CoCalc -- Gauss.xml

GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it
gap4r8 / pkg / Gauss-2017.12.07 / doc / Gauss.xml
⁵⁶³⁵⁴⁹ views
1
<?xml version="1.0" encoding="UTF-8"?>
2

3
<!DOCTYPE Book SYSTEM "gapdoc.dtd"
4
 [ <!ENTITY see '<Alt Only="LaTeX">$\to$</Alt><Alt Not="LaTeX">--&gt;</Alt>'>
5
   <!ENTITY C "<Package>C</Package>">
6
   <!ENTITY Gauss "<Package>Gauss</Package>">
7
   <!ENTITY GaussForHomalg "<Package>GaussForHomalg</Package>">
8
   <!ENTITY homalg "<Package>homalg</Package>">
9
   <!ENTITY RingsForHomalg "<Package>RingsForHomalg</Package>">
10
   <!ENTITY SCO "<Package>SCO</Package>">
11
   <!ENTITY GAPDoc "<Package>GAPDoc</Package>">
12
 ]>
13

14
<Book Name="Gauss">
15

16
<TitlePage>
17
  <Title> The &Gauss; Package Manual</Title>
18
  <Subtitle>Extended Gauss Functionality for &GAP;</Subtitle>
19
  <Version>
20
      Version <#Include SYSTEM "../VERSION">
21
  </Version>
22
  <Author>Simon Goertzen<Alt Only="LaTeX"><Br/></Alt>
23
          <Email>[email protected]</Email>
24
          <Homepage>http://wwwb.math.rwth-aachen.de/goertzen/</Homepage>
25
	  <Address>
26
	  Lehrstuhl B für Mathematik<Br/>
27
	  Templergraben 64<Br/>
28
	  52062 Aachen<Br/>
29
	  (Germany)
30
	  </Address>
31
  </Author>
32
  <Date>March 2013</Date>
33
  <Abstract>This document explains the primary uses of the &Gauss; package.
34
            Included is a documented list of the most important methods
35
            and functions needed to work with sparse matrices and the
36
            algorithms provided by the &Gauss; package.
37
  </Abstract>
38
  <Copyright>&copyright; 2007-2013 by Simon Goertzen<P/>
39
  This package may be distributed under the terms and conditions of
40
  the GNU Public License Version 2.
41
  </Copyright>
42
  <Acknowledgements>The &Gauss; package would not have been possible without the helpful contributions by
43
   <List>
44
    <Item>Max Neunhöffer, University of St Andrews, and</Item>
45
    <Item>Mohamed Barakat, Lehrstuhl B für Mathematik, RWTH Aachen.</Item>
46
   </List>
47
   Many thanks to these two and the Lehrstuhl B für Mathematik in general.
48
   It should be noted that the &GAP; algorithms for
49
   <C>SemiEchelonForm</C> and other methods formed an important and
50
   informative basis for the development of the extended Gaussian
51
   algorithms. This manual was created with the help of the &GAPDoc;
52
   package by F. Lübeck and M. Neunhöffer <Cite Key="GAPDoc"/>.
53
  </Acknowledgements>
54
</TitlePage>
55

56
<TableOfContents/>
57

58
<Body>
59

60
<Chapter Label="chap:intro"><Heading>Introduction</Heading>
61

62
<Section Label="sec:overview">
63
<Heading>Overview over this manual</Heading>
64

65
Chapter <Ref Chap="chap:intro"/> is concerned with the technical details of
66
installing and running this package. Chapter <Ref Chap="chap:EGF"/>
67
answers the question why and how the &GAP; functionality concerning a
68
sparse matrix type and gaussian algorithms was extended. The following
69
chapters are concerned with the workings of the sparse matrix type
70
(<Ref Chap="chap:SM"/>) and sparse Gaussian algorithms (<Ref
71
Chap="chap:Gauss"/>). Included is a documented list of the most
72
important methods and functions needed to work with sparse matrices
73
and the algorithms provided by the &Gauss; package. Anyone interested
74
in source code should just check out the files in the
75
<F>gap/pkg/Gauss/gap/</F> folder (&see; Appendix <Ref Label="FileOverview"/>).
76

