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577 lines
17 KiB
TeX
577 lines
17 KiB
TeX
\section{Iterative Methods}
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\label{sec:methods}
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In this chapter we provide routines for preconditioners and iterative
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methods.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%
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% Krylov Methods driver routine
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%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subroutine{psb\_krylov \label{krylov}}{Krylov Methods Driver Routine}
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This subroutine is a driver that provides a general interface for all
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the Krylov-Subspace family methods implemented in PSBLAS-2.0. The
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stopping criterion is the normwise backward error, in the infinity
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norm, i.e. the iteration is stopped when
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\[ \frac{\|r\|}{(\|A\|\|x\|+\|b\|)} < eps \]
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or
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\[ \frac{\|r_i\|}{\|b\|_2} < eps \]
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according to the value passed through the istop argument (see later).
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\syntax{call psb\_krylov}{method,a,prec,b,x,eps,desc\_a,info,itmax,iter,err,itrace,istop}
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\begin{description}
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\item[\bf On Entry]
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\item[method] a string that defines the iterative method to be
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used. Valid values in PSBLAS-2.0 are:
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\begin{description}
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\item[CG]: the Conjugate gradient method;
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\item[CGS]:the Conjugate Gradient Stabilized method;
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\item[BICG]: the Bi-Conjugate Gradient method;
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\item[BICGSTAB]: the Bi-Conjugate Gradient Stabilized method;
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\item[BICGSTABL]: the Bi-Conjugate Gradient Stabilized method with restarting;
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\item[RGMRES]: the Generalized Minimal Residual method with restarting.
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\end{description}
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\item[a] the local portion of global sparse matrix
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$A$. \\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a structured data of type \spdata.
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\item[prec] The data structure containing the preconditioner.\\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a structured data of type \precdata.
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\item[b] The RHS vector. \\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a rank one array.
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\item[x] The initial guess. \\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a rank one array.
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\item[eps] The stopping tolerance. \\
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Scope: {\bf global} \\
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Type: {\bf required}\\
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Specified as: a real number.
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\item[desc\_a] contains data structures for communications.\\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a structured data of type \descdata.
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\item[itmax] The maximum number of iterations to perform.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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Default: $itmax = 1000$.\\
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Specified as: an integer variable $itmax \ge 1$.
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\item[itrace] If $>0$ print out an informational message about
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convergence every $itrace$ iterations.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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\item[istop] An integer specifying the stopping criterion.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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\item[\bf On Return]
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\item[x] The computed solution. \\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a rank one array.
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\item[iter] The number of iterations performed.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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Returned as: an integer variable.
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\item[err] The error estimate on exit.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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Returned as: a real number.
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\item[info] An error code.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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Returned as: an integer variable.
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\end{description}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%
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% CG
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%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subroutine{psb\_cg \label{cg}}{CG Iterative Method}
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This subroutine implements the CG method with restarting. The
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stopping criterion is the normwise backward error, in the infinity
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norm, i.e. the iteration is stopped when
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\[ \frac{\|r\|}{(\|A\|\|x\|+\|b\|)} < eps \]
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or
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\[ \frac{\|r_i\|}{\|b\|_2} < eps \]
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according to the value passed through the istop argument (see later).
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\syntax{call psb\_cg}{a,prec,b,x,eps,desc\_a,info,itmax,iter,err,itrace,istop}
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\begin{description}
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\item[\bf On Entry]
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\item[a] the local portion of global sparse matrix
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$A$. \\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a structured data of type \spdata.
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\item[prec] The data structure containing the preconditioner.\\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a structured data of type \precdata.
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\item[b] The RHS vector. \\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a rank one array.
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\item[x] The initial guess. \\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a rank one array.
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\item[eps] The stopping tolerance. \\
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Scope: {\bf global} \\
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Type: {\bf required}\\
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Specified as: a real number.
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\item[desc\_a] contains data structures for communications.\\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a structured data of type \descdata.
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\item[itmax] The maximum number of iterations to perform.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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Default: $itmax = 1000$.\\
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Specified as: an integer variable $itmax \ge 1$.
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\item[itrace] If $>0$ print out an informational message about
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convergence every $itrace$ iterations.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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\item[istop] An integer specifying the stopping criterion.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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\item[\bf On Return]
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\item[x] The computed solution. \\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a rank one array.
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\item[iter] The number of iterations performed.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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Returned as: an integer variable.
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\item[err] The error estimate on exit.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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Returned as: a real number.
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\item[info] An error code.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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Returned as: an integer variable.
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\end{description}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%
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% CGS
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%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subroutine{psb\_cgs \label{cgs}}{CGS Iterative Method}
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This subroutine implements the CGS method with restarting. The
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stopping criterion is the normwise backward error, in the infinity
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norm, i.e. the iteration is stopped when
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\[ \frac{\|r\|}{(\|A\|\|x\|+\|b\|)} < eps \]
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or
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\[ \frac{\|r_i\|}{\|b\|_2} < eps \]
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according to the value passed through the istop argument (see later).
