<Emph>(Elementary Divisors and Integer Matrices, by Frank
Lübeck)</Emph><P/>
This chapter describes the functions defined in the &GAP;4 package
<Package>EDIM</Package>. The main functions implement variants of an
algorithm for computing for a given prime <M>p</M> the <M>p</M>-parts of
the elementary divisors of an integer matrix. These algorithms use
a <M>p</M>-adic method and are described by the author in <Cite
Key="L98"/> (see <Ref Func="ElementaryDivisorsPPartRk"/>).<P/>
These functions were already applied to integer matrices of dimension
greater than <M>11000</M> (which had many non-trivial elementary
divisors which were products of small primes).<P/>
Furthermore there are functions for finding the biggest elementary
divisor of an invertible integer matrix and the inverse of a rational
invertible matrix (see <Ref Func="ExponentSquareIntMatFullRank"/> and
<Ref Func="InverseRatMat"/>). These algorithms use <M>p</M>-adic
approximations, explained in <Ref Sect="Sect-InvRatMatAlg"/>.<P/>
Finally we distribute implementations of some other algorithms for
finding elementary divisors or normal forms of integer matrices: A
<M>p</M>-modular algorithm by Havas and Sterling from <Cite
Key="HS79"/> (see <Ref Func="ElementaryDivisorsPPartHavasSterling"/>)
and LLL-based algorithms for extended greatest common divisors of
integers (see <Ref Func="GcdexIntLLL"/>) and for Hermite normal forms
of integer matrices with (very nice) transforming matrices (see <Ref
Func="HermiteIntMatLLL"/>).<P/>
Please, send me an e-mail
(<Email>Frank.Luebeck@Math.RWTH-Aachen.De</Email>) if you have any
questions, remarks, suggestions, etc. concerning this mini-package.
Also, I would like to hear about applications of this package.<P/>
Frank Lübeck
<Section Label="Sect-Install">
<Heading>Installation of the <Package>EDIM</Package> package</Heading>
To install this package first unpack it inside some GAP root directory
in the subdirectory <F>pkg</F> (see
<Ref Sect="Installing a GAP Package" BookName="Reference"/>). Then
the <Package>EDIM</Package> package can already be loaded and used.
But we strongly recommend to compile a kernel function as well during
installation, otherwise
the function <Ref Func="ElementaryDivisorsPPartRkExpSmall"/>
will not be available.<P/>
To install the kernel function go to the directory <F>pkg/EDIM-...</F> to
which the package was extracted and call<P/>
<C>/bin/sh ./configure [path]</C><P/>
where <C>path</C> is a path to the main <Package>GAP</Package> root
directory (if not given, the default <F>../..</F> is assumed). <!-- Adjust later If you have installed several GAP kernels you can compile the corresponding <Package>EDIM</Package> kernel function by specifying the <C>CONFIGNAME</C> that was used to configure that kernel.<P/>
-->
Afterwards call <C>make</C> to compile a binary file. <P/>
If you have installed several GAP kernels repeat these two steps for each
of them. <!-- , using the various values of <C>CONFIGNAME</C>.-->
You can run a test of the installation by typing <C>make test</C>.
<ManSection>
<InfoClass Name="InfoEDIM"/>
<Description>
This is an <K>Info</K> class for the <Package>EDIM</Package>-package.
By <C>SetInfoLevel(InfoEDIM, 1);</C> you can switch on the printing of
some information during the computations of certain
<Package>EDIM</Package>-functions.
</Description>
</ManSection>
</Section>
<!-- ############################################################ -->
<Section Label="Sect-PPElDiv">
<Heading><M>p</M>-Parts of Elementary Divisors</Heading>
Here we explain the main functions of the package.
