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<hr>
<a name="Specialized-Solvers-1"></a>
<h3 class="section">18.5 Specialized Solvers</h3>
<a name="index-matrix_002c-specialized-solvers"></a>
<a name="XREFbicg"></a><dl>
<dt><a name="index-bicg"></a><em><var>x</var> =</em> <strong>bicg</strong> <em>(<var>A</var>, <var>b</var>)</em></dt>
<dt><a name="index-bicg-1"></a><em><var>x</var> =</em> <strong>bicg</strong> <em>(<var>A</var>, <var>b</var>, <var>tol</var>)</em></dt>
<dt><a name="index-bicg-2"></a><em><var>x</var> =</em> <strong>bicg</strong> <em>(<var>A</var>, <var>b</var>, <var>tol</var>, <var>maxit</var>)</em></dt>
<dt><a name="index-bicg-3"></a><em><var>x</var> =</em> <strong>bicg</strong> <em>(<var>A</var>, <var>b</var>, <var>tol</var>, <var>maxit</var>, <var>M</var>)</em></dt>
<dt><a name="index-bicg-4"></a><em><var>x</var> =</em> <strong>bicg</strong> <em>(<var>A</var>, <var>b</var>, <var>tol</var>, <var>maxit</var>, <var>M1</var>, <var>M2</var>)</em></dt>
<dt><a name="index-bicg-5"></a><em><var>x</var> =</em> <strong>bicg</strong> <em>(<var>A</var>, <var>b</var>, <var>tol</var>, <var>maxit</var>, <var>M</var>, [], <var>x0</var>)</em></dt>
<dt><a name="index-bicg-6"></a><em><var>x</var> =</em> <strong>bicg</strong> <em>(<var>A</var>, <var>b</var>, <var>tol</var>, <var>maxit</var>, <var>M1</var>, <var>M2</var>, <var>x0</var>)</em></dt>
<dt><a name="index-bicg-7"></a><em><var>x</var> =</em> <strong>bicg</strong> <em>(<var>A</var>, <var>b</var>, <var>tol</var>, <var>maxit</var>, <var>M</var>, [], <var>x0</var>, …)</em></dt>
<dt><a name="index-bicg-8"></a><em><var>x</var> =</em> <strong>bicg</strong> <em>(<var>A</var>, <var>b</var>, <var>tol</var>, <var>maxit</var>, <var>M1</var>, <var>M2</var>, <var>x0</var>, …)</em></dt>
<dt><a name="index-bicg-9"></a><em>[<var>x</var>, <var>flag</var>, <var>relres</var>, <var>iter</var>, <var>resvec</var>] =</em> <strong>bicg</strong> <em>(<var>A</var>, <var>b</var>, …)</em></dt>
<dd><p>Solve the linear system of equations <code><var>A</var> * <var>x</var> = <var>b</var></code><!-- /@w -->
by means of the Bi-Conjugate Gradient iterative method.
</p>
<p>The input arguments are:
</p>
<ul>
<li> <var>A</var> is the matrix of the linear system and it must be square.
<var>A</var> can be passed as a matrix, function handle, or inline function
<code>Afun</code> such that <code>Afun (x, "notransp") = A * x</code><!-- /@w --> and
<code>Afun (x, "transp") = A' * x</code><!-- /@w -->. Additional parameters to
<code>Afun</code> may be passed after <var>x0</var>.
</li><li> <var>b</var> is the right-hand side vector. It must be a column vector
with the same number of rows as <var>A</var>.
</li><li> <var>tol</var> is the required relative tolerance for the residual error,
<code><var>b</var> <span class="nolinebreak">-</span> <var>A</var> * <var>x</var></code><!-- /@w -->. The iteration stops if
<code>norm (<var>b</var> <span class="nolinebreak">-</span> <var>A</var> * <var>x</var>)</code> ≤ <code><var>tol</var> * norm (<var>b</var>)</code><!-- /@w --><!-- /@w -->.
If <var>tol</var> is omitted or empty, then a tolerance of 1e-6 is used.
</li><li> <var>maxit</var> is the maximum allowed number of iterations; if <var>maxit</var>
is omitted or empty then a value of 20 is used.
</li><li> <var>M1</var>, <var>M2</var> are the preconditioners. The preconditioner <var>M</var> is
given as <code><var>M</var> = <var>M1</var> * <var>M2</var></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 (<var>x</var>, "notransp") = <var>M1</var> \ <var>x</var></code><!-- /@w -->
or <code>g (<var>x</var>, "notransp") = <var>M2</var> \ <var>x</var></code><!-- /@w --> and
<code>g (<var>x</var>, "transp") = <var>M1</var>' \ <var>x</var></code><!-- /@w --> or
<code>g (<var>x</var>, "transp") = <var>M2</var>' \ <var>x</var></code><!-- /@w -->.
If <var>M1</var> is omitted or empty, then preconditioning is not applied.
The preconditioned system is theoretically equivalent to applying the
<code>bicg</code> method to the linear system
<code>inv (<var>M1</var>) * A * inv (<var>M2</var>) * <var>y</var> = inv
(<var>M1</var>) * <var>b</var></code> and
<code>inv (<var>M2'</var>) * A' * inv (<var>M1'</var>) * <var>z</var> =
inv (<var>M2'</var>) * <var>b</var></code> and then setting
<code><var>x</var> = inv (<var>M2</var>) * <var>y</var></code>.
</li><li> <var>x0</var> is the initial guess. If <var>x0</var> is omitted or empty then the
function sets <var>x0</var> to a zero vector by default.
</li></ul>
<p>Any arguments which follow <var>x0</var> are treated as parameters, and passed in
an appropriate manner to any of the functions (<var>Afun</var> or <var>Mfun</var>) or
that have been given to <code>bicg</code>.
