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<a name="Specialized-Solvers"></a>
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Previous: <a rel="previous" accesskey="p" href="Functions-of-a-Matrix.html#Functions-of-a-Matrix">Functions of a Matrix</a>,
Up: <a rel="up" accesskey="u" href="Linear-Algebra.html#Linear-Algebra">Linear Algebra</a>
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<h3 class="section">18.5 Specialized Solvers</h3>
<!-- bicg scripts/sparse/bicg.m -->
<p><a name="doc_002dbicg"></a>
<div class="defun">
— Function File: <var>x</var> = <b>bicg</b> (<var>A, b, rtol, maxit, M1, M2, x0</var>)<var><a name="index-bicg-2111"></a></var><br>
— Function File: <var>x</var> = <b>bicg</b> (<var>A, b, rtol, maxit, P</var>)<var><a name="index-bicg-2112"></a></var><br>
— Function File: [<var>x</var>, <var>flag</var>, <var>relres</var>, <var>iter</var>, <var>resvec</var>] = <b>bicg</b> (<var>A, b, <small class="dots">...</small></var>)<var><a name="index-bicg-2113"></a></var><br>
<blockquote><p>Solve <code>A x = b</code> using the Bi-conjugate gradient iterative method.
<ul>
<li><var>rtol</var> is the relative tolerance, if not given
or set to [] the default value 1e-6 is used.
<li><var>maxit</var> the maximum number of outer iterations,
if not given or set to [] the default value
<code>min (20, numel (b))</code> is used.
<li><var>x0</var> the initial guess, if not given or set to []
the default value <code>zeros (size (b))</code> is used.
</ul>
<p><var>A</var> can be passed as a matrix or as a function handle or
inline function <code>f</code> such that <code>f(x, "notransp") = A*x</code>
and <code>f(x, "transp") = A'*x</code>.
<p>The preconditioner <var>P</var> is given as <code>P = M1 * M2</code>.
Both <var>M1</var> and <var>M2</var> can be passed as a matrix or as
a function handle or inline function <code>g</code> such that
<code>g(x, 'notransp') = M1 \ x</code> or <code>g(x, 'notransp') = M2 \ x</code> and
<code>g(x, 'transp') = M1' \ x</code> or <code>g(x, 'transp') = M2' \ x</code>.
<p>If called with more than one output parameter
<ul>
<li><var>flag</var> indicates the exit status:
<ul>
<li>0: iteration converged to the within the chosen tolerance
<li>1: the maximum number of iterations was reached before convergence
<li>3: the algorithm reached stagnation
</ul>
(the value 2 is unused but skipped for compatibility).
<li><var>relres</var> is the final value of the relative residual.
<li><var>iter</var> is the number of iterations performed.
<li><var>resvec</var> is a vector containing the relative residual at each iteration.
</ul>
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dbicgstab.html#doc_002dbicgstab">bicgstab</a>, <a href="doc_002dcgs.html#doc_002dcgs">cgs</a>, <a href="doc_002dgmres.html#doc_002dgmres">gmres</a>, <a href="doc_002dpcg.html#doc_002dpcg">pcg</a>.
</blockquote></div>
<!-- bicgstab scripts/sparse/bicgstab.m -->
<p><a name="doc_002dbicgstab"></a>
<div class="defun">
— Function File: <var>x</var> = <b>bicgstab</b> (<var>A, b, rtol, maxit, M1, M2, x0</var>)<var><a name="index-bicgstab-2114"></a></var><br>
— Function File: <var>x</var> = <b>bicgstab</b> (<var>A, b, rtol, maxit, P</var>)<var><a name="index-bicgstab-2115"></a></var><br>
— Function File: [<var>x</var>, <var>flag</var>, <var>relres</var>, <var>iter</var>, <var>resvec</var>] = <b>bicgstab</b> (<var>A, b, <small class="dots">...</small></var>)<var><a name="index-bicgstab-2116"></a></var><br>
<blockquote><p>Solve <code>A x = b</code> using the stabilizied Bi-conjugate gradient iterative
method.
<ul>
<li><var>rtol</var> is the relative tolerance, if not given or set to
[] the default value 1e-6 is used.
<li><var>maxit</var> the maximum number of outer iterations, if not
given or set to [] the default value <code>min (20, numel (b))</code> is
used.
<li><var>x0</var> the initial guess, if not given or set to [] the
default value <code>zeros (size (b))</code> is used.
</ul>
<p><var>A</var> can be passed as a matrix or as a function handle or
inline function <code>f</code> such that <code>f(x) = A*x</code>.
<p>The preconditioner <var>P</var> is given as <code>P = M1 * M2</code>.
Both <var>M1</var> and <var>M2</var> can be passed as a matrix or as a function
handle or inline function <code>g</code> such that <code>g(x) = M1 \ x</code> or
<code>g(x) = M2 \ x</code>.
<p>If called with more than one output parameter
<ul>
<li><var>flag</var> indicates the exit status:
<ul>
<li>0: iteration converged to the within the chosen tolerance
<li>1: the maximum number of iterations was reached before convergence
<li>3: the algorithm reached stagnation
</ul>
(the value 2 is unused but skipped for compatibility).
<li><var>relres</var> is the final value of the relative residual.
<li><var>iter</var> is the number of iterations performed.
<li><var>resvec</var> is a vector containing the relative residual at each iteration.