<Description> These functions return a list <M>[m_1, m_2, \ldots,
m_r]</M> where <M>m_i</M> is the number of nonzero elementary divisors
of <A>A</A> divisible by <M><A>p</A>^i</M> (see <Ref BookName="ref"
Func="ElementaryDivisorsMat" /> for a definition of the elementary
divisors).<P/>
The algorithms for these functions are described in <Cite
Key="L98"/>.<P/>
<A>A</A> must be a matrix with integer entries, <A>p</A> a prime, and
<A>rk</A> the rank of <A>A</A> (as rational matrix). In the first version of the command <A>rk</A> is computed, if it is not given. <P/>
The first version of the command delegates its job to the fourth version by trying growing values for <A>exp</A>, see below. <P/>
The second and third versions implement the main algorithm described
in <Cite Key="L98"/> and a variation. Here <Ref
Func="ElementaryDivisorsPPartRkII"/> has a bit more overhead, but can
be advantageous because the intermediate entries during the
computation can be much smaller.<P/>
In the fourth form <A>exp</A> must be an upper bound for the highest
power of <A>p</A> appearing in an elementary divisor of <A>A</A>.
This information allows reduction of matrix entries modulo
<M><A>p</A>^{{<A>exp+1</A>}}</M> during the computation. <P/>
If <A>exp</A> is too small or the given <A>rk</A> is too large the
function returns <K>fail</K>. <P/>
If <M><A>p</A>^{<A>exp</A>}</M> is small enough we use internally a
kernel function which can also be used directly in the fifth form of
the command. There <A>il</A> can be <M>0</M> or <M>1</M> where in the
second case some information is printed during the computation.<P/>
More precisely, <Ref Func="ElementaryDivisorsPPartRkExpSmall"/> is
applicable when <M><A>p</A>^{{<A>exp</A>+1}} < 2^{{b-4}}</M> and
<M>(<A>p</A>^{{<A>exp</A>+1}} - 1)(p-1) < 2^b</M> where <M>b=32</M>
in a 32-bit version of &GAP; and <M>b=64</M> in a 64-bit version of
&GAP;.<P/>
This last form of the function was already succesfully applied to
dense matrices of rank up to <M>11000</M>.<P/>
Note that you have to compile a file (see <Ref Sect="Sect-Install"/>)
while installing this package, if you want to have this kernel
function available.<P/>
<Description>
For an integer matrix <A>A</A> and a prime <A>p</A> this function
returns a list <M>[m_1, m_2, \ldots, m_r]</M> where <M>m_i</M> is the
number of nonzero elementary divisors of <A>A</A> divisible by
<M><A>p</A>^i</M>. <P/>
An upper bound <A>d</A> for the highest power of <A>p</A> appearing in
an elementary divisor of <A>A</A> must be given. Smaller <A>d</A>
improve the performance of the algorithm considerably.<P/>
This is an implementation of the modular algorithm described in <Cite
Key="HS79"/>. <P/>
We added a slight improvement: we divide the considered submatrices by
the <A>p</A>-part of the greatest common divisor of all entries (and
lower the <A>d</A> appropriately). This reduces the size of the
entries and often shortens the pivot search.<P/>
<Section Label="Sect-InvRatMat">
<Heading>Inverse of Rational Matrices</Heading>
<ManSection>
<Func Name="InverseRatMat"
Arg="A[, p ]"
Comm="inverse of a rational matrix"/>
<Description>
This function returns the inverse of an invertible matrix over the
rational numbers.<P/>
It first computes the inverse modulo some prime <A>p</A>, computes
from this a <A>p</A>-adic approximation to the inverse and finally
constructs the rational entries from their <A>p</A>-adic
approximations. See section <Ref Sect="Sect-InvRatMatAlg"/> for more
details.<P/>
This seems to be better than <Package>GAP</Package>'s
standard Gauß algorithm (<C>A^-1</C>) already for small
matrices. (Try, e.g., <C>RandomMat(20,20,[-10000..10000])</C> or
<C>RandomMat(100,100)</C>.)<P/>
The optional argument <A>p</A> should be a prime such that <A>A</A>
modulo <A>p</A> is invertible (default is <M><A>p</A>=251</M>). If