77
</Section>
78

79
<#Include SYSTEM "install.xml"/>
80

81
</Chapter>
82

83
<Chapter Label="chap:EGF"><Heading>Extending Gauss Functionality</Heading>
84

85
 <Section Label="sec:need"><Heading>The need for extended functionality</Heading>
86

87
    &GAP; has a lot of functionality for row echelon forms of
88
    matrices. These can be called by <C>SemiEchelonForm</C> and
89
    similar commands. All of these work for the &GAP; matrix type over
90
    fields. However, these algorithms are not capable of computing a
91
    reduced row echelon form (RREF) of a matrix, there is no way to
92
    "Gauss upwards". While this is not neccessary for things like Rank
93
    or Kernel computations, this was one in a number of missing features
94
    important for the development of the &GAP; package &homalg; by
95
    M. Barakat <Cite Key="homalg-package"/>.<P/><P/>
96
    
97
    Parallel to this development I worked on &SCO; <Cite Key="SCO"/>,
98
    a package for creating simplicial sets and computing the
99
    cohomology of orbifolds, based on the paper "Simplicial Cohomology
100
    of Orbifolds" by I. Moerdijk and D. A. Pronk <Cite
101
    Key="MP_SCO"/>. Very early on it became clear that the cohomology
102
    matrices (with entries in &ZZ; or finite quotients of &ZZ;) would
103
    grow exponentially in size with the cohomology degree. At one
104
    point in time, for example, a 50651 x 1133693 matrix had to be
105
    handled.<P/><P/>
106
    
107
    It should be quite clear that there was a need for a sparse matrix
108
    data type and corresponding Gaussian algorithms. After an
109
    unfruitful search for a computer algebra system capable of this
110
    task, the &Gauss; package was born - to provide not only the
111
    missing RREF algorithms, but also support a new data type,
112
    enabling &GAP; to handle sparse matrices of almost arbritrary
113
    size.<P/><P/>
114
    
115
    I am proud to tell you that, thanks to optimizing the algorithms
116
    for matrices over GF(2), it was possible to compute the GF(2)-Rank
117
    of the matrix mentioned above in less than 20 minutes with a
118
    memory usage of about 3 GB.
119
    
120
 </Section>
121

122
 <Section Label="sec:app"><Heading>The applications of the &Gauss; package algorithms</Heading>
123
    
124
    Please refer to <Cite Key="homalg-project"/>  to find out more about the
125
    &homalg; project and its related packages. Most of the motivation
126
    for the algorithms in the &Gauss; package can be found there. If
127
    you are interested in this project, you might also want to check
128
    out my &GaussForHomalg; <Cite Key="GaussForHomalg"/>  package,
129
    which, just as &RingsForHomalg; <Cite Key="RingsForHomalg"/> does
130
    for external Rings, serves as the connection between &homalg;
131
    and &Gauss;. By allowing &homalg; to delegate computational tasks
132
    to &Gauss; this small package extends &homalg;'s  capabilities to
133
    dense and sparse matrices over fields and rings of the form
134
    <M>&ZZ; / \langle p^n \rangle</M>.<P/>
135
    
136
    For those unfamiliar with the &homalg; project let me explain a
137
    couple of points. As outlined in <Cite Key="BR"/> by D. Robertz
138
    and M. Barakat homological computations can be reduced to three
139
    basic tasks:<P/>
140

141
    <List>
142
     <Item>Computing a row basis of a module (<C>BasisOfRowModule</C>).</Item>
143
     <Item>Reducing a module with a basis (<C>DecideZeroRows</C>).</Item>
144
     <Item>Compute the relations between module elements (<C>SyzygiesGeneratorsOfRows</C>).</Item>
145
    </List>
146
    
147
    In addition to these tasks only relatively easy tools for matrix
148
    manipulation are needed, ranging from addition and multiplication
149
    to finding the zero rows in a matrix. However, to reduce the need for
150
    communication it might be helpful to supply &homalg; with some
151
    more advanced procedures.<P/><P/>
152
    
