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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" CLASS="mathdisplay">
<!-- MATH
 \begin{displaymath}
err = \frac{\|r_i\|}{(\|A\|\|x_i\|+\|b\|)} < eps
\end{displaymath}
 -->

<IMG
 WIDTH="204" HEIGHT="44" BORDER="0"
 SRC="img168.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" CLASS="mathdisplay">
<!-- MATH
 \begin{displaymath}
err = \frac{\|r_i\|}{\|b\|_2} < eps
\end{displaymath}
 -->

<IMG
 WIDTH="121" HEIGHT="44" BORDER="0"
 SRC="img169.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" CLASS="mathdisplay">
<!-- MATH
 \begin{displaymath}
err = \frac{\|r_i\|}{\|r_0\|_2} < eps
\end{displaymath}
 -->

<IMG
 WIDTH="128" HEIGHT="44" BORDER="0"
 SRC="img170.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, <SPAN CLASS="MATH"><IMG
 WIDTH="18" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img171.png"
 ALT="$x_i$"></SPAN> is the tentative solution and
<SPAN CLASS="MATH"><IMG
 WIDTH="91" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img172.png"
 ALT="$r_i=b-Ax_i$"></SPAN> the corresponding residual at the <SPAN CLASS="MATH"><IMG
 WIDTH="9" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="img4.png"
 ALT="$i$"></SPAN>-th iteration. 

<P>
<BR>
<IMG
 WIDTH="482" HEIGHT="35" ALIGN="BOTTOM" BORDER="0"
 SRC="img173.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="tex2html32"
  HREF="footnode.html#foot8318"><SUP><SPAN CLASS="arabic">5</SPAN></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
<SPAN CLASS="MATH"><IMG
 WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img1.png"
 ALT="$A$"></SPAN>. 
<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: <SPAN CLASS="MATH"><IMG
 WIDTH="99" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="img175.png"
 ALT="$itmax = 1000$"></SPAN>.
<BR>
Specified as: an integer variable <SPAN CLASS="MATH"><IMG
 WIDTH="76" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="img176.png"
 ALT="$itmax \ge 1$"></SPAN>.
</DD>
<DT><STRONG>itrace</STRONG></DT>
<DD>If <SPAN CLASS="MATH"><IMG
 WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img120.png"
 ALT="$&gt;0$"></SPAN> print out an informational message about
  convergence  every <SPAN CLASS="MATH"><IMG
 WIDTH="45" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="img177.png"
 ALT="$itrace$"></SPAN> iterations. If <SPAN CLASS="MATH"><IMG
 WIDTH="29" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="img178.png"
 ALT="$=0$"></SPAN> print a message in
  case of convergence failure.
<BR>
Scope: <B>global</B> 
<BR>
Type: <B>optional</B>
<BR>
Intent: <B>in</B>.
<BR>
Default: <SPAN CLASS="MATH"><IMG
 WIDTH="87" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="img179.png"
 ALT="$itrace = -1$"></SPAN>.
<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: <SPAN CLASS="MATH"><IMG
 WIDTH="60" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="img180.png"
 ALT="$irst&gt;0$"></SPAN>. 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 <SPAN CLASS="MATH"><IMG
 WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img1.png"
 ALT="$A$"></SPAN>; only
  available with the <SPAN CLASS="MATH"><IMG
 WIDTH="29" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img181.png"
 ALT="$CG$"></SPAN> 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>

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