394 ], [ 16896, "doc/main.xml", 395 ], 
      [ 16901, "doc/main.xml", 396 ], [ 16906, "doc/main.xml", 397 ], 
      [ 16917, "doc/main.xml", 398 ], [ 16932, "doc/main.xml", 399 ], 
      [ 16946, "doc/main.xml", 400 ], [ 16947, "doc/main.xml", 401 ], 
      [ 16958, "doc/main.xml", 402 ], [ 16959, "doc/main.xml", 403 ], 
      [ 17029, "doc/main.xml", 404 ], [ 17030, "doc/main.xml", 405 ], 
      [ 17065, "doc/main.xml", 406 ], [ 17115, "doc/main.xml", 407 ], 
      [ 17116, "doc/main.xml", 408 ], [ 17187, "doc/main.xml", 409 ], 
      [ 17258, "doc/main.xml", 410 ], [ 17329, "doc/main.xml", 411 ], 
      [ 17400, "doc/main.xml", 412 ], [ 17418, "doc/main.xml", 413 ], 
      [ 17419, "doc/main.xml", 414 ], [ 17490, "doc/main.xml", 415 ], 
      [ 17561, "doc/main.xml", 416 ], [ 17635, "doc/main.xml", 417 ], 
      [ 17706, "doc/main.xml", 418 ], [ 17764, "doc/main.xml", 419 ], 
      [ 17765, "doc/main.xml", 420 ], [ 17780, "doc/main.xml", 421 ], 
      [ 17781, "doc/main.xml", 422 ], [ 17794, "doc/main.xml", 423 ], 
      [ 17819, "doc/main.xml", 424 ], [ 17849, "doc/main.xml", 425 ], 
      [ 17850, "doc/main.xml", 426 ], [ 17866, "doc/main.xml", 427 ], 
      [ 17939, "doc/main.xml", 428 ], [ 18012, "doc/main.xml", 429 ], 
      [ 18085, "doc/main.xml", 430 ], [ 18144, "doc/main.xml", 431 ], 
      [ 18145, "doc/main.xml", 432 ], [ 18155, "doc/main.xml", 433 ], 
      [ 18228, "doc/main.xml", 434 ], [ 18268, "doc/main.xml", 435 ], 
      [ 18279, "doc/main.xml", 436 ], [ 18294, "doc/main.xml", 437 ], 
      [ 18308, "doc/main.xml", 438 ], [ 18309, "doc/main.xml", 439 ], 
      [ 18379, "doc/main.xml", 440 ], [ 18380, "doc/main.xml", 441 ], 
      [ 18393, "doc/main.xml", 442 ], [ 18423, "doc/main.xml", 443 ], 
      [ 18441, "doc/main.xml", 444 ], [ 18442, "doc/main.xml", 445 ], 
      [ 18513, "doc/main.xml", 446 ], [ 18584, "doc/main.xml", 447 ], 
      [ 18599, "doc/main.xml", 448 ], [ 18613, "doc/main.xml", 449 ], 
      [ 18614, "doc/main.xml", 450 ], [ 18684, "doc/main.xml", 451 ], 
      [ 18685, "doc/main.xml", 452 ], [ 18698, "doc/main.xml", 453 ], 
      [ 18733, "doc/main.xml", 454 ], [ 18754, "doc/main.xml", 455 ], 
      [ 18755, "doc/main.xml", 456 ], [ 18770, "doc/main.xml", 457 ], 
      [ 18841, "doc/main.xml", 458 ], [ 18912, "doc/main.xml", 459 ], 
      [ 18982, "doc/main.xml", 460 ], [ 18983, "doc/main.xml", 461 ], 
      [ 18993, "doc/main.xml", 462 ], [ 19032, "doc/main.xml", 463 ], 
      [ 19103, "doc/main.xml", 464 ], [ 19108, "doc/main.xml", 465 ], 
      [ 19185, "doc/main.xml", 466 ], [ 19189, "doc/main.xml", 467 ], 
      [ 19262, "doc/main.xml", 468 ], [ 19267, "doc/main.xml", 469 ], 
      [ 19278, "doc/main.xml", 470 ], [ 19293, "doc/main.xml", 471 ], 
      [ 19307, "doc/main.xml", 472 ], [ 19308, "doc/main.xml", 473 ], 
      [ 19378, "doc/main.xml", 474 ], [ 19379, "doc/main.xml", 475 ], 
      [ 19392, "doc/main.xml", 476 ], [ 19420, "doc/main.xml", 477 ], 
      [ 19441, "doc/main.xml", 478 ], [ 19442, "doc/main.xml", 479 ], 
      [ 19457, "doc/main.xml", 480 ], [ 19528, "doc/main.xml", 481 ], 
      [ 19587, "doc/main.xml", 482 ], [ 19602, "doc/main.xml", 483 ], 
      [ 19616, "doc/main.xml", 484 ], [ 19617, "doc/main.xml", 485 ], 
      [ 19687, "doc/main.xml", 486 ], [ 19688, "doc/main.xml", 487 ], 
      [ 19701, "doc/main.xml", 488 ], [ 19734, "doc/main.xml", 489 ], 
      [ 19755, "doc/main.xml", 490 ], [ 19756, "doc/main.xml", 491 ], 
      [ 19770, "doc/main.xml", 492 ], [ 19841, "doc/main.xml", 493 ], 
      [ 19901, "doc/main.xml", 494 ], [ 19902, "doc/main.xml", 495 ], 
      [ 19973, "doc/main.xml", 496 ], [ 20044, "doc/main.xml", 497 ], 
      [ 20115, "doc/main.xml", 498 ], [ 20144, "doc/main.xml", 499 ], 
      [ 20145, "doc/main.xml", 500 ], [ 20155, "doc/main.xml", 501 ], 
      [ 20194, "doc/main.xml", 502 ], [ 20265, "doc/main.xml", 503 ], 
      [ 20270, "doc/main.xml", 504 ], [ 20309, "doc/main.xml", 505 ], 
      [ 20348, "doc/main.xml", 506 ], [ 20353, "doc/main.xml", 507 ], 
      [ 20364, "doc/main.xml", 508 ], [ 20379, "doc/main.xml", 509 ], 
      [ 20393, "doc/main.xml", 510 ], [ 20394, "doc/main.xml", 511 ], 
      [ 20405, "doc/main.xml", 512 ], [ 20406, "doc/main.xml", 513 ], 
      [ 20407, "doc/main.xml", 514 ], [ 20477, "doc/main.xml", 515 ], 
      [ 20478, "doc/main.xml", 516 ], [ 20506, "doc/main.xml", 517 ], 
      [ 20507, "doc/main.xml", 518 ], [ 20585, "doc/main.xml", 519 ], 
      [ 20586, "doc/main.xml", 520 ], [ 20599, "doc/main.xml", 521 ], 
      [ 20633, "doc/main.xml", 522 ], [ 20660, "doc/main.xml", 523 ], 
      [ 20661, "doc/main.xml", 524 ], [ 20675, "doc/main.xml", 525 ], 
      [ 20740, "doc/main.xml", 526 ], [ 20771, "doc/main.xml", 527 ], 
      [ 20772, "doc/main.xml", 528 ], [ 20843, "doc/main.xml", 529 ], 
      [ 20914, "doc/main.xml", 530 ], [ 20985, "doc/main.xml", 531 ], 
      [ 21056, "doc/main.xml", 532 ], [ 21127, "doc/main.xml", 533 ], 
      [ 21198, "doc/main.xml", 534 ], [ 21269, "doc/main.xml", 535 ], 
      [ 21340, "doc/main.xml", 536 ], [ 21367, "doc/main.xml", 537 ], 
      [ 21382, "doc/main.xml", 538 ], [ 21396, "doc/main.xml", 539 ], 
      [ 21397, "doc/main.xml", 540 ], [ 21467, "doc/main.xml", 541 ], 
      [ 21468, "doc/main.xml", 542 ], [ 21481, "doc/main.xml", 543 ], 
      [ 21511, "doc/main.xml", 544 ], [ 21533, "doc/main.xml", 545 ], 
      [ 21534, "doc/main.xml", 546 ], [ 21548, "doc/main.xml", 547 ], 
      [ 21619, "doc/main.xml", 548 ], [ 21652, "doc/main.xml", 549 ], 
      [ 21653, "doc/main.xml", 550 ], [ 21724, "doc/main.xml", 551 ], 
      [ 21795, "doc/main.xml", 552 ], [ 21866, "doc/main.xml", 553 ], 
      [ 21937, "doc/main.xml", 554 ], [ 21965, "doc/main.xml", 555 ], 
