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Updated description of DOT and friends to account for optional GLOBAL
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<BR>
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<BR></DIV>
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<!--End of Navigation Panel-->
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<H2><A NAME="SECTION000130000000000000000">
|
||||
Bibliography</A>
|
||||
</H2><DL COMPACT><DD><P></P><DT><A NAME="DesPat:11">1</A>
|
||||
<DD>
|
||||
D. Barbieri, V. Cardellini, S. Filippone and D. Rouson
|
||||
<EM>Design Patterns for Scientific Computations on Sparse Matrices</EM>,
|
||||
HPSS 2011, Algorithms and Programming Tools for Next-Generation High-Performance Scientific Software, Bordeaux, Sep. 2011
|
||||
|
||||
<P>
|
||||
<P></P><DT><A NAME="PARA04FOREST">2</A>
|
||||
<DD>
|
||||
G. Bella, S. Filippone, A. De Maio and M. Testa,
|
||||
<EM>A Simulation Model for Forest Fires</EM>,
|
||||
in J. Dongarra, K. Madsen, J. Wasniewski, editors,
|
||||
Proceedings of PARA 04 Workshop on State of the Art
|
||||
in Scientific Computing, pp. 546-553, Lecture Notes in Computer Science,
|
||||
Springer, 2005.
|
||||
<P></P><DT><A NAME="2007d">3</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">4</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>
|
||||
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
|
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<BR><P></P>
|
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<DIV ALIGN="CENTER" CLASS="mathdisplay">
|
||||
<!-- MATH
|
||||
\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="205" HEIGHT="44" BORDER="0"
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SRC="img163.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>
|
||||
<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" CLASS="mathdisplay">
|
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<!-- MATH
|
||||
\begin{displaymath}
|
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err = \frac{\|r_i\|}{\|b\|_2} < eps
|
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\end{displaymath}
|
||||
-->
|
||||
|
||||
<IMG
|
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WIDTH="121" HEIGHT="44" BORDER="0"
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SRC="img164.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>
|
||||
<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
|
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\end{displaymath}
|
||||
-->
|
||||
|
||||
<IMG
|
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WIDTH="129" HEIGHT="44" BORDER="0"
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SRC="img165.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>
|
||||
<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="img166.png"
|
||||
ALT="$x_i$"></SPAN> is the tentative solution and
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="91" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img167.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>
|
||||
<P></P><DT><A NAME="BLAS2">5</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. 14, 1-17, 1988.
|
||||
<P></P><DT><A NAME="BLAS3">6</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. 16, 1-17, 1990.
|
||||
<P></P><DT><A NAME="BLACS">7</A>
|
||||
<DD>
|
||||
J. J. Dongarra and R. C. Whaley,
|
||||
<EM>A User's Guide to the BLACS v. 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">8</A>
|
||||
<DD>
|
||||
I. Duff, M. Marrone, G. Radicati and C. Vittoli,
|
||||
<EM>Level 3 Basic Linear Algebra Subprograms for Sparse Matrices:
|
||||
a User Level Interface</EM>,
|
||||
ACM Transactions on Mathematical Software, 23(3), pp. 379-401, 1997.
|
||||
<P></P><DT><A NAME="sblas02">9</A>
|
||||
<DD>
|
||||
I. Duff, M. Heroux and R. 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. 239-267, 2002.
|
||||
<P></P><DT><A NAME="PSBLAS">10</A>
|
||||
<DD>
|
||||
S. Filippone and M. Colajanni,
|
||||
<EM>PSBLAS: A Library for Parallel Linear Algebra
|
||||
Computation on Sparse Matrices</EM>,
|
||||
<BR>
|
||||
ACM Transactions on Mathematical Software, 26(4), pp. 527-550, 2000.
