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psblas3/docs/pdf/methods.tex

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19 years ago
\section{Iterative Methods}
\label{sec:methods}
In this chapter we provide routines for preconditioners and iterative
methods. The interfaces for Krylov subspace methods are available in
the module \verb|psb_krylov_mod|.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Krylov Methods driver routine
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subroutine{psb\_krylov \label{krylov}}{Krylov Methods Driver Routine}
This subroutine is a driver that provides a general interface for all
the Krylov-Subspace family methods implemented in PSBLAS version 2.
The stopping criterion is the normwise backward error, in the infinity
norm, i.e. the iteration is stopped when
\[ err = \frac{\|r_i\|}{(\|A\|\|x_i\|+\|b\|)} < eps \]
or the 2-norm residual reduction
\[ err = \frac{\|r_i\|}{\|b\|_2} < eps \]
according to the value passed through the istop argument (see
later). In the above formulae, $x_i$ is the tentative solution and
$r_i=b-Ax_i$ the corresponding residual at the $i$-th iteration.
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\syntax{call psb\_krylov}{method,a,prec,b,x,eps,desc\_a,info,itmax,iter,err,itrace,irst,istop}
\begin{description}
\item[\bf On Entry]
\item[method] a string that defines the iterative method to be
used. Supported values are:
\begin{description}
\item[CG]: the Conjugate Gradient method;
\item[CGS]:the Conjugate Gradient Stabilized method;
\item[BICG]: the Bi-Conjugate Gradient method;
\item[BICGSTAB]: the Bi-Conjugate Gradient Stabilized method;
\item[BICGSTABL]: the Bi-Conjugate Gradient Stabilized method with restarting;
\item[RGMRES]: the Generalized Minimal Residual method with restarting.
\end{description}
\item[a] the local portion of global sparse matrix
$A$. \\
Scope: {\bf local} \\
Type: {\bf required}\\
Intent: {\bf in}.\\
Specified as: a structured data of type \spdata.
\item[prec] The data structure containing the preconditioner.\\
Scope: {\bf local} \\
Type: {\bf required}\\
Intent: {\bf in}.\\
Specified as: a structured data of type \precdata.
\item[b] The RHS vector. \\
Scope: {\bf local} \\
Type: {\bf required}\\
Intent: {\bf in}.\\
Specified as: a rank one array.
\item[x] The initial guess. \\
Scope: {\bf local} \\
Type: {\bf required}\\
Intent: {\bf inout}.\\
Specified as: a rank one array.
\item[eps] The stopping tolerance. \\
Scope: {\bf global} \\
Type: {\bf required}\\
Intent: {\bf in}.\\
Specified as: a real number.
\item[desc\_a] contains data structures for communications.\\
Scope: {\bf local} \\
Type: {\bf required}\\
Intent: {\bf in}.\\
Specified as: a structured data of type \descdata.
\item[itmax] The maximum number of iterations to perform.\\
Scope: {\bf global} \\
Type: {\bf optional}\\
Intent: {\bf in}.\\
Default: $itmax = 1000$.\\
Specified as: an integer variable $itmax \ge 1$.
\item[itrace] If $>0$ print out an informational message about
convergence every $itrace$ iterations.\\
Scope: {\bf global} \\
Type: {\bf optional}\\
Intent: {\bf in}.\\
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\item[irst] An integer specifying the restart parameter.\\
Scope: {\bf global} \\
Type: {\bf optional}.\\
Intent: {\bf in}.\\
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Values: $irst>0$. This is employed for the BiCGSTABL or RGMRES
methods, otherwise it is ignored.
\item[istop] An integer specifying the stopping criterion.\\
Scope: {\bf global} \\
Type: {\bf optional}.\\
Intent: {\bf in}.\\
Values: 1: use the normwise backward error, 2: use the scaled 2-norm
of the residual. Default: 1.
\item[\bf On Return]
\item[x] The computed solution. \\
Scope: {\bf local} \\
Type: {\bf required}\\
Intent: {\bf inout}.\\
Specified as: a rank one array.
