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fix documentation bug for NRM2
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Salvatore Filippone
2010-06-22 12:42:39 +00:00
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@@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<H2><A NAME="SECTION000130000000000000000">
Bibliography</A>
</H2><DL COMPACT><DD>
<P>
<P></P><DT><A NAME="PARA04FOREST">1</A>
<DD>
G.&nbsp;Bella, S.&nbsp;Filippone, A.&nbsp;De Maio and M.&nbsp;Testa,
<EM>A Simulation Model for Forest Fires</EM>,
in J.&nbsp;Dongarra, K.&nbsp;Madsen, J.&nbsp;Wasniewski, editors,
Proceedings of PARA&nbsp;04 Workshop on State of the Art
in Scientific Computing, pp.&nbsp;546-553, Lecture Notes in Computer Science,
Springer, 2005.
<P></P><DT><A NAME="2007d">2</A>
<DD> A. Buttari, D. di Serafino, P. D'Ambra, S. Filippone,<BR>
2LEV-D2P4: a package of high-performance preconditioners,<BR>
Applicable Algebra in Engineering, Communications and Computing,
Volume 18, Number 3, May, 2007, pp. 223-239
<P></P><DT><A NAME="2007c">3</A>
<DD> P. D'Ambra, S. Filippone, D. Di Serafino<BR>
On the Development of PSBLAS-based Parallel Two-level Schwarz Preconditioners
<H2><A NAME="SECTION000121000000000000000"></A><A NAME="krylov"></A>
<BR>
Applied Numerical Mathematics, Elsevier Science,
Volume 57, Issues 11-12, November-December 2007, Pages 1181-1196.
psb_krylov -- Krylov Methods Driver
Routine
</H2>
<P>
<P></P><DT><A NAME="BLAS2">4</A>
<DD>
Dongarra, J. J., DuCroz, J., Hammarling, S. and Hanson, R.,
An Extended Set of Fortran Basic Linear Algebra Subprograms,
ACM Trans. Math. Softw. vol.&nbsp;14, 1-17, 1988.
<P></P><DT><A NAME="BLAS3">5</A>
<DD>
Dongarra, J., DuCroz, J., Hammarling, S. and Duff, I.,
A Set of level 3 Basic Linear Algebra Subprograms,
ACM Trans. Math. Softw. vol.&nbsp;16, 1-17, 1990.
<P></P><DT><A NAME="BLACS">6</A>
<DD>
J.&nbsp;J.&nbsp;Dongarra and R.&nbsp;C.&nbsp;Whaley,
<EM>A User's Guide to the BLACS v.&nbsp;1.1</EM>,
Lapack Working Note 94, Tech. Rep. UT-CS-95-281, University of
Tennessee, March 1995 (updated May 1997).
<P></P><DT><A NAME="sblas97">7</A>
<DD>
I.&nbsp;Duff, M.&nbsp;Marrone, G.&nbsp;Radicati and C.&nbsp;Vittoli,
<EM>Level 3 Basic Linear Algebra Subprograms for Sparse Matrices:
a User Level Interface</EM>,
ACM Transactions on Mathematical Software, 23(3), pp.&nbsp;379-401, 1997.
<P></P><DT><A NAME="sblas02">8</A>
<DD>
I.&nbsp;Duff, M.&nbsp;Heroux and R.&nbsp;Pozo,
<EM>An Overview of the Sparse Basic Linear
Algebra Subprograms: the New Standard from the BLAS Technical Forum</EM>,
ACM Transactions on Mathematical Software, 28(2), pp.&nbsp;239-267, 2002.
<P></P><DT><A NAME="PSBLAS">9</A>
<DD>
S.&nbsp;Filippone and M.&nbsp;Colajanni,
<EM>PSBLAS: A Library for Parallel Linear Algebra
Computation on Sparse Matrices</EM>,
<BR>
ACM Transactions on Mathematical Software, 26(4), pp.&nbsp;527-550, 2000.
<P></P><DT><A NAME="KIVA3PSBLAS">10</A>
<DD>
S.&nbsp;Filippone, P.&nbsp;D'Ambra, M.&nbsp;Colajanni,
<EM>Using a Parallel Library of Sparse Linear Algebra in a Fluid Dynamics
Applications Code on Linux Clusters</EM>,
in G.&nbsp;Joubert, A.&nbsp;Murli, F.&nbsp;Peters, M.&nbsp;Vanneschi, editors,
Parallel Computing - Advances &amp; Current Issues,
pp.&nbsp;441-448, Imperial College Press, 2002.
<P></P><DT><A NAME="METIS">11</A>
<DD>
Karypis, G. and Kumar, V.,
<EM>METIS: Unstructured Graph Partitioning and Sparse Matrix
Ordering System</EM>.
Minneapolis, MN 55455: University of Minnesota, Department of
Computer Science, 1995.
Internet Address: <code>http://www.cs.umn.edu/~karypis</code>.
<P></P><DT><A NAME="BLAS1">12</A>
<DD>
Lawson, C., Hanson, R., Kincaid, D. and Krogh, F.,
Basic Linear Algebra Subprograms for Fortran usage,
ACM Trans. Math. Softw. vol.&nbsp;5, 38-329, 1979.
This subroutine is a driver that provides a general interface for all
the Krylov-Subspace family methods implemented in PSBLAS version 2.
<P>
<P></P><DT><A NAME="machiels">13</A>
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="img141.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="img142.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="img143.png"
ALT="$x_i$"> is the tentative solution and
<IMG
WIDTH="90" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img144.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>
<PRE>
call psb_krylov(method,a,prec,b,x,eps,desc_a,info,&amp;
&amp; itmax,iter,err,itrace,irst,istop,cond)
</PRE>
<P>
<DL>
<DT><STRONG>Type:</STRONG></DT>
<DD>Synchronous.
</DD>
<DT><STRONG><B>On Entry</B></STRONG></DT>
<DD>
Machiels, L. and Deville, M.
<EM>Fortran 90: An entry to object-oriented programming for the solution
of partial differential equations.</EM>
ACM Trans. Math. Softw. vol.&nbsp;23, 32-49.
<P></P><DT><A NAME="metcalf">14</A>
</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="img145.png"
ALT="$itmax = 1000$">.
<BR>
Specified as: an integer variable <IMG
WIDTH="75" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img146.png"
ALT="$itmax \ge 1$">.
</DD>
<DT><STRONG>itrace</STRONG></DT>
<DD>If <IMG
WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img107.png"
ALT="$&gt;0$"> print out an informational message about
convergence every <IMG
WIDTH="46" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
SRC="img147.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="img148.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>
Metcalf, M., Reid, J. and Cohen, M.
<EM>Fortran 95/2003 explained.</EM>
Oxford University Press, 2004.
<P></P><DT><A NAME="MPI1">15</A>
<DD>
M.&nbsp;Snir, S.&nbsp;Otto, S.&nbsp;Huss-Lederman, D.&nbsp;Walker and J.&nbsp;Dongarra,
<EM>MPI: The Complete Reference. Volume 1 - The MPI Core</EM>, second edition,
MIT Press, 1998.
</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="img149.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>
<BR><HR>
<P>
<P>
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