153
    While the above tasks can be quite difficult when, for example,
154
    working in noncommutative polynomial rings, in the &Gauss; case
155
    they can all be done as long as you can compute a Reduced Row
156
    Echelon Form. This is clear for <C>BasisOfRowModule</C>, as the
157
    rows of the RREF of the matrix are already a basis of the
158
    module. <Ref Meth="EchelonMat"/> is used to compute RREFs, based
159
    on the &GAP; internal method <C>SemiEchelonMat</C> for Row Echelon
160
    Forms.<P/><P/>
161
    
162
    Lets look at the second point, the basic function
163
    <C>DecideZeroRows</C>: When you face the task of reducing a module
164
    <M>A</M> with a given basis <M>B</M>, you can compute the RREF of
165
    the following block matrix:
166
    <Table Align="|c|c|">
167
     <HorLine/>
168
     <Row>
169
      <Item><Alt Not="LaTeX">Id</Alt>
170
            <Alt Only="LaTeX"><![CDATA[
171
                    $\begin{array}{ccc}
172
                    1&\\
173
                    &\ddots&\\
174
                    &&1\\
175
                    \end{array}$
176
            ]]></Alt></Item>
177
      <Item>A</Item>
178
     </Row>
179
     <HorLine/>
180
     <Row>
181
      <Item>0</Item>
182
      <Item>B</Item>
183
     </Row>
184
     <HorLine/>
185
    </Table>
186
    By computing the RREF (notice how important "Gaussing upwards" is
187
    here) <M>A</M> is reduced with <M>B</M>. However, the left side of
188
    the matrix just serves the single purpose of tricking the Gaussian
189
    algorithms into doing what we want. Therefore, it was a logical
190
    step to implement <Ref Meth="ReduceMat"/>, which does the same
191
    thing but without needing unneccessary columns.<P/>
192
    
193
    Note: When, much later, it became clear that it was important to compute
194
    the transformation matrices of the reduction, <Ref
195
    Meth="ReduceMatTransformation"/> was born, similar to <Ref
196
    Meth="EchelonMatTransformation"/>. This corresponds to the
197
    &homalg; procedure <C>DecideZeroRowsEffectively</C>.<P/><P/>
198
    	
199
    The third procedure, <C>SygygiesGeneratorsOfRows</C>, is concerned with the
200
    relations between rows of a matrix, each row representing a module
201
    element. Over a field these relations are exactly the kernel of
202
    the matrix. One can easily see that this can be achieved by taking
203
    a matrix
204
    <Table Align="|c|c|">
205
     <HorLine/>
206
     <Row>
207
      <Item>A</Item>
208
      <Item><Alt Not="LaTeX">Id</Alt>
209
            <Alt Only="LaTeX"><![CDATA[
210
                    $\begin{array}{ccc}
211
                    1&\\
212
                    &\ddots&\\
213
                    &&1\\
214
                    \end{array}$
215
            ]]></Alt></Item>
216
     </Row>
217
     <HorLine/>
218
    </Table>
219
    and computing its Row Echelon Form. Then the row relations are
220
    generated by the rows to the right of the zero rows of the
221
    REF. There are two problems with this approach: The computation
222
    diagonalizes the  kernel, which might not be wanted, and, much
223
    worse, it does not work at all for rings with zero divisors. For
224
    example, the <M>1 \times 1</M> matrix <M>[2 + 8&ZZ;]</M> has a row
225
    relation <M>[4 + 8&ZZ;]</M> which would not have been found by
226
    this method.<P/>
227
    
228
    Approaching this problem led to the method <Ref
229
    Meth="EchelonMatTransformation"/>, which additionally computes the
230
    transformation matrix <M>T</M>, such that RREF <M>= T \cdot M</M>.
231
    Similar to <C>SemiEchelonMatTransformation</C>, <M>T</M> is split
232
    up into the rows needed to create the basis vectors of the RREF,
233
    and the relations that led to zero rows. Focussing on the
234
    computations over fields, it was an easy step to write <Ref
235
    Meth="KernelMat"/>, which terminates after the REF and returns the
236
    kernel generators.<P/>
237
    