<A>A</A> is not invertible modulo <A>p</A> then <A>p</A> is
automatically replaced by the next prime.<P/>
</Description>
</ManSection>
<!-- ############################################################ -->
<ManSection>
<Func Name="RationalSolutionIntMat"
Arg="A, v[, p[, invA ] ]"
Comm="rational solution of linear system over integers"/>
<Description>
This function returns the solution <M>x</M> of the system of linear
equations <M>x <A>A</A> = <A>v</A></M>.<P/>
Here, <A>A</A> must be a matrix with integer entries which is invertible
over the rationals and <A>v</A> must be a vector with integer entries of
the appropriate length.<P/>
The optional arguments are a prime <A>p</A> such that <M><A>A</A>
\pmod{p}</M> is invertible (if not given, <M>p = 251</M> is assumed)
and the inverse <A>invA</A> of <M><A>A</A> \pmod{p}</M>.<P/>
The solution is computed via <M>p</M>-adic approximation as explained in
<Ref Sect="Sect-InvRatMatAlg"/>.<P/>
</Description>
</ManSection>
<ManSection>
<Func Name="ExponentSquareIntMatFullRank"
Arg=" A[, p[, nr ] ]"
Comm=""/>
<Description>
This function returns the biggest elementary divisor of a square
integer matrix <A>A</A> of full rank.<P/>
For such a matrix <A>A</A> the least common multiple of the
denominators of all entries of the inverse matrix <M><A>A</A>^{-1}</M>
is exactly the biggest elementary divisor of <A>A</A>.<P/>
This function is implemented by a slight modification of <Ref
Func="InverseRatMat"/>. The third argument <A>nr</A> tells the
function to return the least common multiple of the first <A>nr</A>
rows of the rational inverse matrix only. Very often the function
will already return the biggest elementary divisor with
<M><A>nr</A>=2</M> or <M>3</M> (and the command without this argument
would spend most time in checking, that this is correct).<P/>
The optional argument <A>p</A> should be a prime such that <A>A</A>
modulo <A>p</A> is invertible (default is <M><A>p</A>=251</M>). If
<A>A</A> is not invertible modulo <A>p</A> then <A>p</A> is
automatically replaced by the next prime.<P/>
<Example>
gap> ReadPackage("edim", "tst/mat");
Reading 242x242 integer matrix 'mat' with elementary divisors 'eldiv'.
true
gap> inv := InverseRatMat(mat);; time;
840
gap> ExponentSquareIntMatFullRank(mat, 101, 3); # same as without the `3'
115200
</Example>
</Description>
</ManSection>
<Section Label="Sect-ElDivPad">
<Heading>All Elementary Divisors Using p-adic Method</Heading>
In the following two functions we put things together. In particular
we handle the prime parts of the elementary divisors efficiently for
primes appearing with low powers in the highest elementary divisor
respectively determinant divisor.
<ManSection>
<Func Name="ElementaryDivisorsSquareIntMatFullRank"
Arg=" A " />
<Description>
This function returns a list of nonzero elementary divisors of an
integer matrix <A>A</A>.<P/>
Here we start with computing the biggest elementary divisor via <Ref
Func="ExponentSquareIntMatFullRank"/>. If it runs into a problem
because <A>A</A> is singular modulo a choosen prime (it starts by
default with 251) then the prime is automatically replaced by the next
one.<P/>
The rest is done using <Ref Func="ElementaryDivisorsPPartRkExp"/> and
<Ref Func="RankMod"/>.<P/>
The function fails if the biggest elementary divisor cannot be
completely factored and the non-factored part is not a divisor of the
biggest elementary divisor only.<P/>
Note that this function may for many matrices not be the best choice
for computing all elementary divisors. You may first try the standard
<Package>GAP</Package> library routines for Smith normal form instead
of this function. Nevertheless remember <Ref
Func="ElementaryDivisorsSquareIntMatFullRank"/> for hard and big
examples. It is particularly good when the largest elementary divisor
is a very small factor of the determinant.