</p>
<p>The output parameters are:
</p>
<ul>
<li> <var>x</var> is the computed approximation to the solution of
<code><var>A</var> * <var>x</var> = <var>b</var></code><!-- /@w -->. If the algorithm did not converge,
then <var>x</var> is the iteration which has the minimum residual.
</li><li> <var>flag</var> indicates the exit status:
<ul>
<li> 0: The algorithm converged to within the prescribed tolerance.
</li><li> 1: The algorithm did not converge and it reached the maximum number of
iterations.
</li><li> 2: The preconditioner matrix is singular.
</li><li> 3: The algorithm stagnated, i.e., the absolute value of the
difference between the current iteration <var>x</var> and the previous is less
than <code>eps * norm (<var>x</var>,2)</code>.
</li><li> 4: The algorithm could not continue because intermediate values
became too small or too large for reliable computation.
</li></ul>
</li><li> <var>relres</var> is the ratio of the final residual to its initial value,
measured in the Euclidean norm.
</li><li> <var>iter</var> is the iteration which <var>x</var> is computed.
</li><li> <var>resvec</var> is a vector containing the residual at each iteration.
The total number of iterations performed is given by
<code>length (<var>resvec</var>) - 1</code>.
</li></ul>
<p>Consider a trivial problem with a tridiagonal matrix
</p>
<div class="example">
<pre class="example">n = 20;
A = toeplitz (sparse ([1, 1], [1, 2], [2, 1] * n ^ 2, 1, n)) + ...
toeplitz (sparse (1, 2, -1, 1, n) * n / 2, ...
sparse (1, 2, 1, 1, n) * n / 2);
b = A * ones (n, 1);
restart = 5;
[M1, M2] = ilu (A); # in this tridiag case, it corresponds to lu (A)
M = M1 * M2;
Afun = @(x, string) strcmp (string, "notransp") * (A * x) + ...
strcmp (string, "transp") * (A' * x);
Mfun = @(x, string) strcmp (string, "notransp") * (M \ x) + ...
strcmp (string, "transp") * (M' \ x);
M1fun = @(x, string) strcmp (string, "notransp") * (M1 \ x) + ...
strcmp (string, "transp") * (M1' \ x);
M2fun = @(x, string) strcmp (string, "notransp") * (M2 \ x) + ...
strcmp (string, "transp") * (M2' \ x);
</pre></div>
<p><small>EXAMPLE 1:</small> simplest usage of <code>bicg</code>
</p>
<div class="example">
<pre class="example">x = bicg (A, b)
</pre></div>
<p><small>EXAMPLE 2:</small> <code>bicg</code> with a function that computes
<code><var>A</var>*<var>x</var></code> and <code><var>A'</var>*<var>x</var></code>
</p>
<div class="example">
<pre class="example">x = bicg (Afun, b, [], n)
</pre></div>
<p><small>EXAMPLE 3:</small> <code>bicg</code> with a preconditioner matrix <var>M</var>
</p>
<div class="example">
<pre class="example">x = bicg (A, b, 1e-6, n, M)
</pre></div>
<p><small>EXAMPLE 4:</small> <code>bicg</code> with a function as preconditioner
</p>
<div class="example">
<pre class="example">x = bicg (Afun, b, 1e-6, n, Mfun)
</pre></div>
<p><small>EXAMPLE 5:</small> <code>bicg</code> with preconditioner matrices <var>M1</var>
and <var>M2</var>
</p>
<div class="example">
<pre class="example">x = bicg (A, b, 1e-6, n, M1, M2)
</pre></div>
<p><small>EXAMPLE 6:</small> <code>bicg</code> with functions as preconditioners
</p>
<div class="example">
<pre class="example">x = bicg (Afun, b, 1e-6, n, M1fun, M2fun)
</pre></div>
<p><small>EXAMPLE 7:</small> <code>bicg</code> with as input a function requiring an argument
</p>
<div class="example">
<pre class="example">function y = Ap (A, x, string, z)
## compute A^z * x or (A^z)' * x
y = x;
if (strcmp (string, "notransp"))
for i = 1:z
y = A * y;
endfor
elseif (strcmp (string, "transp"))
for i = 1:z
y = A' * y;
endfor
endif
endfunction
Apfun = @(x, string, p) Ap (A, x, string, p);
x = bicg (Apfun, b, [], [], [], [], [], 2);
</pre></div>
<p>References:
</p>
<ol>
<li> Y. Saad, <cite>Iterative Methods for Sparse Linear Systems</cite>,
Second edition, 2003, SIAM.
</li></ol>
<p><strong>See also:</strong> <a href="#XREFbicgstab">bicgstab</a>, <a href="#XREFcgs">cgs</a>, <a href="#XREFgmres">gmres</a>, <a href="Iterative-Techniques.html#XREFpcg">pcg</a>, <a href="#XREFqmr">qmr</a>, <a href="#XREFtfqmr">tfqmr</a>.
</p></dd></dl>
<a name="XREFbicgstab"></a><dl>
<dt><a name="index-bicgstab"></a><em><var>x</var> =</em> <strong>bicgstab</strong> <em>(<var>A</var>, <var>b</var>, <var>tol</var>, <var>maxit</var>, <var>M1</var>, <var>M2</var>, <var>x0</var>, …)</em></dt>
<dt><a name="index-bicgstab-1"></a><em><var>x</var> =</em> <strong>bicgstab</strong> <em>(<var>A</var>, <var>b</var>, <var>tol</var>, <var>maxit</var>, <var>M</var>, [], <var>x0</var>, …)</em></dt>
<dt><a name="index-bicgstab-2"></a><em>[<var>x</var>, <var>flag</var>, <var>relres</var>, <var>iter</var>, <var>resvec</var>] =</em> <strong>bicgstab</strong> <em>(<var>A</var>, <var>b</var>, …)</em></dt>
<dd><p>Solve <code>A x = b</code> using the stabilizied Bi-conjugate gradient iterative
method.