238
    The syzygy computation over <M>&ZZ; / \langle p^n \rangle</M> was solved by
239
    carefully keeping track of basis vectors with a zero-divising
240
    head. If, for <M> v = (0,\ldots,0,h,*,\ldots,*), h \neq 0,</M>
241
    there exists <M>g \neq 0</M> such that <M>g \cdot h = 0</M>, the
242
    vector <M>g \cdot v</M> is regarded as an additional row vector
243
    which has to be reduced and can be reduced with. After some more
244
    work this allowed for the implementation of <Ref
245
    Meth="KernelMat"/> for matrices over <M>&ZZ; / \langle p^n \rangle</M>.<P/>
246
    
247
    This concludes the explanation of the so-called basic tasks
248
    &Gauss; has to handle when called by &homalg; to do matrix
249
    calculations. Here is a tabular overview of the current
250
    capabilities of &Gauss; (<M>p</M> is a prime, <M>n \in &NN;</M>):<P/>
251

252
    <Table Align="|c||c|c|c|c|c|">
253
     <HorLine/>
254
     <Row>
255
      <Item>Matrix Type:</Item>
256
      <Item>Dense</Item>
257
      <Item>Dense</Item>
258
      <Item>Sparse</Item>
259
      <Item>Sparse</Item>
260
      <Item>Sparse</Item>
261
     </Row>
262
     <HorLine/>
263
     <Row>
264
      <Item>Base Ring:</Item>
265
      <Item>Field</Item>
266
      <Item><M>&ZZ; / \langle p^n \rangle</M></Item>
267
      <Item>Field</Item>
268
      <Item>GF(2)</Item>
269
      <Item><M>&ZZ; / \langle p^n \rangle</M></Item>
270
     </Row>
271
     <HorLine/>
272
     <HorLine/>
273
     <Row>
274
      <Item>RankMat</Item>
275
      <Item>&GAP;</Item>
276
      <Item>n.a.</Item>
277
      <Item>+</Item>
278
      <Item>++</Item>
279
      <Item>n.a.</Item>
280
     </Row>
281
     <HorLine/>
282
     <Row>
283
      <Item>EchelonMat</Item>
284
      <Item>+</Item>
285
      <Item>-</Item>
286
      <Item>+</Item>
287
      <Item>++</Item>
288
      <Item>+</Item>
289
     </Row>
290
     <HorLine/>
291
     <Row>
292
      <Item>EchelonMatTransf.</Item>
293
      <Item>+</Item>
294
      <Item>-</Item>
295
      <Item>+</Item>
296
      <Item>++</Item>
297
      <Item>+</Item>
298
     </Row>
299
     <HorLine/>
300
     <Row>
301
      <Item>ReduceMat</Item>
302
      <Item>+</Item>
303
      <Item>-</Item>
304
      <Item>+</Item>
305
      <Item>++</Item>
306
      <Item>+</Item>
307
     </Row>
308
     <HorLine/>
309
     <Row>
310
      <Item>ReduceMatTransf.</Item>
311
      <Item>+</Item>
312
      <Item>-</Item>
313
      <Item>+</Item>
314
      <Item>++</Item>
315
      <Item>+</Item>
316
     </Row>
317
     <HorLine/>
318
     <Row>
319
      <Item>KernelMat</Item>
320
      <Item>+</Item>
321
      <Item>-</Item>
322
      <Item>+</Item>
323
      <Item>++</Item>
324
      <Item>+</Item>
325
     </Row>
326
     <HorLine/>
327
    </Table>
328
    
329
    As you can see, the development of hermite algorithms was not
330
    continued for dense matrices. There are two reasons for that:
331
    &GAP; already has very good algorithms for &ZZ;, and for small
332
    matrices the disadvantage of computing over &ZZ;, potentially
333
    leading to coefficient explosion, is marginal.
334
    
335
 </Section>
336

337
</Chapter>
338

339
<Chapter Label="chap:SM"><Heading>The Sparse Matrix Data Type</Heading>
340

341
 <Section Label="sec:workings"><Heading>The inner workings of &Gauss;
342
 sparse matrices</Heading>
343
    
