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<H2><A NAME="SECTION000121000000000000000"></A><A NAME="krylov"></A>
<BR>
psb_krylov -- Krylov Methods Driver
Routine
</H2>
<P>
This subroutine is a driver that provides a general interface for all
the Krylov-Subspace family methods implemented in PSBLAS version 2.
<P>
The stopping criterion can take the following values:
<DL>
<DT><STRONG>1</STRONG></DT>
<DD>normwise backward error in the infinity
norm; the iteration is stopped when
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
err = \frac{\|r_i\|}{(\|A\|\|x_i\|+\|b\|)} < eps
\end{displaymath}
-->
<IMG
WIDTH="204" HEIGHT="44" BORDER="0"
SRC="img156.png"
ALT="\begin{displaymath}err = \frac{\Vert r_i\Vert}{(\Vert A\Vert\Vert x_i\Vert+\Vert b\Vert)} &lt; eps \end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
</DD>
<DT><STRONG>2</STRONG></DT>
<DD>Relative residual in the 2-norm; the iteration is stopped
when
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
err = \frac{\|r_i\|}{\|b\|_2} < eps
\end{displaymath}
-->
<IMG
WIDTH="121" HEIGHT="44" BORDER="0"
SRC="img157.png"
ALT="\begin{displaymath}err = \frac{\Vert r_i\Vert}{\Vert b\Vert _2} &lt; eps \end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
</DD>
<DT><STRONG>3</STRONG></DT>
<DD>Relative residual reduction in the 2-norm; the iteration is stopped
when
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
err = \frac{\|r_i\|}{\|r_0\|_2} < eps
\end{displaymath}
-->
<IMG
WIDTH="129" HEIGHT="44" BORDER="0"
SRC="img158.png"
ALT="\begin{displaymath}err = \frac{\Vert r_i\Vert}{\Vert r_0\Vert _2} &lt; eps \end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
</DD>
</DL>
The behaviour is controlled by the istop argument (see
later). In the above formulae, <IMG
WIDTH="18" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img159.png"
ALT="$x_i$"> is the tentative solution and
<IMG
WIDTH="91" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img160.png"
ALT="$r_i=b-Ax_i$"> the corresponding residual at the <IMG
WIDTH="9" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
SRC="img4.png"
ALT="$i$">-th iteration.
<P>
<BR>
<IMG
WIDTH="482" HEIGHT="35" ALIGN="BOTTOM" BORDER="0"
SRC="img161.png"
ALT="\begin{lstlisting}
call psb_krylov(method,a,prec,b,x,eps,desc_a,info,&amp;
&amp; itmax,iter,err,itrace,irst,istop,cond)
\end{lstlisting}">
<BR>
<P>
<DL>
<DT><STRONG>Type:</STRONG></DT>
<DD>Synchronous.
</DD>
<DT><STRONG><B>On Entry</B></STRONG></DT>
<DD>
</DD>
<DT><STRONG>method</STRONG></DT>
<DD>a string that defines the iterative method to be
used. Supported values are:
<DL>
<DT><STRONG>CG:</STRONG></DT>
<DD>the Conjugate Gradient method;
</DD>
<DT><STRONG>CGS:</STRONG></DT>
<DD>the Conjugate Gradient Stabilized method;
<P>
</DD>
<DT><STRONG>GCR:</STRONG></DT>
<DD>the Generalized Conjugate Residual method;
</DD>
<DT><STRONG>FCG:</STRONG></DT>
<DD>the Flexible Conjugate Gradient method<A NAME="tex2html31"
HREF="footnode.html#foot7956"><SUP>4</SUP></A>;
<P>
</DD>
<DT><STRONG>BICG:</STRONG></DT>
<DD>the Bi-Conjugate Gradient method;
</DD>
<DT><STRONG>BICGSTAB:</STRONG></DT>
<DD>the Bi-Conjugate Gradient Stabilized method;
</DD>
<DT><STRONG>BICGSTABL:</STRONG></DT>
<DD>the Bi-Conjugate Gradient Stabilized method with restarting;
</DD>
<DT><STRONG>RGMRES:</STRONG></DT>
<DD>the Generalized Minimal Residual method with restarting.
</DD>
</DL>
</DD>
<DT><STRONG>a</STRONG></DT>
<DD>the local portion of global sparse matrix
<IMG
WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img1.png"
ALT="$A$">.
<BR>
Scope: <B>local</B>
<BR>
Type: <B>required</B>
<BR>
Intent: <B>in</B>.
<BR>
Specified as: a structured data of type spdata<TT>psb_Tspmat_type</TT>.
</DD>
<DT><STRONG>prec</STRONG></DT>
<DD>The data structure containing the preconditioner.
<BR>
Scope: <B>local</B>
<BR>
Type: <B>required</B>
<BR>
Intent: <B>in</B>.
<BR>
Specified as: a structured data of type precdata<TT>psb_prec_type</TT>.