</p>
<p>The input parameters are:
</p>
<ul class="no-bullet">
<li>- <var>A</var> is the matrix of the linear system and it must be square.
<var>A</var> can be passed as a matrix, function handle, or inline
function <code>Afun</code> such that <code>Afun(x) = A * x</code>. Additional
parameters to <code>Afun</code> are passed after <var>x0</var>.
</li><li>- <var>b</var> is the right hand side vector. It must be a column vector
with the same number of rows as <var>A</var>.
</li><li>- <var>tol</var> is the required relative tolerance for the residual error,
<code><var>b</var> <span class="nolinebreak">-</span> <var>A</var> * <var>x</var></code><!-- /@w -->. The iteration stops if
<code>norm (<var>b</var> <span class="nolinebreak">-</span> <var>A</var> * <var>x</var>)</code> ≤ <code><var>tol</var> * norm (<var>b</var>)</code><!-- /@w --><!-- /@w -->.
If <var>tol</var> is omitted or empty, then a tolerance of 1e-6 is used.
</li><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><li>- <var>M1</var>, <var>M2</var> are the preconditioners. The preconditioner
<var>M</var> is given as <code><var>M</var> = <var>M1</var> * <var>M2</var></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(<var>x</var>) = <var>M1</var> \ <var>x</var></code> or
<code>g(<var>x</var>) = <var>M2</var> \ <var>x</var></code>.
The technique used is the right preconditioning, i.e., it is
solved <code><var>A</var> * inv (<var>M</var>) * <var>y</var> = <var>b</var></code> and then
<code><var>x</var> = inv (<var>M</var>) * <var>y</var></code>.
</li><li>- <var>x0</var> the initial guess, if not given or set to [] the default
value <code>zeros (size (<var>b</var>))</code> is used.
</li></ul>
<p>The arguments which follow <var>x0</var> are treated as parameters, and passed in
a proper way to any of the functions (<var>A</var> or <var>M</var>) which are passed
to <code>bicstab</code>.
</p>
<p>The output parameters are:
</p>
<ul class="no-bullet">
<li>- <var>x</var> is the approximation computed. If the method doesn’t
converge then it is the iterated with the minimum residual.
</li><li>- <var>flag</var> indicates the exit status:
<ul class="no-bullet">
<li>- 0: iteration converged to the within the chosen tolerance
</li><li>- 1: the maximum number of iterations was reached before convergence
</li><li>- 2: the preconditioner matrix is singular
</li><li>- 3: the algorithm reached stagnation
</li><li>- 4: the algorithm can’t continue due to a division by zero
</li></ul>
</li><li>- <var>relres</var> is the relative residual obtained with as
<code>(<var>A</var>*<var>x</var>-<var>b</var>) / <code>norm(<var>b</var>)</code></code>.
</li><li>- <var>iter</var> is the (possibily half) iteration which <var>x</var> is
computed. If it is an half iteration then it is <code><var>iter</var> + 0.5</code>
</li><li>- <var>resvec</var> is a vector containing the residual of each half and
total iteration (There are also the half iterations since <var>x</var> is
computed in two steps at each iteration).
Doing <code>(length(<var>resvec</var>) - 1) / 2</code> is possible to see the
total number of (total) iterations performed.
</li></ul>
<p>Let us consider a trivial problem with a tridiagonal matrix
</p>
<div class="example">
<pre class="example">n = 20;
A = toeplitz (sparse ([1, 1], [1, 2], [2, 1] * n ^ 2, 1, n)) + ...
toeplitz (sparse (1, 2, -1, 1, n) * n / 2, ...
sparse (1, 2, 1, 1, n) * n / 2);
b = A * ones (n, 1);
restart = 5;
[M1, M2] = ilu (A); # in this tridiag case, it corresponds to lu (A)
M = M1 * M2;
Afun = @(x) A * x;
Mfun = @(x) M \ x;
M1fun = @(x) M1 \ x;
M2fun = @(x) M2 \ x;
</pre></div>
<p><small>EXAMPLE 1:</small> simplest usage of <code>bicgstab</code>
</p>
<div class="example">
<pre class="example">x = bicgstab (A, b, [], n)
</pre></div>
<p><small>EXAMPLE 2:</small> <code>bicgstab</code> with a function which computes
<code><var>A</var> * <var>x</var></code>
</p>
<div class="example">
<pre class="example">x = bicgstab (Afun, b, [], n)
</pre></div>
<p><small>EXAMPLE 3:</small> <code>bicgstab</code> with a preconditioner matrix <var>M</var>
</p>
<div class="example">
<pre class="example">x = bicgstab (A, b, [], 1e-06, n, M)
</pre></div>
<p><small>EXAMPLE 4:</small> <code>bicgstab</code> with a function as preconditioner
</p>
<div class="example">
<pre class="example">x = bicgstab (Afun, b, 1e-6, n, Mfun)
</pre></div>
<p><small>EXAMPLE 5:</small> <code>bicgstab</code> with preconditioner matrices <var>M1</var>
and <var>M2</var>
</p>
<div class="example">
<pre class="example">x = bicgstab (A, b, [], 1e-6, n, M1, M2)
</pre></div>
<p><small>EXAMPLE 6:</small> <code>bicgstab</code> with functions as preconditioners
</p>
<div class="example">
<pre class="example">x = bicgstab (Afun, b, 1e-6, n, M1fun, M2fun)
</pre></div>
<p><small>EXAMPLE 7:</small> <code>bicgstab</code> with as input a function requiring
an argument
</p>
<div class="example">
<pre class="example">function y = Ap (A, x, z) # compute A^z * x
y = x;
for i = 1:z
y = A * y;
endfor
endfunction
Apfun = @(x, string, p) Ap (A, x, string, p);
x = bicgstab (Apfun, b, [], [], [], [], [], 2);
</pre></div>
<p><small>EXAMPLE 8:</small> explicit example to show that <code>bicgstab</code> uses a
right preconditioner
</p>
<div class="example">
<pre class="example">[M1, M2] = ilu (A + 0.1 * eye (n)); # factorization of A perturbed
M = M1 * M2;
## reference solution computed by bicgstab after one iteration
[x_ref, fl] = bicgstab (A, b, [], 1, M)
## right preconditioning
[y, fl] = bicgstab (A / M, b, [], 1)
x = M \ y # compare x and x_ref
</pre></div>
<p>References:
</p>
<ol>
<li> Y. Saad, <cite>Iterative Methods for Sparse Linear
Systems</cite>, Second edition, 2003, SIAM
</li></ol>
<p><strong>See also:</strong> <a href="#XREFbicg">bicg</a>, <a href="#XREFcgs">cgs</a>, <a href="#XREFgmres">gmres</a>, <a href="Iterative-Techniques.html#XREFpcg">pcg</a>, <a href="#XREFqmr">qmr</a>, <a href="#XREFtfqmr">tfqmr</a>.