\item[iter] The number of iterations performed.\\
Scope: {\bf global} \\
Type: {\bf optional}\\
Intent: {\bf out}.\\
Returned as: an integer variable.
\item[err] The convergence estimate on exit.\\
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Scope: {\bf global} \\
Type: {\bf optional}\\
Intent: {\bf out}.\\
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Returned as: a real number.
\item[info] Error code.\\
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Scope: {\bf local} \\
Type: {\bf required} \\
Intent: {\bf out}.\\
An integer value; 0 means no error has been detected.
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\end{description}
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% %
%% % CG
%% %
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% \subroutine{psb\_cg \label{cg}}{CG Iterative Method}
%% This subroutine implements the CG method with restarting. The
%% stopping criterion is the normwise backward error, in the infinity
%% norm, i.e. the iteration is stopped when
%% \[ \frac{\|r\|}{(\|A\|\|x\|+\|b\|)} < eps \]
%% or
%% \[ \frac{\|r_i\|}{\|b\|_2} < eps \]
%% according to the value passed through the istop argument (see later).
%% \syntax{call psb\_cg}{a,prec,b,x,eps,desc\_a,info,itmax,iter,err,itrace,istop}
%% \begin{description}
%% \item[\bf On Entry]
%% \item[a] the local portion of global sparse matrix
%% $A$. \\
%% Scope: {\bf local} \\
%% Type: {\bf required}\\
%% Specified as: a structured data of type \spdata.
%% \item[prec] The data structure containing the preconditioner.\\
%% Scope: {\bf local} \\
%% Type: {\bf required}\\
%% Specified as: a structured data of type \precdata.
%% \item[b] The RHS vector. \\
%% Scope: {\bf local} \\
%% Type: {\bf required}\\
%% Specified as: a rank one array.
%% \item[x] The initial guess. \\
%% Scope: {\bf local} \\
%% Type: {\bf required}\\
%% Specified as: a rank one array.
%% \item[eps] The stopping tolerance. \\
%% Scope: {\bf global} \\
%% Type: {\bf required}\\
%% Specified as: a real number.
%% \item[desc\_a] contains data structures for communications.\\
%% Scope: {\bf local} \\
%% Type: {\bf required}\\
%% Specified as: a structured data of type \descdata.
%% \item[itmax] The maximum number of iterations to perform.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}\\
%% Default: $itmax = 1000$.\\
%% Specified as: an integer variable $itmax \ge 1$.
%% \item[itrace] If $>0$ print out an informational message about
%% convergence every $itrace$ iterations.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}\\
%% \item[istop] An integer specifying the stopping criterion.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}\\
%% \item[\bf On Return]
%% \item[x] The computed solution. \\
%% Scope: {\bf local} \\
%% Type: {\bf required}\\
%% Specified as: a rank one array.
%% \item[iter] The number of iterations performed.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}\\
%% Returned as: an integer variable.
%% \item[err] The error estimate on exit.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}\\
%% Returned as: a real number.
%% \item[info] Error code.\\
%% Scope: {\bf local} \\
%% Type: {\bf required} \\
%% An integer value; 0 means no error has been detected.
%% \end{description}
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% %
%% % CGS
%% %
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% \subroutine{psb\_cgs \label{cgs}}{CGS Iterative Method}
%% This subroutine implements the CGS method with restarting. The
%% stopping criterion is the normwise backward error, in the infinity
%% norm, i.e. the iteration is stopped when
%% \[ \frac{\|r\|}{(\|A\|\|x\|+\|b\|)} < eps \]
%% or
%% \[ \frac{\|r_i\|}{\|b\|_2} < eps \]
%% according to the value passed through the istop argument (see later).
%% \syntax{call psb\_cgs}{a,prec,b,x,eps,desc\_a,info,itmax,iter,err,itrace,istop}
%% \begin{description}
%% \item[\bf On Entry]
%% \item[a] the local portion of global sparse matrix
%% $A$. \\
%% Scope: {\bf local} \\
%% Type: {\bf required}\\
%% Specified as: a structured data of type \spdata.
%% \item[prec] The data structure containing the preconditioner.\\
%% Scope: {\bf local} \\
%% Type: {\bf required}\\
%% Specified as: a structured data of type \precdata.