      [ 21966, "doc/main.xml", 556 ], [ 22037, "doc/main.xml", 557 ], 
      [ 22038, "doc/main.xml", 558 ], [ 22097, "doc/main.xml", 559 ], 
      [ 22112, "doc/main.xml", 560 ], [ 22126, "doc/main.xml", 561 ], 
      [ 22127, "doc/main.xml", 562 ], [ 22197, "doc/main.xml", 563 ], 
      [ 22198, "doc/main.xml", 564 ], [ 22211, "doc/main.xml", 565 ], 
      [ 22244, "doc/main.xml", 566 ], [ 22263, "doc/main.xml", 567 ], 
      [ 22264, "doc/main.xml", 568 ], [ 22278, "doc/main.xml", 569 ], 
      [ 22349, "doc/main.xml", 570 ], [ 22420, "doc/main.xml", 571 ], 
      [ 22490, "doc/main.xml", 572 ], [ 22505, "doc/main.xml", 573 ], 
      [ 22519, "doc/main.xml", 574 ], [ 22520, "doc/main.xml", 575 ], 
      [ 22590, "doc/main.xml", 576 ], [ 22591, "doc/main.xml", 577 ], 
      [ 22604, "doc/main.xml", 578 ], [ 22633, "doc/main.xml", 579 ], 
      [ 22658, "doc/main.xml", 580 ], [ 22659, "doc/main.xml", 581 ], 
      [ 22673, "doc/main.xml", 582 ], [ 22744, "doc/main.xml", 583 ], 
      [ 22815, "doc/main.xml", 584 ], [ 22886, "doc/main.xml", 585 ], 
      [ 22897, "doc/main.xml", 586 ], [ 22898, "doc/main.xml", 587 ], 
      [ 22971, "doc/main.xml", 588 ], [ 23044, "doc/main.xml", 589 ], 
      [ 23088, "doc/main.xml", 590 ], [ 23104, "doc/main.xml", 591 ], 
      [ 23118, "doc/main.xml", 592 ], [ 23119, "doc/main.xml", 593 ], 
      [ 23189, "doc/main.xml", 594 ], [ 23190, "doc/main.xml", 595 ], 
      [ 23203, "doc/main.xml", 596 ], [ 23224, "doc/main.xml", 597 ], 
      [ 23245, "doc/main.xml", 598 ], [ 23246, "doc/main.xml", 599 ], 
      [ 23260, "doc/main.xml", 600 ], [ 23331, "doc/main.xml", 601 ], 
      [ 23402, "doc/main.xml", 602 ], [ 23473, "doc/main.xml", 603 ], 
      [ 23522, "doc/main.xml", 604 ], [ 23523, "doc/main.xml", 605 ], 
      [ 23594, "doc/main.xml", 606 ], [ 23665, "doc/main.xml", 607 ], 
      [ 23736, "doc/main.xml", 608 ], [ 23806, "doc/main.xml", 609 ], 
      [ 23850, "doc/main.xml", 610 ], [ 23851, "doc/main.xml", 611 ], 
      [ 23922, "doc/main.xml", 612 ], [ 23993, "doc/main.xml", 613 ], 
      [ 24041, "doc/main.xml", 614 ], [ 24042, "doc/main.xml", 615 ], 
      [ 24052, "doc/main.xml", 616 ], [ 24090, "doc/main.xml", 617 ], 
      [ 24161, "doc/main.xml", 618 ], [ 24166, "doc/main.xml", 619 ], 
      [ 24188, "doc/main.xml", 620 ], [ 24192, "doc/main.xml", 621 ], 
      [ 24224, "doc/main.xml", 622 ], [ 24267, "doc/main.xml", 623 ], 
      [ 24278, "doc/main.xml", 624 ], [ 24293, "doc/main.xml", 625 ], 
      [ 24307, "doc/main.xml", 626 ], [ 24308, "doc/main.xml", 627 ], 
      [ 24319, "doc/main.xml", 628 ], [ 24320, "doc/main.xml", 629 ], 
      [ 24390, "doc/main.xml", 630 ], [ 24391, "doc/main.xml", 631 ], 
      [ 24427, "doc/main.xml", 632 ], [ 24476, "doc/main.xml", 633 ], 
      [ 24477, "doc/main.xml", 634 ], [ 24478, "doc/main.xml", 635 ], 
      [ 24549, "doc/main.xml", 636 ], [ 24620, "doc/main.xml", 637 ], 
      [ 24693, "doc/main.xml", 638 ], [ 24764, "doc/main.xml", 639 ], 
      [ 24775, "doc/main.xml", 640 ], [ 24776, "doc/main.xml", 641 ], 
      [ 24847, "doc/main.xml", 642 ], [ 24918, "doc/main.xml", 643 ], 
      [ 24932, "doc/main.xml", 644 ], [ 24933, "doc/main.xml", 645 ], 
      [ 25004, "doc/main.xml", 646 ], [ 25075, "doc/main.xml", 647 ], 
      [ 25146, "doc/main.xml", 648 ], [ 25217, "doc/main.xml", 649 ], 
      [ 25288, "doc/main.xml", 650 ], [ 25335, "doc/main.xml", 651 ], 
      [ 25336, "doc/main.xml", 652 ], [ 25407, "doc/main.xml", 653 ], 
      [ 25478, "doc/main.xml", 654 ], [ 25549, "doc/main.xml", 655 ], 
      [ 25620, "doc/main.xml", 656 ], [ 25691, "doc/main.xml", 657 ], 
      [ 25762, "doc/main.xml", 658 ], [ 25822, "doc/main.xml", 659 ], 
      [ 25823, "doc/main.xml", 660 ], [ 25894, "doc/main.xml", 661 ], 
      [ 25965, "doc/main.xml", 662 ], [ 26036, "doc/main.xml", 663 ], 
      [ 26101, "doc/main.xml", 664 ], [ 26112, "doc/main.xml", 665 ], 
      [ 26113, "doc/main.xml", 666 ], [ 26184, "doc/main.xml", 667 ], 
      [ 26255, "doc/main.xml", 668 ], [ 26326, "doc/main.xml", 669 ], 
      [ 26397, "doc/main.xml", 670 ], [ 26468, "doc/main.xml", 671 ], 
      [ 26539, "doc/main.xml", 672 ], [ 26559, "doc/main.xml", 673 ], 
      [ 26560, "doc/main.xml", 674 ], [ 26631, "doc/main.xml", 675 ], 
      [ 26702, "doc/main.xml", 676 ], [ 26773, "doc/main.xml", 677 ], 
      [ 26844, "doc/main.xml", 678 ], [ 26915, "doc/main.xml", 679 ], 
      [ 26929, "doc/main.xml", 680 ], [ 26930, "doc/main.xml", 681 ], 
      [ 27001, "doc/main.xml", 682 ], [ 27072, "doc/main.xml", 683 ], 
      [ 27143, "doc/main.xml", 684 ], [ 27214, "doc/main.xml", 685 ], 
      [ 27285, "doc/main.xml", 686 ], [ 27342, "doc/main.xml", 687 ], 
      [ 27343, "doc/main.xml", 688 ], [ 27414, "doc/main.xml", 689 ], 
      [ 27485, "doc/main.xml", 690 ], [ 27556, "doc/main.xml", 691 ], 
      [ 27599, "doc/main.xml", 692 ], [ 27600, "doc/main.xml", 693 ], 
      [ 27671, "doc/main.xml", 694 ], [ 27742, "doc/main.xml", 695 ], 
      [ 27813, "doc/main.xml", 696 ], [ 27884, "doc/main.xml", 697 ], 
      [ 27955, "doc/main.xml", 698 ], [ 28026, "doc/main.xml", 699 ], 
      [ 28097, "doc/main.xml", 700 ], [ 28112, "doc/main.xml", 701 ], 
      [ 28113, "doc/main.xml", 702 ], [ 28184, "doc/main.xml", 703 ], 
      [ 28255, "doc/main.xml", 704 ], [ 28326, "doc/main.xml", 705 ], 
      [ 28397, "doc/main.xml", 706 ], [ 28468, "doc/main.xml", 707 ], 
      [ 28539, "doc/main.xml", 708 ], [ 28610, "doc/main.xml", 709 ], 
      [ 28681, "doc/main.xml", 710 ], [ 28752, "doc/main.xml", 711 ], 
      [ 28786, "doc/main.xml", 712 ], [ 28787, "doc/main.xml", 713 ], 
      [ 28825, "doc/main.xml", 714 ], [ 28867, "doc/main.xml", 715 ], 
