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<H2><A NAME="SECTION000521000000000000000"></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 is the normwise backward error, in the infinity
norm, i.e. 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="img137.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>
or the 2-norm residual reduction
<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="img138.png"
 ALT="\begin{displaymath}err = \frac{\Vert r_i\Vert}{\Vert b\Vert _2} &lt; eps \end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
according to the value passed through the  istop argument (see
later). In the above formulae, <IMG
 WIDTH="19" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img139.png"
 ALT="$x_i$"> is the tentative solution and
<IMG
 WIDTH="90" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img140.png"
 ALT="$r_i=b-Ax_i$"> the corresponding residual at the <IMG
 WIDTH="10" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="img3.png"
 ALT="$i$">-th iteration. 

<P>
ifstarssyntaxsyntaxcall psb_krylovmethod,a,prec,b,x,eps,desc_a,info,itmax,iter,err,itrace,irst,istop,cond

<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>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_spmat_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.
</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.
</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="img141.png"
 ALT="$itmax = 1000$">.
<BR>
Specified as: an integer variable <IMG
 WIDTH="75" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="img142.png"
 ALT="$itmax \ge 1$">.
</DD>
<DT><STRONG>itrace</STRONG></DT>
<DD>If <IMG
 WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img103.png"
 ALT="$&gt;0$"> print out an informational message about
  convergence  every <IMG
 WIDTH="46" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="img143.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="img144.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. 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.
</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="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img145.png"
 ALT="$CG$"> method.
<BR>
Scope: <B>global</B> 
<BR>
Type: <B>optional</B>
<BR>
Intent: <B>out</B>.
<BR>
Returned  as: a real number.
</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>

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