<ManSection>
<Func Name="ElementaryDivisorsIntMatDeterminant"
Arg=" A, det[, rk] " />
<Description>
This function returns a list of nonzero elementary divisors of an
integer matrix <A>A</A>.<P/>
Here <A>det</A> must be an integer which is a multiple of the biggest
determinant divisor of <A>A</A>. If the matrix does not have full
rank then its rank <A>rk</A> must be given, too.<P/>
The argument <A>det</A> can be given in the form of
<C>Collected(FactorsInt(<A>det</A>))</C>.<P/>
This function handles prime divisors of <A>det</A> with multiplicity
smaller than 4 specially, for the other prime divisors <M>p</M> it
delegates to <Ref Func="ElementaryDivisorsPPartRkExp"/> where the
<A>exp</A> argument is the multiplicity of the <M>p</M> in <A>det</A>.
(Note that this is not very good when <M>p</M> has actually a much
smaller multiplicity in the largest elementary divisor.)<P/>
<Section Label="Sect-NFIntMatLLL">
<Heading>Gcd and Normal Forms Using LLL</Heading>
The <Package>EDIM</Package>-mini package also contains implementations
of an extended Gcd-algorithm for integers and a Hermite and Smith
normal form algorithm for integer matrices using LLL-techiques. They
are well described in the paper <Cite Key="HMM98"/> by Havas, Majewski
and Matthews.<P/>
They are particularly useful if one wants to have the normal forms
together with transforming matrices. These transforming matrices have
spectacularly nice (i.e., <Q>small</Q>) entries in cases of input
matrices which are non-square or not of full rank (otherwise the
transformation to the Hermite normal form is unique).<P/>
<Description>
This function returns for integers <M><A>n1</A>, <A>n2</A>, \ldots</M> a
list <M>[g, [c_1, c_2, \ldots]]</M>, where <M>g = c_1<A>n1</A> +
c_2<A>n2</A> + \ldots</M> is the greatest common divisor of the
<A>ni</A>. Here all the <M>c_i</M> are usually very small.
<Description> This returns the Hermite normal form of an integer
matrix <A>A</A> and uses the LLL-algorithm to avoid entry explosion.
</Description>
</ManSection>
<ManSection>
<Func Name="HermiteIntMatLLLTrans"
Arg=" A " />
<Description>
This function returns a pair of matrices <M>[H, L]</M> where <M>H = L
<A>A</A></M> is the Hermite normal form of an integer matrix <A>A</A>.
The transforming matrix <M>L</M> can have surprisingly small entries.
<ManSection>
<Func Name="SmithIntMatLLL"
Arg=" A " />
<Description>
This function returns the Smith normal form of an integer matrix
<A>A</A> using the LLL-algorithm to avoid entry explosion.
</Description>
</ManSection>
<ManSection>
<Func Name="SmithIntMatLLLTrans"
Arg=" A " />
<Description>
This function returns <M>[S, L, R]</M> where <M>S = L <A>A</A> R</M>
is the Smith normal form of an integer matrix <A>A</A>.<P/>
We apply the algorithm for Hermite normal form several times to get
the Smith normal form, that is not in the paper <Cite
Key="HMM98"/>. The transforming matrices need not be as nice as for
the Hermite normal form.<P/>
<Description>
This function returns <M>r/s = <A>x</A> \pmod{<A>n</A>}</M>, if
it exists. More precisely:<P/>
<A>n</A>, <A>k</A>, <A>l</A> must be positive integers with
<M>2<A>k</A><A>l</A> \leq <A>n</A></M> and <A>x</A> an integer with
<M>-<A>n</A>/2 < <A>x</A> \leq <A>n</A>/2</M>. If it exists
this function returns a rational number <M>r/s</M> with <M>0 <
s < <A>l</A></M>, <M>\gcd(s, <A>n</A>) = 1</M>, <M>-<A>k</A>
< r < <A>k</A></M> and <M>r/s</M> congruent to <M><A>x</A>
\pmod{<A>n</A>}</M> (i.e., <M><A>n</A> \mid r - s <A>x</A></M>). Such
an <M>r/s</M> is unique. The function returns <K>fail</K> if such a
number does not exist.<P/>
</Description>
</ManSection>
<Description>
This function returns an inverse matrix modulo a prime <A>p</A> or
<K>fail</K>. More precisely:<P/>
<A>A</A> must be an integer matrix and <A>p</A> a prime such that
<A>A</A> is invertible modulo <A>p</A>. This function returns an
integer matrix <A>inv</A> with entries in the range <M>]-<A>p</A>/2
\ldots <A>p</A>/2]</M> such that <A>inv</A><A>A</A> reduced modulo p
is the identity matrix.<P/>
It returns <K>fail</K> if the inverse modulo <A>p</A> does not exist.