      [ 28868, "doc/main.xml", 716 ], [ 28941, "doc/main.xml", 717 ], 
      [ 28983, "doc/main.xml", 718 ], [ 28984, "doc/main.xml", 719 ], 
      [ 28991, "doc/main.xml", 720 ], [ 29064, "doc/main.xml", 721 ], 
      [ 29090, "doc/main.xml", 722 ], [ 29163, "doc/main.xml", 723 ], 
      [ 29236, "doc/main.xml", 724 ], [ 29309, "doc/main.xml", 725 ], 
      [ 29382, "doc/main.xml", 726 ], [ 29449, "doc/main.xml", 727 ], 
      [ 29522, "doc/main.xml", 728 ], [ 29555, "doc/main.xml", 729 ], 
      [ 29628, "doc/main.xml", 730 ], [ 29701, "doc/main.xml", 731 ], 
      [ 29720, "doc/main.xml", 732 ], [ 29728, "doc/main.xml", 733 ], 
      [ 29729, "doc/main.xml", 734 ], [ 29802, "doc/main.xml", 735 ], 
      [ 29875, "doc/main.xml", 736 ], [ 29948, "doc/main.xml", 737 ], 
      [ 30021, "doc/main.xml", 738 ], [ 30034, "doc/main.xml", 739 ], 
      [ 30035, "doc/main.xml", 740 ], [ 30049, "doc/main.xml", 741 ], 
      [ 30050, "doc/main.xml", 742 ], [ 30061, "doc/main.xml", 743 ], 
      [ 30062, "doc/main.xml", 744 ], [ 30073, "doc/main.xml", 745 ], 
      [ 30074, "doc/edim.xml", 50 ], [ 30075, "doc/edim.xml", 51 ], 
      [ 30083, "doc/edim.xml", 52 ], [ 30084, "doc/edim.xml", 53 ], 
      [ 30118, "doc/edim.xml", 54 ], [ 30151, "doc/edim.xml", 55 ], 
      [ 30152, "doc/edim.xml", 56 ], [ 30193, "doc/edim.xml", 57 ], 
      [ 30205, "doc/edim.xml", 58 ], [ 30206, "doc/edim.xml", 59 ], 
      [ 30214, "doc/edim.xml", 60 ] ], 
  labels := rec( ("Chap-EDIM") := [ "1", "chap1_mj.html#X7AA826067CC8C395" ], 
      CheapFactorsInt := [ "1.6-4", "chap1_mj.html#X7BAB977C7EB05067" ], 
      ElementaryDivisorsIntMatDeterminant := 
        [ "1.4-2", "chap1_mj.html#X821E30477A5DCE68" ], 
      ElementaryDivisorsPPartHavasSterling := 
        [ "1.2-2", "chap1_mj.html#X82EC4724865F4DF9" ], 
      ElementaryDivisorsPPartRk := 
        [ "1.2-1", "chap1_mj.html#X813B0D73868CD751" ], 
      ElementaryDivisorsPPartRkExp := 
        [ "1.2-1", "chap1_mj.html#X813B0D73868CD751" ], 
      ElementaryDivisorsPPartRkExpSmall := 
        [ "1.2-1", "chap1_mj.html#X813B0D73868CD751" ], 
      ElementaryDivisorsPPartRkI := 
        [ "1.2-1", "chap1_mj.html#X813B0D73868CD751" ], 
      ElementaryDivisorsPPartRkII := 
        [ "1.2-1", "chap1_mj.html#X813B0D73868CD751" ], 
      ElementaryDivisorsSquareIntMatFullRank := 
        [ "1.4-1", "chap1_mj.html#X7B6A3B8486872B0C" ], 
      ExponentSquareIntMatFullRank := 
        [ "1.3-3", "chap1_mj.html#X86DA61D978D2D889" ], 
      GcdexIntLLL := [ "1.5-1", "chap1_mj.html#X799B1A5285D00859" ], 
      HadamardBoundIntMat := [ "1.6-3", "chap1_mj.html#X7CDF3D2081207080" ], 
      HermiteIntMatLLL := [ "1.5-2", "chap1_mj.html#X7C6AE6777B72F9D2" ], 
      HermiteIntMatLLLTrans := [ "1.5-3", "chap1_mj.html#X862962B7878A284F" ],
      InfoEDIM := [ "1.1-1", "chap1_mj.html#X7AF83FEC7CD0311C" ], 
      InverseIntMatMod := [ "1.6-2", "chap1_mj.html#X78518E1D81435762" ], 
      InverseRatMat := [ "1.3-1", "chap1_mj.html#X7A9656D47C4D2D16" ], 
      RankMod := [ "1.6-5", "chap1_mj.html#X8312EDA78209B4EA" ], 
      RatNumberFromModular := [ "1.6-1", "chap1_mj.html#X7EC844F97C469C03" ], 
      RationalSolutionIntMat := [ "1.3-2", "chap1_mj.html#X8302B31E86B3AFDB" ]
        , ("Sect-ElDivPad") := [ "1.4", "chap1_mj.html#X7A6548FA7C837D27" ], 
      ("Sect-Install") := [ "1.1", "chap1_mj.html#X84D3F5E77E3BF046" ], 
      ("Sect-InvRatMat") := [ "1.3", "chap1_mj.html#X80FF39C07E03D7EF" ], 
      ("Sect-InvRatMatAlg") := [ "1.7", "chap1_mj.html#X83FD1AAB7EB2E934" ], 
      ("Sect-NFIntMatLLL") := [ "1.5", "chap1_mj.html#X788047737FA04422" ], 
      ("Sect-PPElDiv") := [ "1.2", "chap1_mj.html#X818BE04687230849" ], 
      ("Sect-Util") := [ "1.6", "chap1_mj.html#X832582857EB36B23" ], 
      SmithIntMatLLL := [ "1.5-4", "chap1_mj.html#X8626F15179C09798" ], 
      SmithIntMatLLLTrans := [ "1.5-5", "chap1_mj.html#X86094B1B7C87EBF6" ], 
      ("Ssect-rankintmat") := [ "1.7-1", "chap1_mj.html#X791ED7D97F87FDFD" ] )
    , 
  labeltexts := 
    rec( 
      ("Chap-EDIM") := "<span class=\"Heading\">The <strong class=\"pkg\">EDIM\
</strong>-Package</span>", 
      ("Sect-ElDivPad") := "<span class=\"Heading\">All Elementary Divisors Us\
ing p-adic Method</span>", 
      ("Sect-Install") := "<span class=\"Heading\">Installation of the <strong\
 class=\"pkg\">EDIM</strong> package</span>", 
      ("Sect-InvRatMat") := "<span class=\"Heading\">Inverse of Rational Matri\
ces</span>", 
      ("Sect-InvRatMatAlg") := "<span class=\"Heading\">InverseRatMat - the Al\
gorithm</span>", 
      ("Sect-NFIntMatLLL") := "<span class=\"Heading\">Gcd and Normal Forms Us\
ing LLL</span>", 
      ("Sect-PPElDiv") := "<span class=\"Heading\"><span class=\"SimpleMath\">\
\\(p\\)</span>-Parts of Elementary Divisors</span>", 
      ("Sect-Util") := "<span class=\"Heading\">Utility Functions from the <st\
rong class=\"pkg\">EDIM</strong>-package</span>", 
      ("Ssect-rankintmat") := "<span class=\"Heading\">Rank of Integer Matrix<\
/span>" ), linelength := 76, mathmode := "MathJax", name := "WHOLEDOCUMENT", 
  next := 30231, root := ~, 
  six := 
    [ 
      [ "Title page", ".", [ 0, 0, 0 ], 1, 1, "title page", 
          "X7D2C85EC87DD46E5" ], 
      [ "Copyright", ".-1", [ 0, 0, 1 ], 28, 2, "copyright", 
          "X81488B807F2A1CF1" ], 
      [ "Table of Contents", ".-2", [ 0, 0, 2 ], 39, 3, "table of contents", 
          "X8537FEB07AF2BEC8" ], 
      [ 
          "\033[1X\033[33X\033[0;-2YThe \033[5XEDIM\033[105X\033[101X\027\033[1X\\
027-Package\033[133X\033[101X", "1", [ 1, 0, 0 ], 1, 4, "the edim-package", 
          "X7AA826067CC8C395" ], 
      [ 