</ul>
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dbicg.html#doc_002dbicg">bicg</a>, <a href="doc_002dcgs.html#doc_002dcgs">cgs</a>, <a href="doc_002dgmres.html#doc_002dgmres">gmres</a>, <a href="doc_002dpcg.html#doc_002dpcg">pcg</a>.
</blockquote></div>
<!-- cgs scripts/sparse/cgs.m -->
<p><a name="doc_002dcgs"></a>
<div class="defun">
— Function File: <var>x</var> = <b>cgs</b> (<var>A, b, rtol, maxit, M1, M2, x0</var>)<var><a name="index-cgs-2117"></a></var><br>
— Function File: <var>x</var> = <b>cgs</b> (<var>A, b, rtol, maxit, P</var>)<var><a name="index-cgs-2118"></a></var><br>
— Function File: [<var>x</var>, <var>flag</var>, <var>relres</var>, <var>iter</var>, <var>resvec</var>] = <b>cgs</b> (<var>A, b, <small class="dots">...</small></var>)<var><a name="index-cgs-2119"></a></var><br>
<blockquote><p>Solve <code>A x = b</code>, where <var>A</var> is a square matrix, using the
Conjugate Gradients Squared method.
<ul>
<li><var>rtol</var> is the relative tolerance, if not given or set to []
the default value 1e-6 is used.
<li><var>maxit</var> the maximum number of outer iterations, if not
given or set to [] the default value <code>min (20, numel (b))</code> is
used.
<li><var>x0</var> the initial guess, if not given or set to [] the
default value <code>zeros (size (b))</code> is used.
</ul>
<p><var>A</var> can be passed as a matrix or as a function handle or
inline function <code>f</code> such that <code>f(x) = A*x</code>.
<p>The preconditioner <var>P</var> is given as <code>P = M1 * M2</code>.
Both <var>M1</var> and <var>M2</var> can be passed as a matrix or as a function
handle or inline function <code>g</code> such that <code>g(x) = M1 \ x</code> or
<code>g(x) = M2 \ x</code>.
<p>If called with more than one output parameter
<ul>
<li><var>flag</var> indicates the exit status:
<ul>
<li>0: iteration converged to the within the chosen tolerance
<li>1: the maximum number of iterations was reached before convergence
<li>3: the algorithm reached stagnation
</ul>
(the value 2 is unused but skipped for compatibility).
<li><var>relres</var> is the final value of the relative residual.
<li><var>iter</var> is the number of iterations performed.
<li><var>resvec</var> is a vector containing the relative residual at
each iteration.
</ul>
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dpcg.html#doc_002dpcg">pcg</a>, <a href="doc_002dbicgstab.html#doc_002dbicgstab">bicgstab</a>, <a href="doc_002dbicg.html#doc_002dbicg">bicg</a>, <a href="doc_002dgmres.html#doc_002dgmres">gmres</a>.
</p></blockquote></div>
<!-- gmres scripts/sparse/gmres.m -->
<p><a name="doc_002dgmres"></a>
<div class="defun">
— Function File: <var>x</var> = <b>gmres</b> (<var>A, b, m, rtol, maxit, M1, M2, x0</var>)<var><a name="index-gmres-2120"></a></var><br>
— Function File: <var>x</var> = <b>gmres</b> (<var>A, b, m, rtol, maxit, P</var>)<var><a name="index-gmres-2121"></a></var><br>
— Function File: [<var>x</var>, <var>flag</var>, <var>relres</var>, <var>iter</var>, <var>resvec</var>] = <b>gmres</b> (<var><small class="dots">...</small></var>)<var><a name="index-gmres-2122"></a></var><br>
<blockquote><p>Solve <code>A x = b</code> using the Preconditioned GMRES iterative method
with restart, a.k.a. PGMRES(m).
<ul>
<li><var>rtol</var> is the relative tolerance,
if not given or set to [] the default value 1e-6 is used.
<li><var>maxit</var> is the maximum number of outer iterations,
if not given or set to [] the default value
<code>min (10, numel (b) / restart)</code> is used.
<li><var>x0</var> is the initial guess,
if not given or set to [] the default value <code>zeros(size (b))</code> is used.
<li><var>m</var> is the restart parameter,
if not given or set to [] the default value <code>numel (b)</code> is used.
</ul>
<p>Argument <var>A</var> can be passed as a matrix, function handle, or
inline function <code>f</code> such that <code>f(x) = A*x</code>.
<p>The preconditioner <var>P</var> is given as <code>P = M1 * M2</code>.
Both <var>M1</var> and <var>M2</var> can be passed as a matrix, function handle, or
inline function <code>g</code> such that <code>g(x) = M1\x</code> or <code>g(x) = M2\x</code>.
<p>Besides the vector <var>x</var>, additional outputs are:
<ul>
<li><var>flag</var> indicates the exit status:
<dl>
<dt>0 : iteration converged to within the specified tolerance
<br><dt>1 : maximum number of iterations exceeded
<br><dt>2 : unused, but skipped for compatibility
<br><dt>3 : algorithm reached stagnation<dd></dl>
<li><var>relres</var> is the final value of the relative residual.
<li><var>iter</var> is a vector containing the number of outer iterations and
total iterations performed.
<li><var>resvec</var> is a vector containing the relative residual at each
iteration.
</ul>
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dbicg.html#doc_002dbicg">bicg</a>, <a href="doc_002dbicgstab.html#doc_002dbicgstab">bicgstab</a>, <a href="doc_002dcgs.html#doc_002dcgs">cgs</a>, <a href="doc_002dpcg.html#doc_002dpcg">pcg</a>.
</p></blockquote></div>
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