%% \item[b] The RHS vector. \\
%% Scope: {\bf local} \\
%% Type: {\bf required}\\
%% Specified as: a rank one array.
%% \item[x] The initial guess. \\
%% Scope: {\bf local} \\
%% Type: {\bf required}\\
%% Specified as: a rank one array.
%% \item[eps] The stopping tolerance. \\
%% Scope: {\bf global} \\
%% Type: {\bf required}\\
%% Specified as: a real number.
%% \item[desc\_a] contains data structures for communications.\\
%% Scope: {\bf local} \\
%% Type: {\bf required}\\
%% Specified as: a structured data of type \descdata.
%% \item[itmax] The maximum number of iterations to perform.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}\\
%% Default: $itmax = 1000$.\\
%% Specified as: an integer variable $itmax \ge 1$.
%% \item[itrace] If $>0$ print out an informational message about
%% convergence every $itrace$ iterations.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}\\
%% \item[istop] An integer specifying the stopping criterion.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}\\
%% \item[\bf On Return]
%% \item[x] The computed solution. \\
%% Scope: {\bf local} \\
%% Type: {\bf required}\\
%% Specified as: a rank one array.
%% \item[iter] The number of iterations performed.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}\\
%% Returned as: an integer variable.
%% \item[err] The error estimate on exit.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}\\
%% Returned as: a real number.
%% \item[info] Error code.\\
%% Scope: {\bf local} \\
%% Type: {\bf required} \\
%% An integer value; 0 means no error has been detected.
%% \end{description}
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% %
%% % BiCG
%% %
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% \subroutine{psb\_bicg \label{bicg}}{BiCG Iterative Method}
%% This subroutine implements the BiCG method with restarting. The
%% stopping criterion is the normwise backward error, in the infinity
%% norm, i.e. the iteration is stopped when
%% \[ \frac{\|r\|}{(\|A\|\|x\|+\|b\|)} < eps \]
%% or
%% \[ \frac{\|r_i\|}{\|b\|_2} < eps \]
%% according to the value passed through the istop argument (see later).
%% \syntax{call psb\_bicg}{a,prec,b,x,eps,desc\_a,info,itmax,iter,err,itrace,istop}
%% \begin{description}
%% \item[\bf On Entry]
%% \item[a] the local portion of global sparse matrix
%% $A$. \\
%% Scope: {\bf local} \\
%% Type: {\bf required}\\
%% Specified as: a structured data of type \spdata.
%% \item[prec] The data structure containing the preconditioner.\\
%% Scope: {\bf local} \\
%% Type: {\bf required}\\
%% Specified as: a structured data of type \precdata.
%% \item[b] The RHS vector. \\
%% Scope: {\bf local} \\
%% Type: {\bf required}\\
%% Specified as: a rank one array.
%% \item[x] The initial guess. \\
%% Scope: {\bf local} \\
%% Type: {\bf required}\\
%% Specified as: a rank one array.
%% \item[eps] The stopping tolerance. \\
%% Scope: {\bf global} \\
%% Type: {\bf required}\\
%% Specified as: a real number.
%% \item[desc\_a] contains data structures for communications.\\
%% Scope: {\bf local} \\
%% Type: {\bf required}\\
%% Specified as: a structured data of type \descdata.
%% \item[itmax] The maximum number of iterations to perform.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}\\
%% Default: $itmax = 1000$.\\
%% Specified as: an integer variable $itmax \ge 1$.
%% \item[itrace] If $>0$ print out an informational message about
%% convergence every $itrace$ iterations.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}\\
%% \item[istop] An integer specifying the stopping criterion.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}\\
%% \item[\bf On Return]
%% \item[x] The computed solution. \\
%% Scope: {\bf local} \\
%% Type: {\bf required}\\
%% Specified as: a rank one array.
%% \item[iter] The number of iterations performed.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}\\
%% Returned as: an integer variable.
%% \item[err] The error estimate on exit.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}\\
%% Returned as: a real number.
%% \item[info] Error code.\\
%% Scope: {\bf local} \\
%% Type: {\bf required} \\
%% An integer value; 0 means no error has been detected.