          "\033[1X\033[33X\033[0;-2YInstallation of the \033[5XEDIM\033[105X\033[\
101X\027\033[1X\027 package\033[133X\033[101X", "1.1", [ 1, 1, 0 ], 39, 4, 
          "installation of the edim package", "X84D3F5E77E3BF046" ], 
      [ 
          "\033[1X\033[33X\033[0;-2Y\033[22Xp\033[122X\033[101X\027\033[1X\027-Pa\
rts of Elementary Divisors\033[133X\033[101X", "1.2", [ 1, 2, 0 ], 72, 5, 
          "p-parts of elementary divisors", "X818BE04687230849" ], 
      [ 
          "\033[1X\033[33X\033[0;-2YInverse of Rational Matrices\033[133X\033[101\
X", "1.3", [ 1, 3, 0 ], 162, 6, "inverse of rational matrices", 
          "X80FF39C07E03D7EF" ], 
      [ 
          "\033[1X\033[33X\033[0;-2YAll Elementary Divisors Using p-adic Method\\
033[133X\033[101X", "1.4", [ 1, 4, 0 ], 232, 7, 
          "all elementary divisors using p-adic method", "X7A6548FA7C837D27" ]
        , 
      [ 
          "\033[1X\033[33X\033[0;-2YGcd and Normal Forms Using LLL\033[133X\033[1\
01X", "1.5", [ 1, 5, 0 ], 309, 8, "gcd and normal forms using lll", 
          "X788047737FA04422" ], 
      [ 
          "\033[1X\033[33X\033[0;-2YUtility Functions from the \033[5XEDIM\033[10\
5X\033[101X\027\033[1X\027-package\033[133X\033[101X", "1.6", [ 1, 6, 0 ], 
          390, 10, "utility functions from the edim-package", 
          "X832582857EB36B23" ], 
      [ 
          "\033[1X\033[33X\033[0;-2YInverseRatMat - the Algorithm\033[133X\033[10\
1X", "1.7", [ 1, 7, 0 ], 466, 11, "inverseratmat - the algorithm", 
          "X83FD1AAB7EB2E934" ], 
      [ "\033[1X\033[33X\033[0;-2YRank of Integer Matrix\033[133X\033[101X", 
          "1.7-1", [ 1, 7, 1 ], 529, 12, "rank of integer matrix", 
          "X791ED7D97F87FDFD" ], 
      [ "Bibliography", "Bib", [ "Bib", 0, 0 ], 1, 13, "bibliography", 
          "X7A6F98FD85F02BFE" ], 
      [ "References", "Bib", [ "Bib", 0, 0 ], 1, 13, "references", 
          "X7A6F98FD85F02BFE" ], 
      [ "Index", "Ind", [ "Ind", 0, 0 ], 1, 14, "index", "X83A0356F839C696F" ]
        , 
      [ "License", ".-1", [ 0, 0, 1 ], 28, 2, "license", "X81488B807F2A1CF1" ]
        , 
      [ "\033[5XEDIM\033[105X", "1.", [ 1, 0, 0 ], 1, 4, "edim", 
          "X7AA826067CC8C395" ], 
      [ "\033[2XInfoEDIM\033[102X", "1.1-1", [ 1, 1, 1 ], 64, 5, "infoedim", 
          "X7AF83FEC7CD0311C" ], 
      [ "\033[2XElementaryDivisorsPPartRk\033[102X", "1.2-1", [ 1, 2, 1 ], 
          77, 5, "elementarydivisorsppartrk", "X813B0D73868CD751" ], 
      [ "\033[2XElementaryDivisorsPPartRkI\033[102X", "1.2-1", [ 1, 2, 1 ], 
          77, 5, "elementarydivisorsppartrki", "X813B0D73868CD751" ], 
      [ "\033[2XElementaryDivisorsPPartRkII\033[102X", "1.2-1", [ 1, 2, 1 ], 
          77, 5, "elementarydivisorsppartrkii", "X813B0D73868CD751" ], 
      [ "\033[2XElementaryDivisorsPPartRkExp\033[102X", "1.2-1", [ 1, 2, 1 ], 
          77, 5, "elementarydivisorsppartrkexp", "X813B0D73868CD751" ], 
      [ "\033[2XElementaryDivisorsPPartRkExpSmall\033[102X", "1.2-1", 
          [ 1, 2, 1 ], 77, 5, "elementarydivisorsppartrkexpsmall", 
          "X813B0D73868CD751" ], 
      [ "\033[2XElementaryDivisorsPPartHavasSterling\033[102X", "1.2-2", 
          [ 1, 2, 2 ], 134, 6, "elementarydivisorspparthavassterling", 
          "X82EC4724865F4DF9" ], 
      [ "\033[2XInverseRatMat\033[102X", "1.3-1", [ 1, 3, 1 ], 165, 6, 
          "inverseratmat", "X7A9656D47C4D2D16" ], 
      [ "\033[2XRationalSolutionIntMat\033[102X", "1.3-2", [ 1, 3, 2 ], 184, 
          7, "rationalsolutionintmat", "X8302B31E86B3AFDB" ], 
      [ "\033[2XExponentSquareIntMatFullRank\033[102X", "1.3-3", [ 1, 3, 3 ], 
          200, 7, "exponentsquareintmatfullrank", "X86DA61D978D2D889" ], 
      [ "\033[2XElementaryDivisorsSquareIntMatFullRank\033[102X", "1.4-1", 
          [ 1, 4, 1 ], 240, 7, "elementarydivisorssquareintmatfullrank", 
          "X7B6A3B8486872B0C" ], 
      [ "\033[2XElementaryDivisorsIntMatDeterminant\033[102X", "1.4-2", 
          [ 1, 4, 2 ], 279, 8, "elementarydivisorsintmatdeterminant", 
          "X821E30477A5DCE68" ], 
      [ "\033[2XGcdexIntLLL\033[102X", "1.5-1", [ 1, 5, 1 ], 325, 9, 
          "gcdexintlll", "X799B1A5285D00859" ], 
      [ "\033[2XHermiteIntMatLLL\033[102X", "1.5-2", [ 1, 5, 2 ], 338, 9, 
          "hermiteintmatlll", "X7C6AE6777B72F9D2" ], 
      [ "\033[2XHermiteIntMatLLLTrans\033[102X", "1.5-3", [ 1, 5, 3 ], 345, 
          9, "hermiteintmatllltrans", "X862962B7878A284F" ], 
      [ "\033[2XSmithIntMatLLL\033[102X", "1.5-4", [ 1, 5, 4 ], 363, 9, 
          "smithintmatlll", "X8626F15179C09798" ], 
      [ "\033[2XSmithIntMatLLLTrans\033[102X", "1.5-5", [ 1, 5, 5 ], 370, 10, 
          "smithintmatllltrans", "X86094B1B7C87EBF6" ], 
      [ "\033[2XRatNumberFromModular\033[102X", "1.6-1", [ 1, 6, 1 ], 393, 
          10, "ratnumberfrommodular", "X7EC844F97C469C03" ], 
      [ "\033[2XInverseIntMatMod\033[102X", "1.6-2", [ 1, 6, 2 ], 405, 10, 
          "inverseintmatmod", "X78518E1D81435762" ], 
      [ "\033[2XHadamardBoundIntMat\033[102X", "1.6-3", [ 1, 6, 3 ], 419, 10, 
          "hadamardboundintmat", "X7CDF3D2081207080" ], 
      [ "\033[2XCheapFactorsInt\033[102X", "1.6-4", [ 1, 6, 4 ], 427, 11, 
          "cheapfactorsint", "X7BAB977C7EB05067" ], 
      [ "\033[2XRankMod\033[102X", "1.6-5", [ 1, 6, 5 ], 439, 11, "rankmod", 
          "X8312EDA78209B4EA" ] ], 
  sixcount := [ [ 0, 0, 0 ], [ 0, 0, 1 ], [ 0, 0, 1 ], [ 0, 0, 2 ], 
      [ 1, 0, 0 ], [ 1, 0, 0 ], [ 1, 1, 0 ], [ 1, 1, 1 ], [ 1, 2, 0 ], 
      [ 1, 2, 1 ], [ 1, 2, 1 ], [ 1, 2, 1 ], [ 1, 2, 1 ], [ 1, 2, 1 ], 
      [ 1, 2, 2 ], [ 1, 3, 0 ], [ 1, 3, 1 ], [ 1, 3, 2 ], [ 1, 3, 3 ], 
      [ 1, 4, 0 ], [ 1, 4, 1 ], [ 1, 4, 2 ], [ 1, 5, 0 ], [ 1, 5, 1 ], 