</p>
</dd></dl>
<a name="XREFcgs"></a><dl>
<dt><a name="index-cgs"></a><em><var>x</var> =</em> <strong>cgs</strong> <em>(<var>A</var>, <var>b</var>, <var>tol</var>, <var>maxit</var>, <var>M1</var>, <var>M2</var>, <var>x0</var>, …)</em></dt>
<dt><a name="index-cgs-1"></a><em><var>x</var> =</em> <strong>cgs</strong> <em>(<var>A</var>, <var>b</var>, <var>tol</var>, <var>maxit</var>, <var>M</var>, [], <var>x0</var>, …)</em></dt>
<dt><a name="index-cgs-2"></a><em>[<var>x</var>, <var>flag</var>, <var>relres</var>, <var>iter</var>, <var>resvec</var>] =</em> <strong>cgs</strong> <em>(<var>A</var>, <var>b</var>, …)</em></dt>
<dd><p>Solve <code>A x = b</code>, where <var>A</var> is a square matrix, using the
Conjugate Gradients Squared method.
</p>
<p>The input arguments are:
</p>
<ul class="no-bullet">
<li>- <var>A</var> is the matrix of the linear system and it must be square.
<var>A</var> can be passed as a matrix, function handle, or inline
function <code>Afun</code> such that <code>Afun(x) = A * x</code>. Additional
parameters to <code>Afun</code> are passed after <var>x0</var>.
</li><li>- <var>b</var> is the right hand side vector. It must be a column vector
with same number of rows of <var>A</var>.
</li><li>- <var>tol</var> is the relative tolerance, if not given or set to [] the
default value 1e-6 is used.
</li><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><li>- <var>M1</var>, <var>M2</var> are the preconditioners. The preconditioner
matrix is given as <code>M = 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>.
If M1 is empty or not passed then no preconditioners are applied.
The technique used is the right preconditioning, i.e., it is solved
<code><var>A</var>*inv(<var>M</var>)*y = b</code> and then <code><var>x</var> = inv(<var>M</var>)*y</code>.
</li><li>- <var>x0</var> the initial guess, if not given or set to [] the default
value <code>zeros (size (b))</code> is used.
</li></ul>
<p>The arguments which follow <var>x0</var> are treated as parameters, and passed in
a proper way to any of the functions (<var>A</var> or <var>P</var>) which are passed
to <code>cgs</code>.
</p>
<p>The output parameters are:
</p>
<ul class="no-bullet">
<li>- <var>x</var> is the approximation computed. If the method doesn’t
converge then it is the iterated with the minimum residual.
</li><li>- <var>flag</var> indicates the exit status:
<ul class="no-bullet">
<li>- 0: iteration converged to the within the chosen tolerance
</li><li>- 1: the maximum number of iterations was reached before convergence
</li><li>- 2: the preconditioner matrix is singular
</li><li>- 3: the algorithm reached stagnation
</li><li>- 4: the algorithm can’t continue due to a division by zero
</li></ul>
</li><li>- <var>relres</var> is the relative residual obtained with as
<code>(<var>A</var>*<var>x</var>-<var>b</var>) / <code>norm(<var>b</var>)</code></code>.
</li><li>- <var>iter</var> is the iteration which <var>x</var> is computed.
</li><li>- <var>resvec</var> is a vector containing the residual at each iteration.
Doing <code>length(<var>resvec</var>) - 1</code> is possible to see the total number
of iterations performed.