This function is particularly fast for primes smaller 256.
</Description>
</ManSection>
<Description>
The Hadamard bound for a square integer matrix <A>A</A> is the product
of Euclidean norms of the nonzero rows (or columns) of <A>A</A>. It is
an upper bound for the absolute value of the determinant of <A>A</A>.
</Description>
</ManSection>
<Description>
This function returns a list of factors of an integer <A>n</A>,
including <Q>small</Q> prime factors - here the optional argument
<A>nr</A> is the number of iterations for `FactorsRho' (default is
2000).<P/>
This is only a slight modification of the library function <Ref
Func="FactorsInt" BookName="ref"/> which avoids an error message when
the number is not completely factored.<P/>
</Description>
</ManSection>
<Description>
This function returns the rank of an integer matrix <A>A</A> modulo
<A>p</A>. Here <A>p</A> must not necessarily be a prime. If it is not
and this function returns an integer, then this is the rank of
<A>A</A> for all prime divisors of <A>p</A>.<P/>
If during the computation a factorisation of <A>p</A> is found
(because some pivot entry has nontrivial greatest common divisor with
<A>p</A>) then the function is recursively applied to the found
factors <C>f_i</C> of <A>p</A>. The result is then given in the form
<C>[[f_1, rk_1], [f_2, rk_2], ...]</C>.<P/>
The idea to make this function useful for non primes was to use it
with large factors of the biggest elementary divisor of <A>A</A> whose
prime factorization cannot be found easily.<P/>
<Section Label="Sect-InvRatMatAlg">
<Heading>InverseRatMat - the Algorithm</Heading>
The idea is to recover a rational matrix from an <M>l</M>-adic
approximation for some prime <M>l</M>. This description came out of
discussions with Jürgen Müller. I thank John Cannon for pointing out
that the basic idea already appeared in the paper <Cite Key="D82"/> of
Dixon.<P/>
Let <M>A</M> be an invertible matrix over the rational numbers. By
multiplying with a constant we may assume that its entries are in fact
integers.<P/>
(1) We first describe how to find an <M>l</M>-adic approximation of
<M>A^{-1}</M>. Find a prime <M>l</M> such that <M>A</M> is invertible
modulo <M>l</M> and let <M>B</M> be the integer matrix with entries in
the range <M>\left]-l/2,l/2\right]</M> such that <M>BA</M> is
congruent to the identity matrix modulo <M>l</M>. (This can be
computed fast by usual Gauß elimination.)<P/>
Now let <M>v \in &ZZ;^r</M> be a row vector. Define two sequences
<M>v_i</M> and <M>x_i</M> of row vectors in <M>&ZZ;^r</M> by: <M>x_0
:= 0 \in &ZZ;^r</M>, <M>v_0 := -v</M> and for <M>i > 0</M> set
<M>x_i</M> to the vector congruent to <M>-v_{i-1} B</M> modulo
<M>l</M> having entries in the range <M>\left]-l/2, l/2\right]</M>.