      [ 1, 5, 2 ], [ 1, 5, 3 ], [ 1, 5, 4 ], [ 1, 5, 5 ], [ 1, 6, 0 ], 
      [ 1, 6, 1 ], [ 1, 6, 2 ], [ 1, 6, 3 ], [ 1, 6, 4 ], [ 1, 6, 5 ], 
      [ 1, 7, 0 ], [ 1, 7, 1 ], [ "Bib", 0, 0 ], [ "Bib", 0, 0 ], 
      [ "Ind", 0, 0 ] ], 
  sixindex := [ 1, 2, 16, 3, 4, 17, 5, 18, 6, 23, 22, 21, 20, 19, 24, 7, 25, 
      26, 27, 8, 28, 29, 9, 30, 31, 32, 33, 34, 10, 35, 36, 37, 38, 39, 11, 
      12, 13, 14, 15 ], start := 0, stop := 30230, 
  toctext := "<div class=\"ContChap\"><a href=\"chap1_mj.html#X7AA826067CC8C39\
5\">1 <span class=\"Heading\">The <strong class=\"pkg\">EDIM</strong>-Package<\
/span></a>\n<div class=\"ContSect\"><span class=\"tocline\"><span class=\"nocs\
s\">&nbsp;</span><a href=\"chap1_mj.html#X84D3F5E77E3BF046\">1.1 <span class=\
\"Heading\">Installation of the <strong class=\"pkg\">EDIM</strong> package</s\
pan></a>\n</span>\n<div class=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><\
span class=\"nocss\">&nbsp;&nbsp;</span><a href=\"chap1_mj.html#X7AF83FEC7CD03\
11C\">1.1-1 InfoEDIM</a></span>\n</div></div>\n<div class=\"ContSect\"><span c\
lass=\"tocline\"><span class=\"nocss\">&nbsp;</span><a href=\"chap1_mj.html#X8\
18BE04687230849\">1.2 <span class=\"Heading\"><span class=\"SimpleMath\">\\(p\
\\)</span>-Parts of Elementary Divisors</span></a>\n</span>\n<div class=\"Cont\
SSBlock\">\n<span class=\"ContSS\"><br /><span class=\"nocss\">&nbsp;&nbsp;</s\
pan><a href=\"chap1_mj.html#X813B0D73868CD751\">1.2-1 ElementaryDivisorsPPartR\
k</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&nbsp;&nbsp;<\
/span><a href=\"chap1_mj.html#X82EC4724865F4DF9\">1.2-2 ElementaryDivisorsPPar\
tHavasSterling</a></span>\n</div></div>\n<div class=\"ContSect\"><span class=\
\"tocline\"><span class=\"nocss\">&nbsp;</span><a href=\"chap1_mj.html#X80FF39\
C07E03D7EF\">1.3 <span class=\"Heading\">Inverse of Rational Matrices</span></\
a>\n</span>\n<div class=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><span c\
lass=\"nocss\">&nbsp;&nbsp;</span><a href=\"chap1_mj.html#X7A9656D47C4D2D16\">\
1.3-1 InverseRatMat</a></span>\n<span class=\"ContSS\"><br /><span class=\"noc\
ss\">&nbsp;&nbsp;</span><a href=\"chap1_mj.html#X8302B31E86B3AFDB\">1.3-2 Rati\
onalSolutionIntMat</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocs\
s\">&nbsp;&nbsp;</span><a href=\"chap1_mj.html#X86DA61D978D2D889\">1.3-3 Expon\
entSquareIntMatFullRank</a></span>\n</div></div>\n<div class=\"ContSect\"><spa\
n class=\"tocline\"><span class=\"nocss\">&nbsp;</span><a href=\"chap1_mj.html\
#X7A6548FA7C837D27\">1.4 <span class=\"Heading\">All Elementary Divisors Using\
 p-adic Method</span></a>\n</span>\n<div class=\"ContSSBlock\">\n<span class=\
\"ContSS\"><br /><span class=\"nocss\">&nbsp;&nbsp;</span><a href=\"chap1_mj.h\
tml#X7B6A3B8486872B0C\">1.4-1 ElementaryDivisorsSquareIntMatFullRank</a></span\
>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&nbsp;&nbsp;</span><a hr\
ef=\"chap1_mj.html#X821E30477A5DCE68\">1.4-2 ElementaryDivisorsIntMatDetermina\
nt</a></span>\n</div></div>\n<div class=\"ContSect\"><span class=\"tocline\"><\
span class=\"nocss\">&nbsp;</span><a href=\"chap1_mj.html#X788047737FA04422\">\
1.5 <span class=\"Heading\">Gcd and Normal Forms Using LLL</span></a>\n</span>\
\n<div class=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><span class=\"nocs\
s\">&nbsp;&nbsp;</span><a href=\"chap1_mj.html#X799B1A5285D00859\">1.5-1 Gcdex\
IntLLL</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&nbsp;&n\
bsp;</span><a href=\"chap1_mj.html#X7C6AE6777B72F9D2\">1.5-2 HermiteIntMatLLL<\
/a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&nbsp;&nbsp;</s\
pan><a href=\"chap1_mj.html#X862962B7878A284F\">1.5-3 HermiteIntMatLLLTrans</a\
></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&nbsp;&nbsp;</spa\
n><a href=\"chap1_mj.html#X8626F15179C09798\">1.5-4 SmithIntMatLLL</a></span>\
\n<span class=\"ContSS\"><br /><span class=\"nocss\">&nbsp;&nbsp;</span><a hre\
f=\"chap1_mj.html#X86094B1B7C87EBF6\">1.5-5 SmithIntMatLLLTrans</a></span>\n</\
div></div>\n<div class=\"ContSect\"><span class=\"tocline\"><span class=\"nocs\
s\">&nbsp;</span><a href=\"chap1_mj.html#X832582857EB36B23\">1.6 <span class=\
\"Heading\">Utility Functions from the <strong class=\"pkg\">EDIM</strong>-pac\
kage</span></a>\n</span>\n<div class=\"ContSSBlock\">\n<span class=\"ContSS\">\
<br /><span class=\"nocss\">&nbsp;&nbsp;</span><a href=\"chap1_mj.html#X7EC844\
F97C469C03\">1.6-1 RatNumberFromModular</a></span>\n<span class=\"ContSS\"><br\
 /><span class=\"nocss\">&nbsp;&nbsp;</span><a href=\"chap1_mj.html#X78518E1D8\
1435762\">1.6-2 InverseIntMatMod</a></span>\n<span class=\"ContSS\"><br /><spa\
n class=\"nocss\">&nbsp;&nbsp;</span><a href=\"chap1_mj.html#X7CDF3D2081207080\
\">1.6-3 HadamardBoundIntMat</a></span>\n<span class=\"ContSS\"><br /><span cl\
ass=\"nocss\">&nbsp;&nbsp;</span><a href=\"chap1_mj.html#X7BAB977C7EB05067\">1\
.6-4 CheapFactorsInt</a></span>\n<span class=\"ContSS\"><br /><span class=\"no\
css\">&nbsp;&nbsp;</span><a href=\"chap1_mj.html#X8312EDA78209B4EA\">1.6-5 Ran\
kMod</a></span>\n</div></div>\n<div class=\"ContSect\"><span class=\"tocline\"\
><span class=\"nocss\">&nbsp;</span><a href=\"chap1_mj.html#X83FD1AAB7EB2E934\
\">1.7 <span class=\"Heading\">InverseRatMat - the Algorithm</span></a>\n</spa\
n>\n<div class=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><span class=\"no\
css\">&nbsp;&nbsp;</span><a href=\"chap1_mj.html#X791ED7D97F87FDFD\">1.7-1 <sp\
an class=\"Heading\">Rank of Integer Matrix</span></a>\n</span>\n</div></div>\
\n</div>\n<div class=\"ContChap\"><a href=\"chapBib_mj.html\"><span class=\"He\
ading\">References</span></a></div>\n<div class=\"ContChap\"><a href=\"chapInd\