|
||||
<P></P><DT><A NAME="Sparse03">11</A>
|
||||
<DD>
|
||||
S. Filippone and A. Buttari,
|
||||
<EM>Object-Oriented Techniques for Sparse Matrix Computations in Fortran 2003</EM>,
|
||||
<IMG
|
||||
WIDTH="482" HEIGHT="35" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img168.png"
|
||||
ALT="\begin{lstlisting}
|
||||
call psb_krylov(method,a,prec,b,x,eps,desc_a,info,&
|
||||
& itmax,iter,err,itrace,irst,istop,cond)
|
||||
\end{lstlisting}">
|
||||
<BR>
|
||||
ACM Transactions on Mathematical Software, 38(4), 2012.
|
||||
<P></P><DT><A NAME="KIVA3PSBLAS">12</A>
|
||||
<DD>
|
||||
S. Filippone, P. D'Ambra, M. Colajanni,
|
||||
<EM>Using a Parallel Library of Sparse Linear Algebra in a Fluid Dynamics
|
||||
Applications Code on Linux Clusters</EM>,
|
||||
in G. Joubert, A. Murli, F. Peters, M. Vanneschi, editors,
|
||||
Parallel Computing - Advances & Current Issues,
|
||||
pp. 441-448, Imperial College Press, 2002.
|
||||
<P></P><DT><A NAME="DesignPatterns">13</A>
|
||||
<DD>
|
||||
Gamma, E., Helm, R., Johnson, R., and Vlissides,
|
||||
J. 1995.
|
||||
<EM>Design Patterns: Elements of Reusable Object-Oriented Software</EM>.
|
||||
Addison-Wesley.
|
||||
|
||||
<P>
|
||||
<P></P><DT><A NAME="METIS">14</A>
|
||||
<DL>
|
||||
<DT><STRONG>Type:</STRONG></DT>
|
||||
<DD>Synchronous.
|
||||
</DD>
|
||||
<DT><STRONG><B>On Entry</B></STRONG></DT>
|
||||
<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">15</A>
|
||||
<DD>
|
||||
Lawson, C., Hanson, R., Kincaid, D. and Krogh, F.,
|
||||
Basic Linear Algebra Subprograms for Fortran usage,
|
||||
ACM Trans. Math. Softw. vol. 5, 38-329, 1979.
|
||||
</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>
|
||||
<P></P><DT><A NAME="machiels">16</A>
|
||||
<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. 23, 32-49.
|
||||
<P></P><DT><A NAME="metcalf">17</A>
|
||||
<DD>
|
||||
Metcalf, M., Reid, J. and Cohen, M.
|
||||
<EM>Fortran 95/2003 explained.</EM>
|
||||
Oxford University Press, 2004.
|
||||
<P></P><DT><A NAME="RouXiaXu:11">18</A>
|
||||
<DD>
|
||||
Rouson, D.W.I., Xia, J., Xu, X.: Scientific Software Design: The
|
||||
Object-Oriented Way. Cambridge University Press (2011)
|
||||
</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#foot8258"><SUP><SPAN CLASS="arabic">5</SPAN></SUP></A>;
|
||||
|
||||
<P>
|
||||
<P></P><DT><A NAME="MPI1">19</A>
|
||||
</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="img170.png"
|
||||
ALT="$itmax = 1000$"></SPAN>.
|
||||
<BR>
|
||||
Specified as: an integer variable <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="76" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img171.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="img115.png"
|
||||
ALT="$>0$"></SPAN> print out an informational message about
|
||||
convergence every <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="45" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img172.png"
|
||||
ALT="$itrace$"></SPAN> 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: <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="60" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img173.png"
|
||||
ALT="$irst>0$"></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>
|
||||
M. Snir, S. Otto, S. Huss-Lederman, D. Walker and J. 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 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="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img174.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>
|
||||
|
||||
<P>
|
||||
<BR><HR>
|
||||
|
||||
<P>
|
||||
|
||||
<P>
|
||||
|
||||
<P>
|
||||
|
||||
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||||
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