</li></ul>
<p>Let us consider a trivial problem with a tridiagonal matrix
</p>
<div class="example">
<pre class="example">n = 20;
A = toeplitz (sparse ([1, 1], [1, 2], [2, 1] * n ^ 2, 1, n)) + ...
toeplitz (sparse (1, 2, -1, 1, n) * n / 2, ...
sparse (1, 2, 1, 1, n) * n / 2);
b = A * ones (n, 1);
restart = 5;
[M1, M2] = ilu (A); # in this tridiag case it corresponds to chol (A)'
M = M1 * M2;
Afun = @(x) A * x;
Mfun = @(x) M \ x;
M1fun = @(x) M1 \ x;
M2fun = @(x) M2 \ x;
</pre></div>
<p><small>EXAMPLE 1:</small> simplest usage of <code>cgs</code>
</p>
<div class="example">
<pre class="example">x = cgs (A, b, [], n)
</pre></div>
<p><small>EXAMPLE 2:</small> <code>cgs</code> with a function which computes
<code><var>A</var> * <var>x</var></code>
</p>
<div class="example">
<pre class="example">x = cgs (Afun, b, [], n)
</pre></div>
<p><small>EXAMPLE 3:</small> <code>cgs</code> with a preconditioner matrix <var>M</var>
</p>
<div class="example">
<pre class="example">x = cgs (A, b, [], 1e-06, n, M)
</pre></div>
<p><small>EXAMPLE 4:</small> <code>cgs</code> with a function as preconditioner
</p>
<div class="example">
<pre class="example">x = cgs (Afun, b, 1e-6, n, Mfun)
</pre></div>
<p><small>EXAMPLE 5:</small> <code>cgs</code> with preconditioner matrices <var>M1</var>
and <var>M2</var>
</p>
<div class="example">
<pre class="example">x = cgs (A, b, [], 1e-6, n, M1, M2)
</pre></div>
<p><small>EXAMPLE 6:</small> <code>cgs</code> with functions as preconditioners
</p>
<div class="example">
<pre class="example">x = cgs (Afun, b, 1e-6, n, M1fun, M2fun)
</pre></div>
<p><small>EXAMPLE 7:</small> <code>cgs</code> with as input a function requiring an argument
</p>
<div class="example">
<pre class="example">function y = Ap (A, x, z) # compute A^z * x
y = x;
for i = 1:z
y = A * y;
endfor
endfunction
Apfun = @(x, string, p) Ap (A, x, string, p);
x = cgs (Apfun, b, [], [], [], [], [], 2);
</pre></div>
<p><small>EXAMPLE 8:</small> explicit example to show that <code>cgs</code> uses a
right preconditioner
</p>
<div class="example">
<pre class="example">[M1, M2] = ilu (A + 0.3 * eye (n)); # factorization of A perturbed
M = M1 * M2;
## reference solution computed by cgs after one iteration
[x_ref, fl] = cgs (A, b, [], 1, M)
## right preconditioning
[y, fl] = cgs (A / M, b, [], 1)
x = M \ y # compare x and x_ref
</pre></div>
<p>References:
</p>
<ol>
<li> Y. Saad, <cite>Iterative Methods for Sparse Linear Systems</cite>,
Second edition, 2003, SIAM
</li></ol>
<p><strong>See also:</strong> <a href="Iterative-Techniques.html#XREFpcg">pcg</a>, <a href="#XREFbicgstab">bicgstab</a>, <a href="#XREFbicg">bicg</a>, <a href="#XREFgmres">gmres</a>, <a href="#XREFqmr">qmr</a>, <a href="#XREFtfqmr">tfqmr</a>.
</p></dd></dl>
<a name="XREFgmres"></a><dl>
<dt><a name="index-gmres"></a><em><var>x</var> =</em> <strong>gmres</strong> <em>(<var>A</var>, <var>b</var>, <var>restart</var>, <var>tol</var>, <var>maxit</var>, <var>M1</var>, <var>M2</var>, <var>x0</var>, …)</em></dt>
<dt><a name="index-gmres-1"></a><em><var>x</var> =</em> <strong>gmres</strong> <em>(<var>A</var>, <var>b</var>, <var>restart</var>, <var>tol</var>, <var>maxit</var>, <var>M</var>, [], <var>x0</var>, …)</em></dt>
<dt><a name="index-gmres-2"></a><em>[<var>x</var>, <var>flag</var>, <var>relres</var>, <var>iter</var>, <var>resvec</var>] =</em> <strong>gmres</strong> <em>(<var>A</var>, <var>b</var>, …)</em></dt>
<dd><p>Solve <code>A x = b</code> using the Preconditioned GMRES iterative method with
restart, a.k.a. PGMRES(restart).
</p>
<p>The input arguments are:
</p>
<ul class="no-bullet">
<li>- <var>A</var> is the matrix of the linear system and it must be square.
<var>A</var> can be passed as a matrix, function handle, or inline
function <code>Afun</code> such that <code>Afun(x) = A * x</code>. Additional
parameters to <code>Afun</code> are passed after <var>x0</var>.
</li><li>- <var>b</var> is the right hand side vector. It must be a column vector
with the same numbers of rows as <var>A</var>.
</li><li>- <var>restart</var> is the number of iterations before that the
method restarts. If it is [] or N = numel (b), then the restart
is not applied.
</li><li>- <var>tol</var> is the required relative tolerance for the
preconditioned residual error,
<code>inv (<var>M</var>) * (<var>b</var> - <var>a</var> * <var>x</var>)</code>. The iteration
stops if <code>norm (inv (<var>M</var>) * (<var>b</var> - <var>a</var> * <var>x</var>))
≤ <var>tol</var> * norm (inv (<var>M</var>) * <var>B</var>)</code>. If <var>tol</var> is
omitted or empty, then a tolerance of 1e-6 is used.
</li><li>- <var>maxit</var> is the maximum number of outer iterations, if not given or
set to [], then the default value <code>min (10, <var>N</var> / <var>restart</var>)</code>
is used.
Note that, if <var>restart</var> is empty, then <var>maxit</var> is the maximum number
of iterations. If <var>restart</var> and <var>maxit</var> are not empty, then
the maximum number of iterations is <code><var>restart</var> * <var>maxit</var></code>.
If both <var>restart</var> and <var>maxit</var> are empty, then the maximum
number of iterations is set to <code>min (10, <var>N</var>)</code>.