_mj.html\"><span class=\"Heading\">Index</span></a></div>\n<div class=\"ContCh\
ap\"><a href=\"chap1_mj.html#X7AA826067CC8C395\">1 <span class=\"Heading\">The\
 <strong class=\"pkg\">EDIM</strong>-Package</span></a>\n<div class=\"ContSect\
\"><span class=\"tocline\"><span class=\"nocss\">&nbsp;</span><a href=\"chap1_\
mj.html#X84D3F5E77E3BF046\">1.1 <span class=\"Heading\">Installation of the <s\
trong class=\"pkg\">EDIM</strong> package</span></a>\n</span>\n<div class=\"Co\
ntSSBlock\">\n<span class=\"ContSS\"><br /><span class=\"nocss\">&nbsp;&nbsp;<\
/span><a href=\"chap1_mj.html#X7AF83FEC7CD0311C\">1.1-1 InfoEDIM</a></span>\n<\
/div></div>\n<div class=\"ContSect\"><span class=\"tocline\"><span class=\"noc\
ss\">&nbsp;</span><a href=\"chap1_mj.html#X818BE04687230849\">1.2 <span class=\
\"Heading\"><span class=\"SimpleMath\">\\(p\\)</span>-Parts of Elementary Divi\
sors</span></a>\n</span>\n<div class=\"ContSSBlock\">\n<span class=\"ContSS\">\
<br /><span class=\"nocss\">&nbsp;&nbsp;</span><a href=\"chap1_mj.html#X813B0D\
73868CD751\">1.2-1 ElementaryDivisorsPPartRk</a></span>\n<span class=\"ContSS\
\"><br /><span class=\"nocss\">&nbsp;&nbsp;</span><a href=\"chap1_mj.html#X82E\
C4724865F4DF9\">1.2-2 ElementaryDivisorsPPartHavasSterling</a></span>\n</div><\
/div>\n<div class=\"ContSect\"><span class=\"tocline\"><span class=\"nocss\">&\
nbsp;</span><a href=\"chap1_mj.html#X80FF39C07E03D7EF\">1.3 <span class=\"Head\
ing\">Inverse of Rational Matrices</span></a>\n</span>\n<div class=\"ContSSBlo\
ck\">\n<span class=\"ContSS\"><br /><span class=\"nocss\">&nbsp;&nbsp;</span><\
a href=\"chap1_mj.html#X7A9656D47C4D2D16\">1.3-1 InverseRatMat</a></span>\n<sp\
an class=\"ContSS\"><br /><span class=\"nocss\">&nbsp;&nbsp;</span><a href=\"c\
hap1_mj.html#X8302B31E86B3AFDB\">1.3-2 RationalSolutionIntMat</a></span>\n<spa\
n class=\"ContSS\"><br /><span class=\"nocss\">&nbsp;&nbsp;</span><a href=\"ch\
ap1_mj.html#X86DA61D978D2D889\">1.3-3 ExponentSquareIntMatFullRank</a></span>\
\n</div></div>\n<div class=\"ContSect\"><span class=\"tocline\"><span class=\"\
nocss\">&nbsp;</span><a href=\"chap1_mj.html#X7A6548FA7C837D27\">1.4 <span cla\
ss=\"Heading\">All Elementary Divisors Using p-adic Method</span></a>\n</span>\
\n<div class=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><span class=\"nocs\
s\">&nbsp;&nbsp;</span><a href=\"chap1_mj.html#X7B6A3B8486872B0C\">1.4-1 Eleme\
ntaryDivisorsSquareIntMatFullRank</a></span>\n<span class=\"ContSS\"><br /><sp\
an class=\"nocss\">&nbsp;&nbsp;</span><a href=\"chap1_mj.html#X821E30477A5DCE6\
8\">1.4-2 ElementaryDivisorsIntMatDeterminant</a></span>\n</div></div>\n<div c\
lass=\"ContSect\"><span class=\"tocline\"><span class=\"nocss\">&nbsp;</span><\
a href=\"chap1_mj.html#X788047737FA04422\">1.5 <span class=\"Heading\">Gcd and\
 Normal Forms Using LLL</span></a>\n</span>\n<div class=\"ContSSBlock\">\n<spa\
n class=\"ContSS\"><br /><span class=\"nocss\">&nbsp;&nbsp;</span><a href=\"ch\
ap1_mj.html#X799B1A5285D00859\">1.5-1 GcdexIntLLL</a></span>\n<span class=\"Co\
ntSS\"><br /><span class=\"nocss\">&nbsp;&nbsp;</span><a href=\"chap1_mj.html#\
X7C6AE6777B72F9D2\">1.5-2 HermiteIntMatLLL</a></span>\n<span class=\"ContSS\">\
<br /><span class=\"nocss\">&nbsp;&nbsp;</span><a href=\"chap1_mj.html#X862962\
B7878A284F\">1.5-3 HermiteIntMatLLLTrans</a></span>\n<span class=\"ContSS\"><b\
r /><span class=\"nocss\">&nbsp;&nbsp;</span><a href=\"chap1_mj.html#X8626F151\
79C09798\">1.5-4 SmithIntMatLLL</a></span>\n<span class=\"ContSS\"><br /><span\
BUILDSTDERR:  class=\"nocss\">&nbsp;&nb+ rm -fr ../../doc ../pkg
+ RPM_EC=0
BUILDSTDERR: ++ jobs -p
+ exit 0
sp;</span><a href=\"chap1_mj.html#X86094B1B7C87EBF6\
\">1.5-5 SmithIntMatLLLTrans</a></span>\n</div></div>\n<div class=\"ContSect\"\
><span class=\"tocline\"><span class=\"nocss\">&nbsp;</span><a href=\"chap1_mj\
.html#X832582857EB36B23\">1.6 <span class=\"Heading\">Utility Functions from t\
he <strong class=\"pkg\">EDIM</strong>-package</span></a>\n</span>\n<div class\
=\"ContSSBlock\">\n<span class=\"ContSS\"><br /><span class=\"nocss\">&nbsp;&n\
bsp;</span><a href=\"chap1_mj.html#X7EC844F97C469C03\">1.6-1 RatNumberFromModu\
lar</a></span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&nbsp;&nbsp\
;</span><a href=\"chap1_mj.html#X78518E1D81435762\">1.6-2 InverseIntMatMod</a>\
</span>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&nbsp;&nbsp;</span\
><a href=\"chap1_mj.html#X7CDF3D2081207080\">1.6-3 HadamardBoundIntMat</a></sp\
an>\n<span class=\"ContSS\"><br /><span class=\"nocss\">&nbsp;&nbsp;</span><a \
href=\"chap1_mj.html#X7BAB977C7EB05067\">1.6-4 CheapFactorsInt</a></span>\n<sp\
an class=\"ContSS\"><br /><span class=\"nocss\">&nbsp;&nbsp;</span><a href=\"c\
hap1_mj.html#X8312EDA78209B4EA\">1.6-5 RankMod</a></span>\n</div></div>\n<div \
class=\"ContSect\"><span class=\"tocline\"><span class=\"nocss\">&nbsp;</span>\
<a href=\"chap1_mj.html#X83FD1AAB7EB2E934\">1.7 <span class=\"Heading\">Invers\
eRatMat - the Algorithm</span></a>\n</span>\n<div class=\"ContSSBlock\">\n<spa\
n class=\"ContSS\"><br /><span class=\"nocss\">&nbsp;&nbsp;</span><a href=\"ch\
ap1_mj.html#X791ED7D97F87FDFD\">1.7-1 <span class=\"Heading\">Rank of Integer \
Matrix</span></a>\n</span>\n</div></div>\n</div>\n<div class=\"ContChap\"><a h\
ref=\"chapBib_mj.html\"><span class=\"Heading\">References</span></a></div>\n<\
div class=\"ContChap\"><a href=\"chapInd_mj.html\"><span class=\"Heading\">Ind\
ex</span></a></div>\n" )
gap> gap> gap> gap> ─────────────────────────────────────────────────────────────────────────────
Loading  EDIM 1.3.3 (Elementary Divisors of Integer Matrices)
by Frank Lübeck (http://www.math.rwth-aachen.de/~Frank.Luebeck).