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\syntax{call psb\_cgs}{a,prec,b,x,eps,desc\_a,info,itmax,iter,err,itrace,istop}
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\begin{description}
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\item[\bf On Entry]
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\item[a] the local portion of global sparse matrix
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$A$. \\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a structured data of type \spdata.
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\item[prec] The data structure containing the preconditioner.\\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a structured data of type \precdata.
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\item[b] The RHS vector. \\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a rank one array.
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\item[x] The initial guess. \\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a rank one array.
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\item[eps] The stopping tolerance. \\
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Scope: {\bf global} \\
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Type: {\bf required}\\
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Specified as: a real number.
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\item[desc\_a] contains data structures for communications.\\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a structured data of type \descdata.
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\item[itmax] The maximum number of iterations to perform.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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Default: $itmax = 1000$.\\
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Specified as: an integer variable $itmax \ge 1$.
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\item[itrace] If $>0$ print out an informational message about
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convergence every $itrace$ iterations.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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\item[istop] An integer specifying the stopping criterion.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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\item[\bf On Return]
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\item[x] The computed solution. \\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a rank one array.
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\item[iter] The number of iterations performed.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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Returned as: an integer variable.
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\item[err] The error estimate on exit.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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Returned as: a real number.
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\item[info] An error code.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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Returned as: an integer variable.
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\end{description}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%
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% BiCG
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%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subroutine{psb\_bicg \label{bicg}}{BiCG Iterative Method}
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This subroutine implements the BiCG method with restarting. The
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stopping criterion is the normwise backward error, in the infinity
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norm, i.e. the iteration is stopped when
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\[ \frac{\|r\|}{(\|A\|\|x\|+\|b\|)} < eps \]
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or
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\[ \frac{\|r_i\|}{\|b\|_2} < eps \]
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according to the value passed through the istop argument (see later).
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\syntax{call psb\_bicg}{a,prec,b,x,eps,desc\_a,info,itmax,iter,err,itrace,istop}
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\begin{description}
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\item[\bf On Entry]
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\item[a] the local portion of global sparse matrix
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$A$. \\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a structured data of type \spdata.
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\item[prec] The data structure containing the preconditioner.\\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a structured data of type \precdata.
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\item[b] The RHS vector. \\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a rank one array.
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\item[x] The initial guess. \\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a rank one array.
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\item[eps] The stopping tolerance. \\
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Scope: {\bf global} \\
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Type: {\bf required}\\
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Specified as: a real number.
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\item[desc\_a] contains data structures for communications.\\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a structured data of type \descdata.
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\item[itmax] The maximum number of iterations to perform.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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Default: $itmax = 1000$.\\
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Specified as: an integer variable $itmax \ge 1$.
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\item[itrace] If $>0$ print out an informational message about
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convergence every $itrace$ iterations.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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\item[istop] An integer specifying the stopping criterion.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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\item[\bf On Return]
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\item[x] The computed solution. \\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a rank one array.
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\item[iter] The number of iterations performed.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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Returned as: an integer variable.
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\item[err] The error estimate on exit.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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Returned as: a real number.
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\item[info] An error code.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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Returned as: an integer variable.
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\end{description}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%
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% BiCGSTAB
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%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subroutine{psb\_bicgstab \label{bicgstab}}{BiCGSTAB Iterative Method}
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This subroutine implements the BiCGSTAB method with restarting. The
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stopping criterion is the normwise backward error, in the infinity
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norm, i.e. the iteration is stopped when
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\[ \frac{\|r\|}{(\|A\|\|x\|+\|b\|)} < eps \]
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or
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\[ \frac{\|r_i\|}{\|b\|_2} < eps \]
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according to the value passed through the istop argument (see later).
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\syntax{call psb\_bicgstab}{a,prec,b,x,eps,desc\_a,info,itmax,iter,err,itrace,istop}
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\begin{description}
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\item[\bf On Entry]
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\item[a] the local portion of global sparse matrix
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$A$. \\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a structured data of type \spdata.
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\item[prec] The data structure containing the preconditioner.\\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a structured data of type \precdata.
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\item[b] The RHS vector. \\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a rank one array.
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\item[x] The initial guess. \\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a rank one array.
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\item[eps] The stopping tolerance. \\
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Scope: {\bf global} \\
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Type: {\bf required}\\
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Specified as: a real number.
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\item[desc\_a] contains data structures for communications.\\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a structured data of type \descdata.
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\item[itmax] The maximum number of iterations to perform.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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Default: $itmax = 1000$.\\
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Specified as: an integer variable $itmax \ge 1$.
