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# Gauss

## Extended Gauss functionality for GAP

2022.08-04

17 August 2022

Simon Görtzen
Email: simon.goertzen@rwth-aachen.de
Homepage: https://www.linkedin.com/in/simongoertzen/
Address:
Simon Görtzen
Lehrstuhl B fuer Mathematik, RWTH Aachen
Templergraben 64
52062 Aachen
Germany

### Abstract

This document explains the primary uses of the Gauss package. 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.

### Copyright

© 2007-2013 by Simon Goertzen

This package may be distributed under the terms and conditions of the GNU Public License Version 2 or (at your option) any later version.

### Acknowledgements

The Gauss package would not have been possible without the helpful contributions by

- Max Neunhöffer, University of St Andrews, and

- Mohamed Barakat, Lehrstuhl B für Mathematik, RWTH Aachen.

Many thanks to these two and the Lehrstuhl B für Mathematik in general. It should be noted that the GAP algorithms for `SemiEchelonForm` and other methods formed an important and informative basis for the development of the extended Gaussian algorithms. This manual was created with the help of the GAPDoc package by F. Lübeck and M. Neunhöffer [LN08].

### Contents

1 Introduction
1.1 Overview over this manual
1.2 Installation of the Gauss Package
2 Extending Gauss Functionality
2.1 The need for extended functionality
2.2 The applications of the Gauss package algorithms
3 The Sparse Matrix Data Type
3.1 The inner workings of Gauss sparse matrices
3.1-1 A special case: GF(2)
3.2 Methods and functions for sparse matrices
3.2-1 SparseMatrix
3.2-2 ConvertSparseMatrixToMatrix
3.2-3 CopyMat
3.2-4 GetEntry
3.2-5 SetEntry
3.2-6 AddToEntry
3.2-7 SparseZeroMatrix
3.2-8 SparseIdentityMatrix
3.2-9 TransposedSparseMat
3.2-10 CertainRows
3.2-11 CertainColumns
3.2-12 SparseUnionOfRows
3.2-13 SparseUnionOfColumns
3.2-14 SparseDiagMat
3.2-15 Nrows
3.2-16 Ncols
3.2-17 IndicesOfSparseMatrix
3.2-18 EntriesOfSparseMatrix
3.2-19 RingOfDefinition
4 Gaussian Algorithms
4.1 A list of the available algorithms
4.2 Methods and Functions for Gaussian algorithms
4.2-1 EchelonMat
4.2-2 EchelonMatTransformation
4.2-3 ReduceMat
4.2-4 ReduceMatTransformation
4.2-5 KernelMat
4.2-6 Rank
A An Overview of the Gauss package source code
References
Index
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