344
    When doing any kind of computation there is a constant conflict
345
    between memory load and speed. On the one hand, memory usage is
346
    bounded by the total available memory, on the other hand,
347
    computation time should also not exceed certain
348
    proportions. Memory usage and CPU time are generally
349
    inversely proportional, because the computer needs more time to
350
    perform operations on a compactified data structure. The
351
    idea of sparse matrices mirrors exactly the need for less memory
352
    load, therefore it is natural that sparse algorithms take more
353
    time than dense ones. However, if the matrix is sufficiently large
354
    and sparse at the same time, sparse algorithms can easily be
355
    faster than dense ones while maintaining minimal memory load.<P/>
356
    
357
    It should be noted that, although matrices that appear naturally
358
    in homological algebra are almost always sparse, they do not have
359
    to stay sparse under (R)REF algorithms, especially when the
360
    computation is concerned with transformation matrices. Therefore,
361
    in a perfect world there should be ways implemented to not only
362
    find out which data structure to use, but also at what point to
363
    convert from one to the other. This was, however, not the aim of
364
    the &Gauss; package and is just one of many points in which this
365
    package could be optimized or extended.
366
    
367
    Take a look at this matrix <M>M</M>:
368

369
    <Table Align="|ccccc|">
370
     <HorLine/>
371
     <Row>
372
      <Item>0</Item><Item>0</Item><Item>2</Item><Item>9</Item><Item>0</Item>
373
     </Row>
374
     <Row>
375
      <Item>0</Item><Item>5</Item><Item>0</Item><Item>0</Item><Item>0</Item>
376
     </Row>
377
     <Row>
378
      <Item>0</Item><Item>0</Item><Item>0</Item><Item>1</Item><Item>0</Item>
379
     </Row>
380
     <HorLine/>
381
    </Table>
382
    
383
    The matrix <M>M</M> carries the same information as the following table,
384
    if and only if you know how many rows and columns the matrix
385
    has. There is also the matter of the base ring, but this is not
386
    important for now:
387
    
388
    <Table Align="|cc|">
389
     <HorLine/>
390
     <Row><Item>(i,j)</Item><Item>Entry</Item></Row>
391
     <HorLine/>
392
     <Row><Item>(1,3)</Item><Item>2</Item></Row>
393
     <Row><Item>(1,4)</Item><Item>9</Item></Row>
394
     <Row><Item>(2,2)</Item><Item>5</Item></Row>
395
     <Row><Item>(3,4)</Item><Item>1</Item></Row>
396
     <HorLine/>
397
    </Table>
398
    
399
    This table relates each index tuple to its nonzero entry, all
400
    other matrix entries are defined to be zero. This only works for
401
    known dimensions of the matrix, otherwise trailing zero rows and
402
    columns could get lost (notice how the table gives no hint about
403
    the existence of a 5th column). To convert the above table into a
404
    sparse data structure, one could list the table entries in this
405
    way:<P/>
406

407
    <Table Align="c">
408
    <Row><Item><M>[ [ 1, 3, 2 ], [ 1, 4, 9 ], [ 2, 2, 5 ], [ 3, 4, 1 ] ]</M></Item></Row>
409
    </Table>
410
    
411
    However, this data structure would not be very efficient. Whenever
412
    you are interested in a row <M>i</M> of <M>M</M> (this happens all the time
413
    when performing Gaussian elimination) the whole list would have
414
    to be searched for 3-tuples of the form <M>[ i, *, *
415
    ]</M>. This is why I tried to manage the row index by putting the
416
    tuples into the corresponding list entry:<Br/>
417

418
    <Table Align = "l">
419
    <Row><Item><M>[ [ 3, 2 ], [ 4, 9 ] ],</M></Item></Row>
420
    <Row><Item><M>[ [ 2, 5 ] ],</M></Item></Row>
421
    <Row><Item><M>[ [ 4, 1 ] ] ]</M></Item></Row>
422
    </Table>
423
    
424
    As you can see, this looks fairly complicated. However, the same
425
    information can be stored in this form, which would become the
426
    final data structure for &Gauss; sparse matrices:
427

