<Section Label="sec:overview">
<Heading>Overview over this manual</Heading>
Chapter <Ref Chap="chap:intro"/> is concerned with the technical details of
installing and running this package. Chapter <Ref Chap="chap:EGF"/>
answers the question why and how the &GAP; functionality concerning a
sparse matrix type and gaussian algorithms was extended. The following
chapters are concerned with the workings of the sparse matrix type
(<Ref Chap="chap:SM"/>) and sparse Gaussian algorithms (<Ref
Chap="chap:Gauss"/>). Included is a documented list of the most
important methods and functions needed to work with sparse matrices
and the algorithms provided by the &Gauss; package. Anyone interested
in source code should just check out the files in the
<F>gap/pkg/Gauss/gap/</F> folder (&see; Appendix <Ref Label="FileOverview"/>).
<Section Label="sec:need"><Heading>The need for extended functionality</Heading>
&GAP; has a lot of functionality for row echelon forms of
matrices. These can be called by <C>SemiEchelonForm</C> and
similar commands. All of these work for the &GAP; matrix type over
fields. However, these algorithms are not capable of computing a
reduced row echelon form (RREF) of a matrix, there is no way to "Gauss upwards". While this is not neccessary for things like Rank
or Kernel computations, this was one in a number of missing features
important for the development of the &GAP; package &homalg; by
M. Barakat <Cite Key="homalg-package"/>.<P/><P/>
Parallel to this development I worked on &SCO; <Cite Key="SCO"/>,
a package for creating simplicial sets and computing the
cohomology of orbifolds, based on the paper "Simplicial Cohomology
of Orbifolds" by I. Moerdijk and D. A. Pronk
Key="MP_SCO"/>. Very early on it became clear that the cohomology
matrices (with entries in &ZZ; or finite quotients of &ZZ;) would
grow exponentially in size with the cohomology degree. At one
point in time, for example, a 50651 x 1133693 matrix had to be
handled.<P/><P/>
It should be quite clear that there was a need for a sparse matrix
data type and corresponding Gaussian algorithms. After an
unfruitful search for a computer algebra system capable of this
task, the &Gauss; package was born - to provide not only the
missing RREF algorithms, but also support a new data type,
enabling &GAP; to handle sparse matrices of almost arbritrary
size.<P/><P/>
I am proud to tell you that, thanks to optimizing the algorithms
for matrices over GF(2), it was possible to compute the GF(2)-Rank
of the matrix mentioned above in less than 20 minutes with a
memory usage of about 3 GB.
</Section>
<Section Label="sec:app"><Heading>The applications of the &Gauss; package algorithms</Heading>
Please refer to <Cite Key="homalg-project"/> to find out more about the
&homalg; project and its related packages. Most of the motivation
for the algorithms in the &Gauss; package can be found there. If
you are interested in this project, you might also want to check
out my &GaussForHomalg; <Cite Key="GaussForHomalg"/> package,
which, just as &RingsForHomalg; <Cite Key="RingsForHomalg"/> does
for external Rings, serves as the connection between &homalg;
and &Gauss;. By allowing &homalg; to delegate computational tasks
to &Gauss; this small package extends &homalg;'s capabilities to
dense and sparse matrices over fields and rings of the form
<M>&ZZ; / \langle p^n \rangle</M>.<P/>
For those unfamiliar with the &homalg; project let me explain a
couple of points. As outlined in <Cite Key="BR"/> by D. Robertz
and M. Barakat homological computations can be reduced to three
basic tasks:<P/>
<List>
<Item>Computing a row basis of a module (<C>BasisOfRowModule</C>).</Item>
<Item>Reducing a module with a basis (<C>DecideZeroRows</C>).</Item>
<Item>Compute the relations between module elements (<C>SyzygiesGeneratorsOfRows</C>).</Item>
</List>
In addition to these tasks only relatively easy tools for matrix
manipulation are needed, ranging from addition and multiplication
to finding the zero rows in a matrix. However, to reduce the need for
communication it might be helpful to supply &homalg; with some
more advanced procedures.<P/><P/>