</DD>
<DT><STRONG>b</STRONG></DT>
<DD>The RHS vector.
<BR>
Scope: <B>local</B>
<BR>
Type: <B>required</B>
<BR>
Intent: <B>in</B>.
<BR>
Specified as: a rank one array or an object of type vdata<TT>psb_T_vect_type</TT>.
</DD>
<DT><STRONG>x</STRONG></DT>
<DD>The initial guess.
<BR>
Scope: <B>local</B>
<BR>
Type: <B>required</B>
<BR>
Intent: <B>inout</B>.
<BR>
Specified as: a rank one array or an object of type vdata<TT>psb_T_vect_type</TT>.
</DD>
<DT><STRONG>eps</STRONG></DT>
<DD>The stopping tolerance.
<BR>
Scope: <B>global</B>
<BR>
Type: <B>required</B>
<BR>
Intent: <B>in</B>.
<BR>
Specified as: a real number.
</DD>
<DT><STRONG>desc_a</STRONG></DT>
<DD>contains data structures for communications.
<BR>
Scope: <B>local</B>
<BR>
Type: <B>required</B>
<BR>
Intent: <B>in</B>.
<BR>
Specified as: a structured data of type descdata<TT>psb_desc_type</TT>.
</DD>
<DT><STRONG>itmax</STRONG></DT>
<DD>The maximum number of iterations to perform.
<BR>
Scope: <B>global</B>
<BR>
Type: <B>optional</B>
<BR>
Intent: <B>in</B>.
<BR>
Default: <IMG
WIDTH="100" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
SRC="img163.png"
ALT="$itmax = 1000$">.
<BR>
Specified as: an integer variable <IMG
WIDTH="75" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img164.png"
ALT="$itmax \ge 1$">.
</DD>
<DT><STRONG>itrace</STRONG></DT>
<DD>If <IMG
WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img111.png"
ALT="$&gt;0$"> print out an informational message about
convergence every <IMG
WIDTH="46" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
SRC="img165.png"
ALT="$itrace$"> iterations.
<BR>
Scope: <B>global</B>
<BR>
Type: <B>optional</B>
<BR>
Intent: <B>in</B>.
<BR></DD>
<DT><STRONG>irst</STRONG></DT>
<DD>An integer specifying the restart parameter.
<BR>
Scope: <B>global</B>
<BR>
Type: <B>optional</B>.
<BR>
Intent: <B>in</B>.
<BR>
Values: <IMG
WIDTH="59" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img166.png"
ALT="$irst&gt;0$">. This is employed for the BiCGSTABL or RGMRES
methods, otherwise it is ignored.
<P>
</DD>
<DT><STRONG>istop</STRONG></DT>
<DD>An integer specifying the stopping criterion.
<BR>
Scope: <B>global</B>
<BR>
Type: <B>optional</B>.
<BR>
Intent: <B>in</B>.
<BR>
Values: 1: use the normwise backward error, 2: use the scaled 2-norm
of the residual, 3: use the residual reduction in the 2-norm. Default: 2.
</DD>
<DT><STRONG><B>On Return</B></STRONG></DT>
<DD>
</DD>
<DT><STRONG>x</STRONG></DT>
<DD>The computed solution.
<BR>
Scope: <B>local</B>
<BR>
Type: <B>required</B>
<BR>
Intent: <B>inout</B>.
<BR>
Specified as: a rank one array or an object of type vdata<TT>psb_T_vect_type</TT>.
</DD>
<DT><STRONG>iter</STRONG></DT>
<DD>The number of iterations performed.
<BR>
Scope: <B>global</B>
<BR>
Type: <B>optional</B>
<BR>
Intent: <B>out</B>.
<BR>
Returned as: an integer variable.
</DD>
<DT><STRONG>err</STRONG></DT>
<DD>The convergence estimate on exit.
<BR>
Scope: <B>global</B>
<BR>
Type: <B>optional</B>
<BR>
Intent: <B>out</B>.
<BR>
Returned as: a real number.
</DD>
<DT><STRONG>cond</STRONG></DT>
<DD>An estimate of the condition number of matrix <IMG
WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img1.png"
ALT="$A$">; only
available with the <IMG
WIDTH="29" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img167.png"
ALT="$CG$"> method on real data.
<BR>
Scope: <B>global</B>
<BR>
Type: <B>optional</B>
<BR>
Intent: <B>out</B>.
<BR>
Returned as: a real number. A correct result will be greater than or
equal to one; if specified for non-real data, or an error occurred,
zero is returned.
</DD>
<DT><STRONG>info</STRONG></DT>
<DD>Error code.
<BR>
Scope: <B>local</B>
<BR>
Type: <B>required</B>
<BR>
Intent: <B>out</B>.
<BR>
An integer value; 0 means no error has been detected.
</DD>
</DL>
<P>
<P>
<P>
<P>
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