Then all entries of <M>x_i A + v_{i-1}</M> are divisible by <M>l</M>
and we set <M>v_i := (1/l) \cdot (x_i A + v_{i-1})</M>.<P/>
Induction shows that for <M>y_i := \sum_{k=1}^{i} l^{k-1} x_k</M> we
have <M>y_i A = v + l^i v_i</M> for all <M>i \geq 0</M>. Hence the
sequence <M>y_i</M>, <M>i \geq 0</M>, gives an <M>l</M>-adic
approximation to the vector <M>y \in &QQ;^r</M> with <M>y A =
v</M>.<P/>
(2) The second point is to show how we can get the vector <M>y</M>
from a sufficiently good approximation <M>y_i</M>. Note that the
sequence of <M>y_i</M> becomes constant for <M>i \geq i_0</M> if all
entries of <M>y</M> are integers of absolute value smaller than
<M>l^{i_0} / 2</M> because of our choice of representatives of residue
classes modulo <M>l</M> in the interval <M>\left]-l/2,
l/2\right]</M>.<P/>
More generally consider <M>a / b \in &QQ;</M> with <M>b > 0</M> and
<M>a, b</M> coprime. Then there is for each <M>n \in &NN;</M> which is
coprime to <M>b</M> a unique <M>c \in &ZZ;</M> with <M>-n / 2 < c
\leq n / 2</M> and <M>a \equiv c b \pmod{n}</M>. This <M>c</M> can be
computed via the extended Euclidean algorithm applied to <M>b</M> and
<M>n</M>.<P/>
Now let <M>n, \alpha, \beta \in &NN;</M> with <M>2 \alpha \beta \leq
n</M>. Then the map <M>\{a/b \in &QQ; \mid -\alpha \leq a \leq \alpha,
1 \leq b < \beta \} \rightarrow \left]-n/2, n/2\right]</M>, <M>a/b
\mapsto c</M> (defined as above) is injective (since for <M>a/b</M>,
<M>a'/b'</M> in the above set we have <M>a b' - a' b \equiv 0
\pmod{n}</M> if and only if <M>a b' - a' b = 0</M>).<P/>
In practice we can use for any <M>c \in \left]-n/2, n/2\right]</M> a
certain extended Euclidean algorithm applied to <M>n</M> and <M>c</M>
to decide if <M>c</M> is in the image of the above map and to find the
corresponding <M>a/b</M> if it exists.<P/>
(3) To put things together we apply (2) to the entries of the vectors
<M>y_i</M> constructed in (1), choosing <M>n = l^i</M>, <M>\alpha =
\sqrt{n}/2</M> and <M>\beta = \sqrt{n}</M>. If we have found this way
a candidate for <M>y</M> we can easily check if it is correct by
computing <M>y A</M>. If <M>\mu</M> is the maximal absolute value of
all numerators and denominators of the entries of <M>y</M> it is clear
from (2) that we will find <M>y</M> from <M>y_i</M> if <M>l^i > 2
\mu^2</M>.<P/>
(4) If we take as <M>v</M> in (1) to(3) all standard unit vectors we
clearly get the rows of <M>A^{-1}</M>. But we can do it
better. Namely we can take as <M>v</M> the standard unit vectors
multiplied by the least common multiple <M>\epsilon</M> of the
denominators of the already computed entries of <M>A^{-1}</M>. In
many examples this <M>\epsilon</M> actually equals <M>\epsilon_r</M>
after the computation of the first or first few rows. Therefore we
will often find quickly the next row of <M>A^{-1}</M> already in (1),
because we find a <M>v_i = 0</M> such that the sequence of <M>y_i</M>
becomes constant (<M>=y</M>).<P/>
<Subsection Label="Ssect-rankintmat">
<Heading>Rank of Integer Matrix</Heading>
The following strategy has shown to be useful in proving that some very
big integer matrix is not invertible.<P/>
<List>
<Item>Check the rank modulo some small primes, say with <Ref
Func="RankMod"/>. </Item>
<Item>If the rank seems less than the number of rows choose a prime
<M>p</M>, a collection of lines which is linearly independent modulo
<M>p</M>, and another line linearly dependend on these. Guess that this
last line is also linearly dependend on the chosen collection over the
rational numbers (maybe check modulo several small primes).</Item>
<Item>Find columns of the collection of lines which give an invertible
matrix modulo some prime.</Item>
<Item>Then use <Ref Func="RationalSolutionIntMat"/> with the invertible
submatrix and corresponding entries of the linearly dependend row to
prove this.</Item>
</List>
Guessing the rank of a matrix from the rank modulo several primes,
chosing a maximal set of lines which are linearly independent modulo
some primes, and using <Ref Func="RationalSolutionIntMat"/> with the
remaining lines, one may also find the exact rank of a huge integer
matrix. <P/>
</Subsection>
</Section>
</Chapter>
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