While the above tasks can be quite difficult when, for example,
working in noncommutative polynomial rings, in the &Gauss; case
they can all be done as long as you can compute a Reduced Row
Echelon Form. This is clear for <C>BasisOfRowModule</C>, as the
rows of the RREF of the matrix are already a basis of the
module. <Ref Meth="EchelonMat"/> is used to compute RREFs, based
on the &GAP; internal method <C>SemiEchelonMat</C> for Row Echelon
Forms.<P/><P/>
Lets look at the second point, the basic function
<C>DecideZeroRows</C>: When you face the task of reducing a module
<M>A</M> with a given basis <M>B</M>, you can compute the RREF of
the following block matrix:
<Table Align="|c|c|">
<HorLine/>
<Row>
<Item><Alt Not="LaTeX">Id</Alt>
<Alt Only="LaTeX"><![CDATA[
$\begin{array}{ccc}
1&\\
&\ddots&\\
&&1\\
\end{array}$
]]></Alt></Item>
<Item>A</Item>
</Row>
<HorLine/>
<Row>
<Item>0</Item>
<Item>B</Item>
</Row>
<HorLine/>
</Table>
By computing the RREF (notice how important "Gaussing upwards" is
here) <M>A</M> is reduced with <M>B</M>. However, the left side of
the matrix just serves the single purpose of tricking the Gaussian
algorithms into doing what we want. Therefore, it was a logical
step to implement <Ref Meth="ReduceMat"/>, which does the same
thing but without needing unneccessary columns.<P/>
Note: When, much later, it became clear that it was important to compute
the transformation matrices of the reduction, <Ref
Meth="ReduceMatTransformation"/> was born, similar to <Ref
Meth="EchelonMatTransformation"/>. This corresponds to the
&homalg; procedure <C>DecideZeroRowsEffectively</C>.<P/><P/>
The third procedure, <C>SygygiesGeneratorsOfRows</C>, is concerned with the
relations between rows of a matrix, each row representing a module element. Over a field these relations are exactly the kernel of
the matrix. One can easily see that this can be achieved by taking
a matrix
<Table Align="|c|c|">
<HorLine/>
<Row>
<Item>A</Item>
<Item><Alt Not="LaTeX">Id</Alt>
<Alt Only="LaTeX"><![CDATA[
$\begin{array}{ccc}
1&\\
&\ddots&\\
&&1\\
\end{array}$
]]></Alt></Item>
</Row>
<HorLine/>
</Table>
and computing its Row Echelon Form. Then the row relations are
generated by the rows to the right of the zero rows of the
REF. There are two problems with this approach: The computation
diagonalizes the kernel, which might not be wanted, and, much
worse, it does not work at all for rings with zero divisors. For
example, the <M>1 \times 1</M> matrix <M>[2 + 8&ZZ;]</M> has a row
relation <M>[4 + 8&ZZ;]</M> which would not have been found by
this method.<P/>
Approaching this problem led to the method <Ref
Meth="EchelonMatTransformation"/>, which additionally computes the
transformation matrix <M>T</M>, such that RREF <M>= T \cdot M</M>.
Similar to <C>SemiEchelonMatTransformation</C>, <M>T</M> is split
up into the rows needed to create the basis vectors of the RREF,
and the relations that led to zero rows. Focussing on the
computations over fields, it was an easy step to write <Ref
Meth="KernelMat"/>, which terminates after the REF and returns the
kernel generators.<P/>
The syzygy computation over <M>&ZZ; / \langle p^n \rangle</M> was solved by
carefully keeping track of basis vectors with a zero-divising
head. If, for <M> v = (0,\ldots,0,h,*,\ldots,*), h \neq 0,</M>
there exists <M>g \neq 0</M> such that <M>g \cdot h = 0</M>, the
vector <M>g \cdot v</M> is regarded as an additional row vector
which has to be reduced and can be reduced with. After some more
work this allowed for the implementation of <Ref
Meth="KernelMat"/> for matrices over <M>&ZZ; / \langle p^n \rangle</M>.<P/>
This concludes the explanation of the so-called basic tasks
&Gauss; has to handle when called by &homalg; to do matrix
calculations. Here is a tabular overview of the current
capabilities of &Gauss; (<M>p</M> is a prime, <M>n \in &NN;</M>):<P/>
As you can see, the development of hermite algorithms was not
continued for dense matrices. There are two reasons for that:
&GAP; already has very good algorithms for &ZZ;, and for small
matrices the disadvantage of computing over &ZZ;, potentially
leading to coefficient explosion, is marginal.
