mirror of
https://github.com/sfilippone/amg4psblas.git
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mld2p4:
config/pac.m4 configure docs/html/WARNINGS docs/html/images.log docs/html/images.pl docs/html/images.tex docs/html/img100.png docs/html/img101.png docs/html/img92.png docs/html/img93.png docs/html/img94.png docs/html/img95.png docs/html/img96.png docs/html/img97.png docs/html/img98.png docs/html/img99.png docs/html/index.html docs/html/internals.pl docs/html/labels.pl docs/html/node1.html docs/html/node10.html docs/html/node11.html docs/html/node12.html docs/html/node13.html docs/html/node14.html docs/html/node15.html docs/html/node16.html docs/html/node17.html docs/html/node18.html docs/html/node19.html docs/html/node2.html docs/html/node20.html docs/html/node21.html docs/html/node22.html docs/html/node23.html docs/html/node24.html docs/html/node25.html docs/html/node26.html docs/html/node27.html docs/html/node28.html docs/html/node29.html docs/html/node3.html docs/html/node30.html docs/html/node31.html docs/html/node4.html docs/html/node5.html docs/html/node6.html docs/html/node7.html docs/html/node8.html docs/html/node9.html docs/html/userhtml.html docs/pdf/Makefile docs/pdf/abstract.tex docs/pdf/background.tex docs/pdf/bibliography.tex docs/pdf/building.tex docs/pdf/conventions.tex docs/pdf/distribution.tex docs/pdf/errors.tex docs/pdf/gettingstarted.tex docs/pdf/overview.tex docs/pdf/title.tex docs/pdf/userguide.tex docs/pdf/userhtml.tex docs/pdf/userinterface.tex Configure minro fix: require psblas 2.3 Doc fixes: dual version of title for pdf/html, fixed tables.
This commit is contained in:
Salvatore Filippone
+1
-1
@@ -290,7 +290,7 @@ AC_DEFUN([PAC_ARG_WITH_PSBLAS],
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[
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AC_ARG_WITH(psblas,
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AC_HELP_STRING([--with-psblas], [The source directory for PSBLAS, for example,
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--with-psblas=/opt/packages/psblas-2.2]),
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--with-psblas=/opt/packages/psblas-2.3]),
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[pac_cv_psblas_dir=$withval],
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[pac_cv_psblas_dir=''])
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]
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@@ -1268,7 +1268,7 @@ Optional Packages:
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--with-PACKAGE[=ARG] use PACKAGE [ARG=yes]
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--without-PACKAGE do not use PACKAGE (same as --with-PACKAGE=no)
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--with-psblas The source directory for PSBLAS, for example,
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--with-psblas=/opt/packages/psblas-2.2
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--with-psblas=/opt/packages/psblas-2.3
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--with-libs List additional link flags here. For example,
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--with-libs=-lspecial_system_lib or
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--with-libs=-L/path/to/libs
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+3
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@@ -8,30 +8,11 @@ No implementation found for style `rotating'
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redefining command \thepage
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The userhtml.aux file was not found, so sections will not be numbered
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and cross-references will be shown as icons.
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? brace missing for \
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Substitution of arg to newlabelxx delayed.
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? brace missing for \setlength
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No number for "Applicationofthemulti-levelhybridpost-smoothedpreconditioner."
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No number for "Parametersdefiningthecoarse-spacecorrectionatthecoarsestlevel."
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couldn't convert character ecaron into available encodings
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...set $ACCENT_IMAGES to get an image
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This is pdfTeXk, Version 3.141592-1.40.3 (Web2C 7.5.6) (format=latex 2008.5.20) 23 JUL 2008 10:16
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entering extended mode
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%&-line parsing enabled.
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**./images.tex
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@@ -28,8 +28,10 @@ File: size11.clo 2005/09/16 v1.4f Standard LaTeX file (size option)
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\abovecaptionskip=\skip41
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\belowcaptionskip=\skip42
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\bibindent=\dimen102
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Package: pstricks 2006/08/10 v0.32 LaTeX wrapper for `PSTricks' (RN,HV)
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`PSTricks' v1.15 <2006/12/22> (tvz)
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\pst@dima=\dimen103
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File: pstricks.tex 2006/12/22 v1.15 `PSTricks' (tvz)
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Package: xcolor 2007/01/21 v2.11 LaTeX color extensions (UK)
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File: color.cfg 2007/01/18 v1.5 color configuration of teTeX/TeXLive
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Package xcolor Info: Package option `override' ignored on input line 216.