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\item[itrace] If $>0$ print out an informational message about
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convergence every $itrace$ iterations.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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\item[istop] An integer specifying the stopping criterion.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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\item[\bf On Return]
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\item[x] The computed solution. \\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a rank one array.
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\item[iter] The number of iterations performed.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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Returned as: an integer variable.
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\item[err] The error estimate on exit.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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Returned as: a real number.
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\item[info] An error code.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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Returned as: an integer variable.
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\end{description}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%
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% BiCGSTAB(L)
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%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subroutine{psb\_bicgstabl \label{bicgstabl}}{BiCGSTAB-$l$ Iterative Method}
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This subroutine implements the BiCGSTAB-$l$ method with restarting. The
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stopping criterion is the normwise backward error, in the infinity
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norm, i.e. the iteration is stopped when
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\[ \frac{\|r\|}{(\|A\|\|x\|+\|b\|)} < eps \]
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or
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\[ \frac{\|r_i\|}{\|b\|_2} < eps \]
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according to the value passed through the istop argument (see later).
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|
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\syntax{call psb\_bicgstab}{a,prec,b,x,eps,desc\_a,info,itmax,iter,err,itrace,irst,istop}
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\begin{description}
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\item[\bf On Entry]
|
|
\item[a] the local portion of global sparse matrix
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$A$. \\
|
|
Scope: {\bf local} \\
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|
Type: {\bf required}\\
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|
Specified as: a structured data of type \spdata.
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\item[prec] The data structure containing the preconditioner.\\
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|
Scope: {\bf local} \\
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|
Type: {\bf required}\\
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|
Specified as: a structured data of type \precdata.
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\item[b] The RHS vector. \\
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|
Scope: {\bf local} \\
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|
Type: {\bf required}\\
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Specified as: a rank one array.
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\item[x] The initial guess. \\
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|
Scope: {\bf local} \\
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|
Type: {\bf required}\\
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Specified as: a rank one array.
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\item[eps] The stopping tolerance. \\
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Scope: {\bf global} \\
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Type: {\bf required}\\
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Specified as: a real number.
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\item[desc\_a] contains data structures for communications.\\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a structured data of type \descdata.
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\item[itmax] The maximum number of iterations to perform.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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Default: $itmax = 1000$.\\
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Specified as: an integer variable $itmax \ge 1$.
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\item[itrace] If $>0$ print out an informational message about
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convergence every $itrace$ iterations.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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\item[irst] An integer specifying the restarting iteration.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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\item[istop] An integer specifying the stopping criterion.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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\item[\bf On Return]
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\item[x] The computed solution. \\
|
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Scope: {\bf local} \\
|
|
Type: {\bf required}\\
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|
Specified as: a rank one array.
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\item[iter] The number of iterations performed.\\
|
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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Returned as: an integer variable.
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\item[err] The error estimate on exit.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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Returned as: a real number.
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\item[info] An error code.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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Returned as: an integer variable.
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\end{description}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%
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% GMRES
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%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subroutine{psb\_gmres \label{gmres}}{GMRES Iterative Method}
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This subroutine implements the GMRES method with restarting. The
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stopping criterion is the normwise backward error, in the infinity
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norm, i.e. the iteration is stopped when
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\[ \frac{\|r\|}{(\|A\|\|x\|+\|b\|)} < eps \]
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or
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\[ \frac{\|r_i\|}{\|b\|_2} < eps \]
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according to the value passed through the istop argument (see later).
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\syntax{call psb\_gmres}{a,prec,b,x,eps,desc\_a,info,itmax,iter,err,itrace,irst,istop}
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\begin{description}
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\item[\bf On Entry]
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\item[a] the local portion of global sparse matrix
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$A$. \\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a structured data of type \spdata.
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\item[prec] The data structure containing the preconditioner.\\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a structured data of type \precdata.
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\item[b] The RHS vector. \\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a rank one array.
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\item[x] The initial guess. \\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a rank one array.
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\item[eps] The stopping tolerance. \\
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Scope: {\bf global} \\
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Type: {\bf required}\\
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Specified as: a real number.
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\item[desc\_a] contains data structures for communications.\\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a structured data of type \descdata.
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\item[itmax] The maximum number of iterations to perform.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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Default: $itmax = 1000$.\\
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Specified as: an integer variable $itmax \ge 1$.
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\item[itrace] If $>0$ print out an informational message about
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convergence every $itrace$ iterations.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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\item[irst] An integer specifying the restart iteration.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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\item[istop] An integer specifying the stopping criterion.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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\item[\bf On Return]
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\item[x] The computed solution. \\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a rank one array.
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\item[iter] The number of iterations performed.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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Returned as: an integer variable.
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\item[err] The error estimate on exit.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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Returned as: a real number.
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\item[info] An error code.\\
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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Returned as: an integer variable.
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\end{description}
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: "userguide"
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%%% End:
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