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