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File: dvips.def 1999/02/16 v3.0i Driver-dependant file (DPC,SPQR)
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Package: fancybox 2000/09/19 1.3
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Style option: `fancybox' v1.3 <2000/09/19> (tvz)
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\@fancybox=\box29
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\shadowsize=\dimen118
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\symAMSb=\mathgroup5
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LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold'
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(Font) U/euf/m/n --> U/euf/b/n on input line 132.
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Package: ifpdf 2006/02/20 v1.4 Provides the ifpdf switch (HO)
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Package ifpdf Info: pdfTeX in pdf mode not detected.
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Package: hyperref 2007/02/07 v6.75r Hypertext links for LaTeX
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Package: keyval 1999/03/16 v1.13 key=value parser (DPC)
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\KV@toks@=\toks22
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\@linkdim=\dimen119
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\Hy@linkcounter=\count93
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\Hy@pagecounter=\count94
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File: pd1enc.def 2007/02/07 v6.75r Hyperref: PDFDocEncoding definition (HO)
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File: hyperref.cfg 2002/06/06 v1.2 hyperref configuration of TeXLive
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Package: kvoptions 2006/08/22 v2.4 Connects package keyval with LaTeX options (
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HO)
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Package hyperref Info: Hyper index ON on input line 2296.
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Package hyperref Info: Plain pages OFF on input line 2303.
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Package hyperref Info: Backreferencing OFF on input line 2308.
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Implicit mode ON; LaTeX internals redefined
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Package hyperref Info: Bookmarks ON on input line 2444.
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(/usr/share/texmf/tex/latex/html/url.sty
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*hyperref using default driver hdvips*
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File: hdvips.def 2007/02/07 v6.75r Hyperref driver for dvips
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File: pdfmark.def 2007/02/07 v6.75r Hyperref definitions for pdfmark specials
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\pdf@docset=\toks25
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\pdf@toks=\toks26
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\pdf@defaulttoks=\toks27
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Package: ifthen 2001/05/26 v1.1c Standard LaTeX ifthen package (DPC)
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Package: graphicx 1999/02/16 v1.0f Enhanced LaTeX Graphics (DPC,SPQR)
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Package: graphics 2006/02/20 v1.0o Standard LaTeX Graphics (DPC,SPQR)
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Package: trig 1999/03/16 v1.09 sin cos tan (DPC)
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File: graphics.cfg 2007/01/18 v1.5 graphics configuration of teTeX/TeXLive
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Package graphics Info: Driver file: dvips.def on input line 90.
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\c@subroutine=\count103
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\c@example=\count104
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Package pstcol Warning: ************************************
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(pstcol) The package `pstcol' is obsolet!
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(pstcol) You should use `pstricks' directly:
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Package: inputenc 2006/05/05 v1.1b Input encoding file
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\lthtmlwrite=\write4
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l2hSize :tex2html_wrap_inline1324:7.07185pt::0.0pt::8.68147pt.
|
||||
l2hSize :tex2html_wrap_inline1316:7.07185pt::0.0pt::8.68147pt.
|
||||
[87
|
||||
|
||||
|
||||
]
|
||||
l2hSize :tex2html_wrap_inline1328:7.57185pt::7.57185pt::17.58084pt.
|
||||
l2hSize :tex2html_wrap_inline1320:7.57185pt::7.57185pt::17.58084pt.
|
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[88
|
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|
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|
||||
]
|
||||
l2hSize :tex2html_wrap_inline1342:7.07185pt::0.0pt::4.50171pt.
|
||||
l2hSize :tex2html_wrap_inline1334:7.07185pt::0.0pt::4.50171pt.
|
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[89
|
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|
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|
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]
|
||||
l2hSize :tex2html_wrap_inline1346:8.7125pt::8.7125pt::32.78915pt.
|
||||
l2hSize :tex2html_wrap_inline1338:8.7125pt::8.7125pt::32.78915pt.
|
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[90
|
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|
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|
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]
|
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l2hSize :tex2html_wrap_inline1348:7.60416pt::0.0pt::5.99211pt.
|
||||
l2hSize :tex2html_wrap_inline1340:7.60416pt::0.0pt::5.99211pt.
|
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[91
|
||||
|
||||
|
||||
]
|
||||
l2hSize :tex2html_wrap_inline1352:7.57185pt::7.57185pt::17.58084pt.
|
||||
l2hSize :tex2html_wrap_inline1346:9.62923pt::9.62923pt::75.59969pt.
|
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[92
|
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|
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|
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]
|
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l2hSize :tex2html_wrap_inline1374:9.62923pt::9.62923pt::71.50383pt.
|
||||
l2hSize :tex2html_wrap_inline1362:7.57185pt::7.57185pt::17.58084pt.