</li><li>- <var>M1</var>, <var>M2</var> are the preconditioners. The preconditioner
<var>M</var> is given as <code>M = 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>. If <var>M1</var> is [] or not
given, then the preconditioner is not applied.
The technique used is the left-preconditioning, i.e., it is solved
<code>inv(<var>M</var>) * <var>A</var> * <var>x</var> = inv(<var>M</var>) * <var>b</var></code> instead of
<code><var>A</var> * <var>x</var> = <var>b</var></code>.
</li><li>- <var>x0</var> is the initial guess,
if not given or set to [], then the default value
<code>zeros (size (<var>b</var>))</code> is used.
</li></ul>
<p>The arguments which follow <var>x0</var> are treated as parameters, and passed in
a proper way to any of the functions (<var>A</var> or <var>M</var> or
<var>M1</var> or <var>M2</var>) which are passed to <code>gmres</code>.
</p>
<p>The outputs are:
</p>
<ul class="no-bullet">
<li>- <var>x</var> the computed approximation. If the method does not
converge, then it is the iterated with minimum residual.
</li><li>- <var>flag</var> indicates the exit status:
<dl compact="compact">
<dt>0 : iteration converged to within the specified tolerance</dt>
<dt>1 : maximum number of iterations exceeded</dt>
<dt>2 : the preconditioner matrix is singular</dt>
<dt>3 : algorithm reached stagnation (the relative difference between two</dt>
<dd><p>consecutive iterations is less than eps)
</p></dd>
</dl>
</li><li>- <var>relres</var> is the value of the relative preconditioned
residual of the approximation <var>x</var>.
</li><li>- <var>iter</var> is a vector containing the number of outer iterations and
inner iterations performed to compute <var>x</var>. That is:
<ul>
<li> <var>iter(1)</var>: number of outer iterations, i.e., how many
times the method restarted. (if <var>restart</var> is empty or <var>N</var>,
then it is 1, if not 1 ≤ <var>iter(1)</var> ≤ <var>maxit</var>).
</li><li> <var>iter(2)</var>: the number of iterations performed before the
restart, i.e., the method restarts when
<code><var>iter(2)</var> = <var>restart</var></code>. If <var>restart</var> is empty or
<var>N</var>, then 1 ≤ <var>iter(2)</var> ≤ <var>maxit</var>.
</li></ul>
<p>To be more clear, the approximation <var>x</var> is computed at the iteration
<code>(<var>iter(1)</var> - 1) * <var>restart</var> + <var>iter(2)</var></code>.
Since the output <var>x</var> corresponds to the minimal preconditioned
residual solution, the total number of iterations that
the method performed is given by <code>length (resvec) - 1</code>.
</p>
</li><li>- <var>resvec</var> is a vector containing the preconditioned
relative residual at each iteration, including the 0-th iteration
<code>norm (<var>A</var> * <var>x0</var> - <var>b</var>)</code>.
</li></ul>
<p>Let us consider a trivial problem with a tridiagonal matrix
</p>
<div class="example">
<pre class="example">n = 20;
A = toeplitz (sparse ([1, 1], [1, 2], [2, 1] * n ^ 2, 1, n)) + ...
toeplitz (sparse (1, 2, -1, 1, n) * n / 2, ...
sparse (1, 2, 1, 1, n) * n / 2);
b = A * ones (n, 1);
restart = 5;
[M1, M2] = ilu (A); # in this tridiag case, it corresponds to lu (A)
M = M1 * M2;
Afun = @(x) A * x;
Mfun = @(x) M \ x;
M1fun = @(x) M1 \ x;
M2fun = @(x) M2 \ x;
</pre></div>
<p><small>EXAMPLE 1:</small> simplest usage of <code>gmres</code>
</p>
<div class="example">
<pre class="example">x = gmres (A, b, [], [], n)
</pre></div>
<p><small>EXAMPLE 2:</small> <code>gmres</code> with a function which computes
<code><var>A</var> * <var>x</var></code>
</p>
<div class="example">
<pre class="example">x = gmres (Afun, b, [], [], n)
</pre></div>
<p><small>EXAMPLE 3:</small> usage of <code>gmres</code> with the restart
</p>
<div class="example">
<pre class="example">x = gmres (A, b, restart);
</pre></div>
<p><small>EXAMPLE 4:</small> <code>gmres</code> with a preconditioner matrix <var>M</var>
with and without restart
</p>
<div class="example">
<pre class="example">x = gmres (A, b, [], 1e-06, n, M)
x = gmres (A, b, restart, 1e-06, n, M)
</pre></div>
<p><small>EXAMPLE 5:</small> <code>gmres</code> with a function as preconditioner
</p>
<div class="example">
<pre class="example">x = gmres (Afun, b, [], 1e-6, n, Mfun)
</pre></div>
<p><small>EXAMPLE 6:</small> <code>gmres</code> with preconditioner matrices <var>M1</var>
and <var>M2</var>
</p>
<div class="example">
<pre class="example">x = gmres (A, b, [], 1e-6, n, M1, M2)
</pre></div>
<p><small>EXAMPLE 7:</small> <code>gmres</code> with functions as preconditioners
</p>
<div class="example">
<pre class="example">x = gmres (Afun, b, 1e-6, n, M1fun, M2fun)
</pre></div>
<p><small>EXAMPLE 8:</small> <code>gmres</code> with as input a function requiring an argument
</p>
<div class="example">
<pre class="example"> function y = Ap (A, x, p) # compute A^p * x
y = x;
for i = 1:p
y = A * y;
endfor
endfunction
Apfun = @(x, p) Ap (A, x, p);
x = gmres (Apfun, b, [], [], [], [], [], [], 2);
</pre></div>
<p><small>EXAMPLE 9:</small> explicit example to show that <code>gmres</code> uses a
left preconditioner
</p>
<div class="example">
<pre class="example">[M1, M2] = ilu (A + 0.1 * eye (n)); # factorization of A perturbed
M = M1 * M2;
## reference solution computed by gmres after two iterations
[x_ref, fl] = gmres (A, b, [], [], 1, M)
## left preconditioning
[x, fl] = gmres (M \ A, M \ b, [], [], 1)
x # compare x and x_ref
</pre></div>
<p>References:
</p>
<ol>
<li> Y. Saad, <cite>Iterative Methods for Sparse Linear
Systems</cite>, Second edition, 2003, SIAM
</li></ol>
<p><strong>See also:</strong> <a href="#XREFbicg">bicg</a>, <a href="#XREFbicgstab">bicgstab</a>, <a href="#XREFcgs">cgs</a>, <a href="Iterative-Techniques.html#XREFpcg">pcg</a>, <a href="Iterative-Techniques.html#XREFpcr">pcr</a>, <a href="#XREFqmr">qmr</a>, <a href="#XREFtfqmr">tfqmr</a>.