%% \end{description}
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% %
%% % BiCGSTAB
%% %
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% \subroutine{psb\_bicgstab \label{bicgstab}}{BiCGSTAB Iterative Method}
%% This subroutine implements the BiCGSTAB method with restarting. The
%% stopping criterion is the normwise backward error, in the infinity
%% norm, i.e. the iteration is stopped when
%% \[ \frac{\|r\|}{(\|A\|\|x\|+\|b\|)} < eps \]
%% or
%% \[ \frac{\|r_i\|}{\|b\|_2} < eps \]
%% according to the value passed through the istop argument (see later).
%% \syntax{call psb\_bicgstab}{a,prec,b,x,eps,desc\_a,info,itmax,iter,err,itrace,istop}
%% \begin{description}
%% \item[\bf On Entry]
%% \item[a] the local portion of global sparse matrix
%% $A$. \\
%% Scope: {\bf local} \\
%% Type: {\bf required}\\
%% Specified as: a structured data of type \spdata.
%% \item[prec] The data structure containing the preconditioner.\\
%% Scope: {\bf local} \\
%% Type: {\bf required}\\
%% Specified as: a structured data of type \precdata.
%% \item[b] The RHS vector. \\
%% Scope: {\bf local} \\
%% Type: {\bf required}\\
%% Specified as: a rank one array.
%% \item[x] The initial guess. \\
%% Scope: {\bf local} \\
%% Type: {\bf required}\\
%% Specified as: a rank one array.
%% \item[eps] The stopping tolerance. \\
%% Scope: {\bf global} \\
%% Type: {\bf required}\\
%% Specified as: a real number.
%% \item[desc\_a] contains data structures for communications.\\
%% Scope: {\bf local} \\
%% Type: {\bf required}\\
%% Specified as: a structured data of type \descdata.
%% \item[itmax] The maximum number of iterations to perform.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}\\
%% Default: $itmax = 1000$.\\
%% Specified as: an integer variable $itmax \ge 1$.
%% \item[itrace] If $>0$ print out an informational message about
%% convergence every $itrace$ iterations.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}\\
%% \item[istop] An integer specifying the stopping criterion.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}\\
%% \item[\bf On Return]
%% \item[x] The computed solution. \\
%% Scope: {\bf local} \\
%% Type: {\bf required}\\
%% Specified as: a rank one array.
%% \item[iter] The number of iterations performed.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}\\
%% Returned as: an integer variable.
%% \item[err] The error estimate on exit.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}\\
%% Returned as: a real number.
%% \item[info] Error code.\\
%% Scope: {\bf local} \\
%% Type: {\bf required} \\
%% An integer value; 0 means no error has been detected.
%% \end{description}
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% %
%% % BiCGSTAB(L)
%% %
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% \subroutine{psb\_bicgstabl \label{bicgstabl}}{BiCGSTAB-$l$ Iterative Method}
%% This subroutine implements the BiCGSTAB-$l$ method with restarting. The
%% stopping criterion is the normwise backward error, in the infinity
%% norm, i.e. the iteration is stopped when
%% \[ \frac{\|r\|}{(\|A\|\|x\|+\|b\|)} < eps \]
%% or
%% \[ \frac{\|r_i\|}{\|b\|_2} < eps \]
%% according to the value passed through the istop argument (see later).
%% \syntax{call psb\_bicgstab}{a,prec,b,x,eps,desc\_a,info,itmax,iter,err,itrace,irst,istop}
%% \begin{description}
%% \item[\bf On Entry]
%% \item[a] the local portion of global sparse matrix
%% $A$. \\
%% Scope: {\bf local} \\
%% Type: {\bf required}\\
%% Specified as: a structured data of type \spdata.
%% \item[prec] The data structure containing the preconditioner.\\
%% Scope: {\bf local} \\
%% Type: {\bf required}\\
%% Specified as: a structured data of type \precdata.
%% \item[b] The RHS vector. \\
%% Scope: {\bf local} \\
%% Type: {\bf required}\\
%% Specified as: a rank one array.