428
    <Table Align = "clcl">
429
     <Row><Item>indices :=</Item><Item>[ [ 3, 4 ],</Item><Item>entries:=</Item><Item>[ [ 2, 9 ],</Item></Row>
430
     <Row><Item></Item><Item>  [ 2 ],</Item><Item></Item><Item>  [ 5 ],</Item></Row>
431
     <Row><Item></Item><Item>  [ 4 ] ]</Item><Item></Item><Item>  [ 1 ] ]</Item></Row>
432
    </Table>
433
    
434
    Although now the number of rows is equal to the Length of both
435
    `indices' and `entries', it is still stored in the sparse
436
    matrix. Here is the full data structure (&see;
437
    <Ref Func="SparseMatrix" Label="constructor using gap matrices"/>):
438
    
439
    <Listing Type="from SparseMatrix.gi">
440
    DeclareRepresentation( "IsSparseMatrixRep",
441
         IsSparseMatrix, [ "nrows", "ncols", "indices", "entries", "ring" ] );
442
    </Listing>
443
    
444
    As you can see, the matrix stores its ring to be on the safe side.
445
    This is especially important for zero matrices, as there is no way
446
    to determine the base ring from the sparse matrix structure. For
447
    further information on sparse matrix construction and converting,
448
    refer to <Ref Func="SparseMatrix" Label="constructor using gap matrices"/>.
449
    
450
 <Subsection Label="sub:gf2"><Heading>A special case: GF(2)</Heading>
451

452
    <Listing Type="from SparseMatrix.gi">
453
    DeclareRepresentation( "IsSparseMatrixGF2Rep",
454
         IsSparseMatrix, [ "nrows", "ncols", "indices", "ring" ] );
455
    </Listing>
456

457
    Because the nonzero entries of a matrix over GF(2) are all "1",
458
    the entries of M are not stored at all. It is of course crucial
459
    that all operations and algorithms make 100% sure that all
460
    appearing zero entries are deleted from the `indices' as well as
461
    the `entries' list as they arise.
462
    
463
 </Subsection>
464

465
 </Section>
466

467
 <Section Label="sec:mfSM"><Heading>Methods and functions for sparse matrices</Heading>
468
     <#Include Label="SparseMatrix">
469
     <#Include Label="ConvertSparseMatrixToMatrix">
470
     <#Include Label="CopyMat">
471
     <#Include Label="GetEntry">
472
     <#Include Label="SetEntry">
473
     <#Include Label="AddToEntry">
474
     <#Include Label="SparseZeroMatrix">
475
     <#Include Label="SparseIdentityMatrix">
476
     <#Include Label="TransposedSparseMat">
477
     <#Include Label="CertainRows">
478
     <#Include Label="CertainColumns">
479
     <#Include Label="UnionOfRows">
480
     <#Include Label="UnionOfColumns">
481
     <#Include Label="SparseDiagMat">
482
     <#Include Label="Nrows">
483
     <#Include Label="Ncols">
484
     <#Include Label="IndicesOfSparseMatrix">
485
     <#Include Label="EntriesOfSparseMatrix">
486
     <#Include Label="RingOfDefinition">
487
 </Section>
488

489
</Chapter>
490

491
<Chapter Label="chap:Gauss"><Heading>Gaussian Algorithms</Heading>
492

493
 <Section Label="sec:list"><Heading>A list of the available algorithms</Heading>
494
  
495
  As decribed earlier, the main functions of &Gauss; are <Ref
496
  Meth="EchelonMat"/> and <Ref Meth="EchelonMatTransformation"/>,
497
  <Ref Meth="ReduceMat"/> and <Ref Meth="ReduceMatTransformation"/>,
498
  <Ref Meth="KernelMat"/> and, additionally <Ref Meth="Rank"/>.
499

500
  These are all documented in the next section, but of course rely on
501
  specific algorithms depending on the base ring of the matrix. These
502
  are not fully documented but it should be very easy to find out how
503
  they work based on the documentation of the main functions.
504