Homepage: http://www.math.rwth-aachen.de/~Frank.Luebeck/EDIM
─────────────────────────────────────────────────────────────────────────────
#I  File: /builddir/build/BUILD/EDIM-1.3.3/../pkg/EDIM-1.3.3/doc/manual.lab written.
gap> gap> gap> gap> Executing(%install): /bin/sh -e /var/tmp/rpm-tmp.esI1Zp
+ umask 022
+ cd /builddir/build/BUILD
+ '[' /builddir/build/BUILDROOT/gap-pkg-edim-1.3.3-6.fc31.riscv64 '!=' / ']'
+ rm -rf /builddir/build/BUILDROOT/gap-pkg-edim-1.3.3-6.fc31.riscv64
BUILDSTDERR: ++ dirname /builddir/build/BUILDROOT/gap-pkg-edim-1.3.3-6.fc31.riscv64
+ mkdir -p /builddir/build/BUILDROOT
+ mkdir /builddir/build/BUILDROOT/gap-pkg-edim-1.3.3-6.fc31.riscv64
+ cd EDIM-1.3.3
+ mkdir -p /builddir/build/BUILDROOT/gap-pkg-edim-1.3.3-6.fc31.riscv64/usr/lib/gap/pkg/EDIM-1.3.3/bin/riscv64-redhat-linux-gnu-default64-kv3
+ cp -p bin/riscv64-redhat-linux-gnu-default64-kv3/.libs/ediv.so /builddir/build/BUILDROOT/gap-pkg-edim-1.3.3-6.fc31.riscv64/usr/lib/gap/pkg/EDIM-1.3.3/bin/riscv64-redhat-linux-gnu-default64-kv3
+ cp -a doc lib tst VERSION PackageInfo.g init.g makedocrel.g read.g /builddir/build/BUILDROOT/gap-pkg-edim-1.3.3-6.fc31.riscv64/usr/lib/gap/pkg/EDIM-1.3.3
+ rm -f /builddir/build/BUILDROOT/gap-pkg-edim-1.3.3-6.fc31.riscv64/usr/lib/gap/pkg/EDIM-1.3.3/doc/clean
+ rm -f /builddir/build/BUILDROOT/gap-pkg-edim-1.3.3-6.fc31.riscv64/usr/lib/gap/pkg/EDIM-1.3.3/doc/edim.aux /builddir/build/BUILDROOT/gap-pkg-edim-1.3.3-6.fc31.riscv64/usr/lib/gap/pkg/EDIM-1.3.3/doc/edim.bbl /builddir/build/BUILDROOT/gap-pkg-edim-1.3.3-6.fc31.riscv64/usr/lib/gap/pkg/EDIM-1.3.3/doc/edim.blg /builddir/build/BUILDROOT/gap-pkg-edim-1.3.3-6.fc31.riscv64/usr/lib/gap/pkg/EDIM-1.3.3/doc/edim.brf /builddir/build/BUILDROOT/gap-pkg-edim-1.3.3-6.fc31.riscv64/usr/lib/gap/pkg/EDIM-1.3.3/doc/edim.idx /builddir/build/BUILDROOT/gap-pkg-edim-1.3.3-6.fc31.riscv64/usr/lib/gap/pkg/EDIM-1.3.3/doc/edim.ilg /builddir/build/BUILDROOT/gap-pkg-edim-1.3.3-6.fc31.riscv64/usr/lib/gap/pkg/EDIM-1.3.3/doc/edim.ind /builddir/build/BUILDROOT/gap-pkg-edim-1.3.3-6.fc31.riscv64/usr/lib/gap/pkg/EDIM-1.3.3/doc/edim.log /builddir/build/BUILDROOT/gap-pkg-edim-1.3.3-6.fc31.riscv64/usr/lib/gap/pkg/EDIM-1.3.3/doc/edim.out /builddir/build/BUILDROOT/gap-pkg-edim-1.3.3-6.fc31.riscv64/usr/lib/gap/pkg/EDIM-1.3.3/doc/edim.pnr
+ /usr/lib/rpm/find-debuginfo.sh -j4 --strict-build-id -m -i --build-id-seed 1.3.3-6.fc31 --unique-debug-suffix -1.3.3-6.fc31.riscv64 --unique-debug-src-base gap-pkg-edim-1.3.3-6.fc31.riscv64 --run-dwz --dwz-low-mem-die-limit 10000000 --dwz-max-die-limit 50000000 -S debugsourcefiles.list /builddir/build/BUILD/EDIM-1.3.3
BUILDSTDERR: 17 blocks
+ /usr/lib/rpm/check-buildroot
+ /usr/lib/rpm/redhat/brp-ldconfig
+ /usr/lib/rpm/brp-compress
+ /usr/lib/rpm/brp-strip-static-archive /usr/bin/strip
+ /usr/lib/rpm/redhat/brp-python-bytecompile /usr/bin/python 1 0
+ /usr/lib/rpm/brp-python-hardlink
+ /usr/lib/rpm/redhat/brp-mangle-shebangs
explicitly decompress any DWARF compressed ELF sections in /builddir/build/BUILDROOT/gap-pkg-edim-1.3.3-6.fc31.riscv64/usr/lib/gap/pkg/EDIM-1.3.3/bin/riscv64-redhat-linux-gnu-default64-kv3/ediv.so
extracting debug info from /builddir/build/BUILDROOT/gap-pkg-edim-1.3.3-6.fc31.riscv64/usr/lib/gap/pkg/EDIM-1.3.3/bin/riscv64-redhat-linux-gnu-default64-kv3/ediv.so
original debug info size: 84kB, size after compression: 88kB
/usr/lib/rpm/sepdebugcrcfix: Updated 1 CRC32s, 0 CRC32s did match.