%% \item[x] The initial guess. \\
%% Scope: {\bf local} \\
%% Type: {\bf required}\\
%% Specified as: a rank one array.
%% \item[eps] The stopping tolerance. \\
%% Scope: {\bf global} \\
%% Type: {\bf required}\\
%% Specified as: a real number.
%% \item[desc\_a] contains data structures for communications.\\
%% Scope: {\bf local} \\
%% Type: {\bf required}\\
%% Specified as: a structured data of type \descdata.
%% \item[itmax] The maximum number of iterations to perform.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}\\
%% Default: $itmax = 1000$.\\
%% Specified as: an integer variable $itmax \ge 1$.
%% \item[itrace] If $>0$ print out an informational message about
%% convergence every $itrace$ iterations.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}\\
%% \item[irst] An integer specifying the restarting iteration.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}\\
%% \item[istop] An integer specifying the stopping criterion.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}\\
%% \item[\bf On Return]
%% \item[x] The computed solution. \\
%% Scope: {\bf local} \\
%% Type: {\bf required}\\
%% Specified as: a rank one array.
%% \item[iter] The number of iterations performed.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}\\
%% Returned as: an integer variable.
%% \item[err] The error estimate on exit.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}\\
%% Returned as: a real number.
%% \item[info] Error code.\\
%% Scope: {\bf local} \\
%% Type: {\bf required} \\
%% An integer value; 0 means no error has been detected.
%% \end{description}
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% %
%% % GMRES
%% %
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% \subroutine{psb\_gmres \label{gmres}}{GMRES Iterative Method}
%% This subroutine implements the GMRES method with restarting. The
%% stopping criterion is the normwise backward error, in the infinity
%% norm, i.e. the iteration is stopped when
%% \[ \frac{\|r\|}{(\|A\|\|x\|+\|b\|)} < eps \]
%% or
%% \[ \frac{\|r_i\|}{\|b\|_2} < eps \]
%% according to the value passed through the istop argument (see later).
%% \syntax{call psb\_gmres}{a,prec,b,x,eps,desc\_a,info,itmax,iter,err,itrace,irst,istop}
%% \begin{description}
%% \item[\bf On Entry]
%% \item[a] the local portion of global sparse matrix
%% $A$. \\
%% Scope: {\bf local} \\
%% Type: {\bf required}\\
%% Specified as: a structured data of type \spdata.
%% \item[prec] The data structure containing the preconditioner.\\
%% Scope: {\bf local} \\
%% Type: {\bf required}\\
%% Specified as: a structured data of type \precdata.
%% \item[b] The RHS vector. \\
%% Scope: {\bf local} \\
%% Type: {\bf required}\\
%% Specified as: a rank one array.
%% \item[x] The initial guess. \\
%% Scope: {\bf local} \\
%% Type: {\bf required}\\
%% Specified as: a rank one array.
%% \item[eps] The stopping tolerance. \\
%% Scope: {\bf global} \\
%% Type: {\bf required}\\
%% Specified as: a real number.
%% \item[desc\_a] contains data structures for communications.\\
%% Scope: {\bf local} \\
%% Type: {\bf required}\\
%% Specified as: a structured data of type \descdata.
%% \item[itmax] The maximum number of iterations to perform.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}\\
%% Default: $itmax = 1000$.\\
%% Specified as: an integer variable $itmax \ge 1$.
%% \item[itrace] If $>0$ print out an informational message about
%% convergence every $itrace$ iterations.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}\\
%% \item[irst] An integer specifying the restart iteration.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}\\
%% \item[istop] An integer specifying the stopping criterion.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}\\
%% \item[\bf On Return]
%% \item[x] The computed solution. \\
%% Scope: {\bf local} \\
%% Type: {\bf required}\\
%% Specified as: a rank one array.
%% \item[iter] The number of iterations performed.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}\\
%% Returned as: an integer variable.
%% \item[err] The error estimate on exit.\\
%% Scope: {\bf global} \\
%% Type: {\bf optional}\\
%% Returned as: a real number.
%% \item[info] Error code.\\
%% Scope: {\bf local} \\
%% Type: {\bf required} \\
%% An integer value; 0 means no error has been detected.
%% \end{description}
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