505
  <Table Align="lll">
506
   <Row><Item>EchelonMat</Item></Row>
507
   <Row><Item></Item><Item>Field:</Item><Item><C>EchelonMatDestructive</C></Item></Row>
508
   <Row><Item></Item><Item>Ring:</Item><Item><C>HermiteMatDestructive</C></Item></Row>
509
   <Row><Item>EchelonMatTransformation</Item></Row>
510
   <Row><Item></Item><Item>Field:</Item><Item><C>EchelonMatTransformationDestructive</C></Item></Row>
511
   <Row><Item></Item><Item>Ring:</Item><Item><C>HermiteMatTransformationDestructive</C></Item></Row>
512
   <Row><Item>ReduceMat</Item></Row>
513
   <Row><Item></Item><Item>Field:</Item><Item><C>ReduceMatWithEchelonMat</C></Item></Row>
514
   <Row><Item></Item><Item>Ring:</Item><Item><C>ReduceMatWithHermiteMat</C></Item></Row>
515
   <Row><Item>ReduceMatTransformation</Item></Row>
516
   <Row><Item></Item><Item>Field:</Item><Item><C>ReduceMatWithEchelonMatTransformation</C></Item></Row>
517
   <Row><Item></Item><Item>Ring:</Item><Item><C>ReduceMatWithHermiteMatTransformation</C></Item></Row>
518
   <Row><Item>KernelMat</Item></Row>
519
   <Row><Item></Item><Item>Field:</Item><Item><C>KernelEchelonMatDestructive</C></Item></Row>
520
   <Row><Item></Item><Item>Ring:</Item><Item><C>KernelHermiteMatDestructive</C></Item></Row>
521
   <Row><Item>Rank</Item></Row>
522
   <Row><Item></Item><Item>Field (dense):</Item><Item><C>Rank</C> (&GAP; method)</Item></Row>
523
   <Row><Item></Item><Item>Field (sparse):</Item><Item><C>RankDestructive</C></Item></Row>
524
   <Row><Item></Item><Item>GF(2) (sparse):</Item><Item><C>RankOfIndicesListList</C></Item></Row>
525
   <Row><Item></Item><Item>Ring:</Item><Item>n.a.</Item></Row>
526
  </Table>
527
  
528
 </Section>
529

530
 <Section Label="sec:mfGauss"><Heading>Methods and Functions for &Gauss;ian algorithms</Heading>
531
     <#Include Label="EchelonMat">
532
     <#Include Label="EchelonMatTransformation">
533
     <#Include Label="ReduceMat">
534
     <#Include Label="ReduceMatTransformation">
535
     <#Include Label="KernelMat">
536
     <#Include Label="Rank">
537
 </Section>
538

539

540
</Chapter>
541

542
</Body>
543

544
<Appendix Label="FileOverview">
545
<Heading>An Overview of the &Gauss; package source code</Heading>
546
<Table Align="l|l">
547
<Caption><E>The &Gauss; package files.</E></Caption>
548
<Row><Item>Filename</Item><Item>Content</Item></Row>
549
<HorLine/>
550
<Row><Item>SparseMatrix.gi</Item><Item>Definitions and methods for
551
the sparse matrix type</Item></Row>
552
<Row><Item>SparseMatrixGF2.gi</Item><Item>Special case GF(2): no
553
matrix entries needed</Item></Row>
554
<Row><Item>GaussDense.gi</Item><Item>Gaussian elmination for &GAP;
555
matrices over fields</Item></Row>
556
<Row><Item>Sparse.gi</Item><Item>Documentation and forking depending
557
on the base ring</Item></Row>
558
<Row><Item>GaussSparse.gi</Item><Item>Gaussian elimination for sparse
559
matrices over fields</Item></Row>
560
<Row><Item>HermiteSparse.gi</Item><Item>Hermite elimination for sparse
561
matrices over <M>&ZZ; / \langle p^n \rangle</M></Item></Row>
562
</Table>
563
</Appendix>
564

565
<Bibliography Databases="GaussBib.xml"/>
566

567
<TheIndex/>
568

569
</Book>
570

571
Product

Resources

Company
