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1 Introduction

### 1 Introduction

The LinBox C++ library (http://www.linalg.org) performs exact linear algebra and provides a set of routines for the solution of linear algebra problems such as rank, determinant, and the solution of linear systems. It provides representations for both sparse and dense matrices over integers and finite fields. It has a particular emphasis on black-box matrix methods (which are very efficient over sparse matrices), but increasingly also provides elimination-based routines for dense matrices using the industry-standard BLAS numeric routines.

GAP (http://www.gap-system.org) is a system for computational discrete algebra, with particular emphasis on Computational Group Theory. It provides good implementations of exact linear algebra routines on dense matrices over all common fields and the integers. Typically, GAP's versions are faster than LinBox for small finite fields (i.e. order less than 256), but LinBox is much faster for larger finite fields and the integers.

The linboxing (LinBox-in-GAP) package provides an interface to the LinBox C++ library from GAP. It provides alternative versions of GAP linear algebra routines which are mapped through to the equivalent LinBox library routines at the GAP kernel level. The result is linear algebra routines in GAP that are, in the majority of cases, considerably faster than the native GAP versions, and which scale better with matrix size. See the linboxing website at http://www.maths.nuigalway.ie/~pas/CHA/linboxing/index.shtml for timing examples and speedup estimates. As is typical, this speed is at the expense of memory, since the GAP matrices and vector must be copied into a memory format that LinBox can use.

The functions provided by the linboxing package are named the same as the GAP equivalents, but are all contained within the `LinBox` record, and so are prefixed with ``LinBox.`'. The functions provided are

- `LinBox.Determinant` (3.1-1)

- `LinBox.Rank` (3.1-2)

- `LinBox.Trace` (3.1-3)

- `LinBox.SolutionMat` (3.1-4)

over the integers and prime fields.

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