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<p>
Next: <a href="Sparse-Iterative-Solvers-Types.html#Sparse-Iterative-Solvers-Types" accesskey="n" rel="next">Sparse Iterative Solvers Types</a>, Up: <a href="Sparse-Iterative-Solvers.html#Sparse-Iterative-Solvers" accesskey="u" rel="up">Sparse Iterative Solvers</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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<a name="Overview-9"></a>
<h4 class="subsection">43.2.1 Overview</h4>

<p>Many practical iterative methods of solving large <em>n</em>-by-<em>n</em>
sparse linear systems involve projecting an approximate solution for
<var>x</var> onto a subspace of <em>{\bf R}^n</em>. If we define a <em>m</em>-dimensional
subspace <em>{\cal K}</em> as the subspace of approximations to the solution
<var>x</var>, then <em>m</em> constraints must be imposed to determine
the next approximation. These <em>m</em> constraints define another
<em>m</em>-dimensional subspace denoted by <em>{\cal L}</em>. The
subspace dimension <em>m</em> is typically chosen to be much smaller than
<em>n</em> in order to reduce the computational
effort needed to generate the next approximate solution vector.
The many iterative algorithms which exist differ mainly
in their choice of <em>{\cal K}</em> and <em>{\cal L}</em>.
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