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<div class="ChapSects"><a href="chap1.html#X7DFB63A97E67C0A1">1 <span class="Heading">Introduction</span></a>
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<h3>1 <span class="Heading">Introduction</span></h3>
<p>The LinBox C++ library (<span class="URL"><a href="http://www.linalg.org">http://www.linalg.org</a></span>) 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.</p>
<p><strong class="pkg">GAP</strong> (<span class="URL"><a href="http://www.gap-system.org">http://www.gap-system.org</a></span>) 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, <strong class="pkg">GAP</strong>'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.</p>
<p>The <strong class="pkg">linboxing</strong> (LinBox-in-GAP) package provides an interface to the LinBox C++ library from <strong class="pkg">GAP</strong>. It provides alternative versions of <strong class="pkg">GAP</strong> linear algebra routines which are mapped through to the equivalent LinBox library routines at the <strong class="pkg">GAP</strong> kernel level. The result is linear algebra routines in <strong class="pkg">GAP</strong> that are, in the majority of cases, considerably faster than the native <strong class="pkg">GAP</strong> versions, and which scale better with matrix size. See the <strong class="pkg">linboxing</strong> website at <span class="URL"><a href="http://www.maths.nuigalway.ie/~pas/CHA/linboxing/index.shtml">http://www.maths.nuigalway.ie/~pas/CHA/linboxing/index.shtml</a></span> for timing examples and speedup estimates. As is typical, this speed is at the expense of memory, since the <strong class="pkg">GAP</strong> matrices and vector must be copied into a memory format that LinBox can use.</p>
<p>The functions provided by the <strong class="pkg">linboxing</strong> package are named the same as the <strong class="pkg">GAP</strong> equivalents, but are all contained within the <code class="code">LinBox</code> record, and so are prefixed with `<code class="code">LinBox.</code>'. The functions provided are</p>
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<li><p><code class="func">LinBox.Determinant</code> (<a href="chap3.html#X7ACB12B684D2F52F"><b>3.1-1</b></a>)</p>
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<li><p><code class="func">LinBox.Rank</code> (<a href="chap3.html#X781B821B7E4D2CD4"><b>3.1-2</b></a>)</p>
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<li><p><code class="func">LinBox.Trace</code> (<a href="chap3.html#X7F68AD6A7A002514"><b>3.1-3</b></a>)</p>
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<li><p><code class="func">LinBox.SolutionMat</code> (<a href="chap3.html#X7AE7EDC878216815"><b>3.1-4</b></a>)</p>
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<p>over the integers and prime fields.</p>
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