Executing(%check): /bin/sh -e /var/tmp/rpm-tmp.eiePNo
+ umask 022
+ cd /builddir/build/BUILD
+ cd EDIM-1.3.3
+ export LC_ALL=C.UTF-8
+ LC_ALL=C.UTF-8
+ gap -q -l '/builddir/build/BUILDROOT/gap-pkg-edim-1.3.3-6.fc31.riscv64/usr/lib/gap;/usr/lib/gap'
+ tee log
+ grep -Fq false log
+ rm -f log
+ RPM_EC=0
BUILDSTDERR: ++ jobs -p
+ exit 0
#I  autpgrp package is not available. Check that the name is correct
#I  and it is present in one of the GAP root directories (see '??RootPaths')
#I  alnuth package is not available. Check that the name is correct
#I  and it is present in one of the GAP root directories (see '??RootPaths')
#I  crisp package is not available. Check that the name is correct
#I  and it is present in one of the GAP root directories (see '??RootPaths')
#I  ctbllib package is not available. Check that the name is correct
#I  and it is present in one of the GAP root directories (see '??RootPaths')
#I  factint package is not available. Check that the name is correct
#I  and it is present in one of the GAP root directories (see '??RootPaths')
#I  fga package is not available. Check that the name is correct
#I  and it is present in one of the GAP root directories (see '??RootPaths')
#I  irredsol package is not available. Check that the name is correct
#I  and it is present in one of the GAP root directories (see '??RootPaths')
#I  laguna package is not available. Check that the name is correct
#I  and it is present in one of the GAP root directories (see '??RootPaths')
#I  polenta package is not available. Check that the name is correct
#I  and it is present in one of the GAP root directories (see '??RootPaths')
#I  polycyclic package is not available. Check that the name is correct
#I  and it is present in one of the GAP root directories (see '??RootPaths')
#I  resclasses package is not available. Check that the name is correct
#I  and it is present in one of the GAP root directories (see '??RootPaths')
#I  sophus package is not available. Check that the name is correct
#I  and it is present in one of the GAP root directories (see '??RootPaths')
#I  tomlib package is not available. Check that the name is correct
#I  and it is present in one of the GAP root directories (see '??RootPaths')
Reading 34x34 integer matrix 'mat2' with elementary divisors 'eldiv2'.
true
From here you should only see a sequence of `true' lines,
until the session quits: (after a few minutes)
true
true
true
true
true
true
true
true
true
true
true
true
true
true
true
true
true
true
true
true
true
true
true
true
true
true
true
true
true
true
true
true
true
true
true
true
Processing files: gap-pkg-edim-1.3.3-6.fc31.riscv64
Executing(%doc): /bin/sh -e /var/tmp/rpm-tmp.YP7K4n
+ umask 022
+ cd /builddir/build/BUILD
+ cd EDIM-1.3.3
+ DOCDIR=/builddir/build/BUILDROOT/gap-pkg-edim-1.3.3-6.fc31.riscv64/usr/share/doc/gap-pkg-edim
+ export LC_ALL=C
+ LC_ALL=C
+ export DOCDIR
+ /usr/bin/mkdir -p /builddir/build/BUILDROOT/gap-pkg-edim-1.3.3-6.fc31.riscv64/usr/share/doc/gap-pkg-edim
+ cp -pr CHANGES /builddir/build/BUILDROOT/gap-pkg-edim-1.3.3-6.fc31.riscv64/usr/share/doc/gap-pkg-edim
+ cp -pr README /builddir/build/BUILDROOT/gap-pkg-edim-1.3.3-6.fc31.riscv64/usr/share/doc/gap-pkg-edim
+ cp -pr TODO /builddir/build/BUILDROOT/gap-pkg-edim-1.3.3-6.fc31.riscv64/usr/share/doc/gap-pkg-edim
+ RPM_EC=0
BUILDSTDERR: ++ jobs -p
+ exit 0
Executing(%license): /bin/sh -e /var/tmp/rpm-tmp.xXhr6m
+ umask 022
+ cd /builddir/build/BUILD
+ cd EDIM-1.3.3
+ LICENSEDIR=/builddir/build/BUILDROOT/gap-pkg-edim-1.3.3-6.fc31.riscv64/usr/share/licenses/gap-pkg-edim
+ export LC_ALL=C
+ LC_ALL=C
+ export LICENSEDIR
+ /usr/bin/mkdir -p /builddir/build/BUILDROOT/gap-pkg-edim-1.3.3-6.fc31.riscv64/usr/share/licenses/gap-pkg-edim
+ cp -pr GPL /builddir/build/BUILDROOT/gap-pkg-edim-1.3.3-6.fc31.riscv64/usr/share/licenses/gap-pkg-edim
+ RPM_EC=0
BUILDSTDERR: ++ jobs -p
+ exit 0
Provides: gap-pkg-edim = 1.3.3-6.fc31 gap-pkg-edim(riscv-64) = 1.3.3-6.fc31
Requires(rpmlib): rpmlib(CompressedFileNames) <= 3.0.4-1 rpmlib(FileDigests) <= 4.6.0-1 rpmlib(PayloadFilesHavePrefix) <= 4.0-1
Requires: libc.so.6()(64bit) libc.so.6(GLIBC_2.27)(64bit) rtld(GNU_HASH)
Processing files: gap-pkg-edim-doc-1.3.3-6.fc31.noarch
Provides: gap-pkg-edim-doc = 1.3.3-6.fc31
Requires(rpmlib): rpmlib(CompressedFileNames) <= 3.0.4-1 rpmlib(FileDigests) <= 4.6.0-1 rpmlib(PayloadFilesHavePrefix) <= 4.0-1
Processing files: gap-pkg-edim-debugsource-1.3.3-6.fc31.riscv64
Provides: gap-pkg-edim-debugsource = 1.3.3-6.fc31 gap-pkg-edim-debugsource(riscv-64) = 1.3.3-6.fc31
Requires(rpmlib): rpmlib(CompressedFileNames) <= 3.0.4-1 rpmlib(FileDigests) <= 4.6.0-1 rpmlib(PayloadFilesHavePrefix) <= 4.0-1
Processing files: gap-pkg-edim-debuginfo-1.3.3-6.fc31.riscv64
Provides: debuginfo(build-id) = 2d7a56d745ecd60de2fa0b3eb2f7a8e18083d14d gap-pkg-edim-debuginfo = 1.3.3-6.fc31 gap-pkg-edim-debuginfo(riscv-64) = 1.3.3-6.fc31
Requires(rpmlib): rpmlib(CompressedFileNames) <= 3.0.4-1 rpmlib(FileDigests) <= 4.6.0-1 rpmlib(PayloadFilesHavePrefix) <= 4.0-1
Recommends: gap-pkg-edim-debugsource(riscv-64) = 1.3.3-6.fc31
Checking for unpackaged file(s): /usr/lib/rpm/check-files /builddir/build/BUILDROOT/gap-pkg-edim-1.3.3-6.fc31.riscv64
Wrote: /builddir/build/RPMS/gap-pkg-edim-debugsource-1.3.3-6.fc31.riscv64.rpm
Wrote: /builddir/build/RPMS/gap-pkg-edim-debuginfo-1.3.3-6.fc31.riscv64.rpm
Wrote: /builddir/build/RPMS/gap-pkg-edim-1.3.3-6.fc31.riscv64.rpm
Wrote: /builddir/build/RPMS/gap-pkg-edim-doc-1.3.3-6.fc31.noarch.rpm
Executing(%clean): /bin/sh -e /var/tmp/rpm-tmp.HS3v4n
+ umask 022
+ cd /builddir/build/BUILD
+ cd EDIM-1.3.3
+ /usr/bin/rm -rf /builddir/build/BUILDROOT/gap-pkg-edim-1.3.3-6.fc31.riscv64
+ RPM_EC=0
BUILDSTDERR: ++ jobs -p
+ exit 0
Child return code was: 0