</p></dd></dl>
<a name="XREFqmr"></a><dl>
<dt><a name="index-qmr"></a><em><var>x</var> =</em> <strong>qmr</strong> <em>(<var>A</var>, <var>b</var>, <var>rtol</var>, <var>maxit</var>, <var>M1</var>, <var>M2</var>, <var>x0</var>)</em></dt>
<dt><a name="index-qmr-1"></a><em><var>x</var> =</em> <strong>qmr</strong> <em>(<var>A</var>, <var>b</var>, <var>rtol</var>, <var>maxit</var>, <var>P</var>)</em></dt>
<dt><a name="index-qmr-2"></a><em>[<var>x</var>, <var>flag</var>, <var>relres</var>, <var>iter</var>, <var>resvec</var>] =</em> <strong>qmr</strong> <em>(<var>A</var>, <var>b</var>, …)</em></dt>
<dd><p>Solve <code>A x = b</code> using the Quasi-Minimal Residual iterative method
(without look-ahead).
</p>
<ul class="no-bullet">
<li>- <var>rtol</var> is the relative tolerance, if not given or set to [] the
default value 1e-6 is used.
</li><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><li>- <var>x0</var> the initial guess, if not given or set to [] the default
value <code>zeros (size (b))</code> is used.
</li></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>
<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>
<p>If called with more than one output parameter
</p>
<ul class="no-bullet">
<li>- <var>flag</var> indicates the exit status:
<ul class="no-bullet">
<li>- 0: iteration converged to the within the chosen tolerance
</li><li>- 1: the maximum number of iterations was reached before convergence
</li><li>- 3: the algorithm reached stagnation
</li></ul>
<p>(the value 2 is unused but skipped for compatibility).
</p>
</li><li>- <var>relres</var> is the final value of the relative residual.
</li><li>- <var>iter</var> is the number of iterations performed.
</li><li>- <var>resvec</var> is a vector containing the residual norms at each
iteration.
</li></ul>
<p>References:
</p>
<ol>
<li> R. Freund and N. Nachtigal, <cite>QMR: a quasi-minimal residual
method for non-Hermitian linear systems</cite>, Numerische Mathematik,
1991, 60, pp. 315-339.
</li><li> R. Barrett, M. Berry, T. Chan, J. Demmel, J. Donato, J. Dongarra,
V. Eijkhour, R. Pozo, C. Romine, and H. van der Vorst,
<cite>Templates for the solution of linear systems: Building blocks
for iterative methods</cite>, SIAM, 2nd ed., 1994.
</li></ol>
<p><strong>See also:</strong> <a href="#XREFbicg">bicg</a>, <a href="#XREFbicgstab">bicgstab</a>, <a href="#XREFcgs">cgs</a>, <a href="#XREFgmres">gmres</a>, <a href="Iterative-Techniques.html#XREFpcg">pcg</a>.
</p></dd></dl>
<a name="XREFtfqmr"></a><dl>
<dt><a name="index-tfqmr"></a><em><var>x</var> =</em> <strong>tfqmr</strong> <em>(<var>A</var>, <var>b</var>, <var>tol</var>, <var>maxit</var>, <var>M1</var>, <var>M2</var>, <var>x0</var>, …)</em></dt>
<dt><a name="index-tfqmr-1"></a><em><var>x</var> =</em> <strong>tfqmr</strong> <em>(<var>A</var>, <var>b</var>, <var>tol</var>, <var>maxit</var>, <var>M</var>, [], <var>x0</var>, …)</em></dt>
<dt><a name="index-tfqmr-2"></a><em>[<var>x</var>, <var>flag</var>, <var>relres</var>, <var>iter</var>, <var>resvec</var>] =</em> <strong>tfqmr</strong> <em>(<var>A</var>, <var>b</var>, …)</em></dt>
<dd><p>Solve <code>A x = b</code> using the Transpose-Tree qmr method, based on the cgs.
</p>
<p>The input parameters are:
</p>
<ul class="no-bullet">
<li>- <var>A</var> is the matrix of the linear system and it must be square.
<var>A</var> can be passed as a matrix, function handle, or inline
function <code>Afun</code> such that <code>Afun(x) = A * x</code>. Additional
parameters to <code>Afun</code> are passed after <var>x0</var>.
</li><li>- <var>b</var> is the right hand side vector. It must be a column vector
with the same number of rows as <var>A</var>.
</li><li>- <var>tol</var> is the relative tolerance, if not given or set to [] the
default value 1e-6 is used.
