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<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Линейная алгебра</title><meta name="generator" content="DocBook XSL Stylesheets Vsnapshot"><link rel="home" href="index.html" title="Руководство пользователя Genius"><link rel="up" href="ch08.html" title="Chapter 8. Матрицы в GEL"><link rel="prev" href="ch08s02.html" title="Conjugate Transpose and Transpose Operator"><link rel="next" href="ch09.html" title="Chapter 9. Многочлены в GEL"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3" align="center">Линейная алгебра</th></tr><tr><td width="20%" align="left"><a accesskey="p" href="ch08s02.html">Prev</a> </td><th width="60%" align="center">Chapter 8. Матрицы в GEL</th><td width="20%" align="right"> <a accesskey="n" href="ch09.html">Next</a></td></tr></table><hr></div><div class="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a name="genius-gel-matrix-linalg"></a>Линейная алгебра</h2></div></div></div><p lang="en">
Genius implements many useful linear algebra and matrix manipulation
routines. See the <a class="link" href="ch11s09.html" title="Линейная алгебра">Linear Algebra</a> and
<a class="link" href="ch11s08.html" title="Операции с матрицами">Matrix Manipulation</a>
sections of the GEL function listing.
</p><p lang="en">
The linear algebra routines implemented in GEL do not currently come
from a well tested numerical package, and thus should not be used for critical
numerical computation. On the other hand, Genius implements very well many
linear algebra operations with rational and integer coefficients. These are
inherently exact and in fact will give you much better results than common
double precision routines for linear algebra.
</p><p lang="en">
For example, it is pointless to compute the rank and nullspace of a
floating point matrix since for all practical purposes, we need to consider the
matrix as having some slight errors. You are likely to get a different result
than you expect. The problem is that under a small perturbation every matrix
is of full rank and invertible. If the matrix however is of rational numbers,
then the rank and nullspace are always exact.
</p><p lang="en">
In general when Genius computes the basis of a certain vectorspace
(for example with the <a class="link" href="ch11s09.html#gel-function-NullSpace"><code class="function">NullSpace</code></a>) it will give the basis as
a matrix, in which the columns are the vectors of the basis. That is, when
Genius talks of a linear subspace it means a matrix whose column space is
the given linear subspace.
</p><p lang="en">
It should be noted that Genius can remember certain properties of a
matrix. For example, it will remember that a matrix is in row reduced form.
If many calls are made to functions that internally use row reduced form of
the matrix, we can just row reduce the matrix beforehand once. Successive
calls to <a class="link" href="ch11s09.html#gel-function-rref"><code class="function">rref</code></a> will be very fast.
</p></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch08s02.html">Prev</a> </td><td width="20%" align="center"><a accesskey="u" href="ch08.html">Up</a></td><td width="40%" align="right"> <a accesskey="n" href="ch09.html">Next</a></td></tr><tr><td width="40%" align="left" valign="top">Conjugate Transpose and Transpose Operator </td><td width="20%" align="center"><a accesskey="h" href="index.html">Home</a></td><td width="40%" align="right" valign="top"> Chapter 9. Многочлены в GEL</td></tr></table></div></body></html>
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