</Section>
</Chapter>
<Chapter Label="chap:SM"><Heading>The Sparse Matrix Data Type</Heading>
<Section Label="sec:workings"><Heading>The inner workings of &Gauss;
sparse matrices</Heading>
When doing any kind of computation there is a constant conflict
between memory load and speed. On the one hand, memory usage is
bounded by the total available memory, on the other hand,
computation time should also not exceed certain
proportions. Memory usage and CPU time are generally
inversely proportional, because the computer needs more time to
perform operations on a compactified data structure. The
idea of sparse matrices mirrors exactly the need for less memory
load, therefore it is natural that sparse algorithms take more
time than dense ones. However, if the matrix is sufficiently large
and sparse at the same time, sparse algorithms can easily be
faster than dense ones while maintaining minimal memory load.<P/>
It should be noted that, although matrices that appear naturally
in homological algebra are almost always sparse, they do not have
to stay sparse under (R)REF algorithms, especially when the
computation is concerned with transformation matrices. Therefore,
in a perfect world there should be ways implemented to not only
find out which data structure to use, but also at what point to
convert from one to the other. This was, however, not the aim of
the &Gauss; package and is just one of many points in which this
package could be optimized or extended.
The matrix <M>M</M> carries the same information as the following table,
if and only if you know how many rows and columns the matrix
has. There is also the matter of the base ring, but this is not
important for now:
This table relates each index tuple to its nonzero entry, all
other matrix entries are defined to be zero. This only works for
known dimensions of the matrix, otherwise trailing zero rows and
columns could get lost (notice how the table gives no hint about
the existence of a 5th column). To convert the above table into a
sparse data structure, one could list the table entries in this
way:<P/>
However, this data structure would not be very efficient. Whenever
you are interested in a row <M>i</M> of <M>M</M> (this happens all the time
when performing Gaussian elimination) the whole list would have
to be searched for 3-tuples of the form <M>[ i, *, *
]</M>. This is why I tried to manage the row index by putting the
tuples into the corresponding list entry:<Br/>
As you can see, this looks fairly complicated. However, the same
information can be stored in this form, which would become the
final data structure for &Gauss; sparse matrices:
Although now the number of rows is equal to the Length of both
`indices' and `entries', it is still stored in the sparse
matrix. Here is the full data structure (&see;
<Ref Func="SparseMatrix" Label="constructor using gap matrices"/>):
As you can see, the matrix stores its ring to be on the safe side.
This is especially important for zero matrices, as there is no way
to determine the base ring from the sparse matrix structure. For
further information on sparse matrix construction and converting,
refer to <Ref Func="SparseMatrix" Label="constructor using gap matrices"/>.
<Subsection Label="sub:gf2"><Heading>A special case: GF(2)</Heading>
Because the nonzero entries of a matrix over GF(2) are all "1",
the entries of M are not stored at all. It is of course crucial
that all operations and algorithms make 100% sure that all
appearing zero entries are deleted from the `indices' as well as
the `entries' list as they arise.
<Section Label="sec:list"><Heading>A list of the available algorithms</Heading>
As decribed earlier, the main functions of &Gauss; are <Ref
Meth="EchelonMat"/> and <Ref Meth="EchelonMatTransformation"/>,
<Ref Meth="ReduceMat"/> and <Ref Meth="ReduceMatTransformation"/>,
<Ref Meth="KernelMat"/> and, additionally <Ref Meth="Rank"/>.
These are all documented in the next section, but of course rely on
specific algorithms depending on the base ring of the matrix. These
are not fully documented but it should be very easy to find out how
they work based on the documentation of the main functions.
<Section Label="sec:mfGauss"><Heading>Methods and Functions for &Gauss;ian algorithms</Heading>
<#Include Label="EchelonMat">
<#Include Label="EchelonMatTransformation">
<#Include Label="ReduceMat">
<#Include Label="ReduceMatTransformation">
<#Include Label="KernelMat">
<#Include Label="Rank">
</Section>
</Chapter>
</Body>
<Appendix Label="FileOverview">
<Heading>An Overview of the &Gauss; package source code</Heading>
<Table Align="l|l">
<Caption><E>The &Gauss; package files.</E></Caption>
<Row><Item>Filename</Item><Item>Content</Item></Row>
<HorLine/>
<Row><Item>SparseMatrix.gi</Item><Item>Definitions and methods for
the sparse matrix type</Item></Row>
<Row><Item>SparseMatrixGF2.gi</Item><Item>Special case GF(2): no
matrix entries needed</Item></Row>
<Row><Item>GaussDense.gi</Item><Item>Gaussian elmination for &GAP;
matrices over fields</Item></Row>
<Row><Item>Sparse.gi</Item><Item>Documentation and forking depending
on the base ring</Item></Row>
<Row><Item>GaussSparse.gi</Item><Item>Gaussian elimination for sparse
matrices over fields</Item></Row>
<Row><Item>HermiteSparse.gi</Item><Item>Hermite elimination for sparse
matrices over <M>&ZZ; / \langle p^n \rangle</M></Item></Row>
</Table>
</Appendix>
<Bibliography Databases="GaussBib.xml"/>
<TheIndex/>
</Book>
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