</li><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. To be
compatible, since the method as different behaviors in the iteration
number is odd or even, is considered as iteration in <code>tfqmr</code> the
entire odd-even cycle. That is, to make an entire iteration, the algorithm
performs two sub-iterations: the odd one and the even one.
</li><li>- <var>M1</var>, <var>M2</var> are the preconditioners. The preconditioner
<var>M</var> is given as <code>M = 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>.
The technique used is the right-preconditioning, i.e., it is solved
<code>A*inv(M)*y = b</code> and then <code>x = inv(M)*y</code>, instead of
<code>A x = b</code>.
</li><li>- <var>x0</var> the initial guess, if not given or set to [] the default
value <code>zeros (size (b))</code> is used.
</li></ul>
<p>The arguments which follow <var>x0</var> are treated as parameters, and passed in
a proper way to any of the functions (<var>A</var> or <var>M</var>) which are passed
to <code>tfqmr</code>.
</p>
<p>The output parameters are:
</p>
<ul class="no-bullet">
<li>- <var>x</var> is the approximation computed. If the method doesn’t
converge then it is the iterated with the minimum residual.
</li><li>- <var>flag</var> indicates the exit status:
<ul class="no-bullet">
<li>- 0: iteration converged to the within the chosen tolerance
</li><li>- 1: the maximum number of iterations was reached before convergence
</li><li>- 2: the preconditioner matrix is singular
</li><li>- 3: the algorithm reached stagnation
</li><li>- 4: the algorithm can’t continue due to a division by zero
</li></ul>
</li><li>- <var>relres</var> is the relative residual obtained as
<code>(<var>A</var>*<var>x</var>-<var>b</var>) / <code>norm (<var>b</var>)</code></code>.
</li><li>- <var>iter</var> is the iteration which <var>x</var> is
computed.
</li><li>- <var>resvec</var> is a vector containing the residual at each iteration
(including <code>norm (b - A x0)</code>).
Doing <code>length (<var>resvec</var>) - 1</code> is possible to see the
total number of iterations performed.
</li></ul>
<p>Let us consider a trivial problem with a tridiagonal matrix
</p>
<div class="example">
<pre class="example">n = 20;
A = toeplitz (sparse ([1, 1], [1, 2], [2, 1] * n ^ 2, 1, n)) + ...
toeplitz (sparse (1, 2, -1, 1, n) * n / 2, ...
sparse (1, 2, 1, 1, n) * n / 2);
b = A * ones (n, 1);
restart = 5;
[M1, M2] = ilu (A); # in this tridiag case it corresponds to chol (A)'
M = M1 * M2;
Afun = @(x) A * x;
Mfun = @(x) M \ x;
M1fun = @(x) M1 \ x;
M2fun = @(x) M2 \ x;
</pre></div>
<p><small>EXAMPLE 1:</small> simplest usage of <code>tfqmr</code>
</p>
<div class="example">
<pre class="example">x = tfqmr (A, b, [], n)
</pre></div>
<p><small>EXAMPLE 2:</small> <code>tfqmr</code> with a function which computes
<code><var>A</var> * <var>x</var></code>
</p>
<div class="example">
<pre class="example">x = tfqmr (Afun, b, [], n)
</pre></div>
<p><small>EXAMPLE 3:</small> <code>tfqmr</code> with a preconditioner matrix <var>M</var>
</p>
<div class="example">
<pre class="example">x = tfqmr (A, b, [], 1e-06, n, M)
</pre></div>
<p><small>EXAMPLE 4:</small> <code>tfqmr</code> with a function as preconditioner
</p>
<div class="example">
<pre class="example">x = tfqmr (Afun, b, 1e-6, n, Mfun)
</pre></div>
<p><small>EXAMPLE 5:</small> <code>tfqmr</code> with preconditioner matrices <var>M1</var>
and <var>M2</var>
</p>
<div class="example">
<pre class="example">x = tfqmr (A, b, [], 1e-6, n, M1, M2)
</pre></div>
<p><small>EXAMPLE 6:</small> <code>tfmqr</code> with functions as preconditioners
</p>
<div class="example">
<pre class="example">x = tfqmr (Afun, b, 1e-6, n, M1fun, M2fun)
</pre></div>
<p><small>EXAMPLE 7:</small> <code>tfqmr</code> with as input a function requiring an argument
</p>
<div class="example">
<pre class="example">function y = Ap (A, x, z) # compute A^z * x
y = x;
for i = 1:z
y = A * y;
endfor
endfunction
Apfun = @(x, string, p) Ap (A, x, string, p);
x = tfqmr (Apfun, b, [], [], [], [], [], 2);
</pre></div>
<p><small>EXAMPLE 8:</small> explicit example to show that <code>tfqmr</code> uses a
right preconditioner
</p>
<div class="example">
<pre class="example">[M1, M2] = ilu (A + 0.3 * eye (n)); # factorization of A perturbed
M = M1 * M2;
## reference solution computed by tfqmr after one iteration
[x_ref, fl] = tfqmr (A, b, [], 1, M)
## right preconditioning
[y, fl] = tfqmr (A / M, b, [], 1)
x = M \ y # compare x and x_ref
</pre></div>
<p>References:
</p>
<ol>
<li> Y. Saad, <cite>Iterative Methods for Sparse Linear Systems</cite>,
Second edition, 2003, SIAM
</li></ol>
<p><strong>See also:</strong> <a href="#XREFbicg">bicg</a>, <a href="#XREFbicgstab">bicgstab</a>, <a href="#XREFcgs">cgs</a>, <a href="#XREFgmres">gmres</a>, <a href="Iterative-Techniques.html#XREFpcg">pcg</a>, <a href="#XREFqmr">qmr</a>, <a href="Iterative-Techniques.html#XREFpcr">pcr</a>.
</p>
</dd></dl>
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