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[93
|
||||
|
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|
||||
]
|
||||
l2hSize :tex2html_wrap_inline1378:7.57185pt::7.57185pt::11.36446pt.
|
||||
l2hSize :tex2html_wrap_inline1374:9.62923pt::9.62923pt::71.50383pt.
|
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[94
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|
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|
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]
|
||||
l2hSize :tex2html_wrap_inline1382:7.07185pt::0.0pt::6.80576pt.
|
||||
l2hSize :tex2html_wrap_inline1378:7.57185pt::7.57185pt::11.36446pt.
|
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[95
|
||||
|
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|
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]
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l2hSize :tex2html_wrap_inline1384:7.57185pt::7.57185pt::6.30898pt.
|
||||
l2hSize :tex2html_wrap_inline1382:7.07185pt::0.0pt::6.80576pt.
|
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[96
|
||||
|
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|
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]
|
||||
l2hSize :tex2html_wrap_inline1386:9.62923pt::9.62923pt::80.83733pt.
|
||||
l2hSize :tex2html_wrap_inline1384:7.57185pt::7.57185pt::6.30898pt.
|
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[97
|
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|
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|
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]
|
||||
l2hSize :tex2html_wrap_inline1388:9.94032pt::9.94032pt::82.71652pt.
|
||||
l2hSize :tex2html_wrap_inline1386:9.62923pt::9.62923pt::80.83733pt.
|
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[98
|
||||
|
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|
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]
|
||||
l2hSize :tex2html_wrap_inline1390:9.62923pt::9.62923pt::27.98401pt.
|
||||
l2hSize :tex2html_wrap_inline1388:9.94032pt::9.94032pt::82.71652pt.
|
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[99
|
||||
|
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|
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]
|
||||
l2hSize :tex2html_wrap_inline1392:9.94032pt::9.94032pt::83.21068pt.
|
||||
l2hSize :tex2html_wrap_inline1390:9.62923pt::9.62923pt::27.98401pt.
|
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[100
|
||||
|
||||
|
||||
]
|
||||
l2hSize :tex2html_wrap_inline1392:9.94032pt::9.94032pt::83.21068pt.
|
||||
[101
|
||||
|
||||
|
||||
] (./images.aux) )
|
||||
Here is how much of TeX's memory you used:
|
||||
4265 strings out of 94074
|
||||
55276 string characters out of 1167173
|
||||
134209 words of memory out of 1500000
|
||||
7519 multiletter control sequences out of 10000+50000
|
||||
4373 strings out of 94074
|
||||
57907 string characters out of 1167173
|
||||
141384 words of memory out of 1500000
|
||||
7590 multiletter control sequences out of 10000+50000
|
||||
10304 words of font info for 41 fonts, out of 1200000 for 2000
|
||||
645 hyphenation exceptions out of 8191
|
||||
30i,8n,37p,306b,390s stack positions out of 5000i,500n,6000p,200000b,10000s
|
||||
|
||||
Output written on images.dvi (100 pages, 72052 bytes).
|
||||
Output written on images.dvi (101 pages, 72868 bytes).
|
||||
|
||||
+33
-27
@@ -85,7 +85,7 @@ $cached_env_img{$key} = q|<IMG
|
||||
$key = q/op(M^{-1})=M^{-T};MSF=1.6;LFS=11;AAT/;
|
||||
$cached_env_img{$key} = q|<IMG
|
||||
WIDTH="135" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="|."$dir".q|img98.png"
|
||||
SRC="|."$dir".q|img99.png"
|
||||
ALT="$op(M^{-1}) = M^{-T}$">|;
|
||||
|
||||
$key = q/W_i^{delta-1};MSF=1.6;LFS=11;AAT/;
|
||||
@@ -140,8 +140,8 @@ $cached_env_img{$key} = q|<IMG
|
||||
|
||||
$key = q/y;MSF=1.6;LFS=11;AAT/;
|
||||
$cached_env_img{$key} = q|<IMG
|
||||
WIDTH="14" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="|."$dir".q|img96.png"
|
||||
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="|."$dir".q|img97.png"
|
||||
ALT="$y$">|;
|
||||
|
||||
$key = q/{displaymath}A_C=R_CAR_C^T{displaymath};MSF=1.6;LFS=11;AAT/;
|
||||
@@ -205,6 +205,12 @@ $cached_env_img{$key} = q|<IMG
|
||||
SRC="|."$dir".q|img63.png"
|
||||
ALT="$A_l$">|;
|
||||
|
||||
$key = q/A_i^delta;MSF=1.6;LFS=11;AAT/;
|
||||
$cached_env_img{$key} = q|<IMG
|
||||
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="|."$dir".q|img28.png"
|
||||
ALT="$A_i^\delta$">|;
|
||||
|
||||
$key = q/n_C;MSF=1.6;LFS=11;AAT/;
|
||||
$cached_env_img{$key} = q|<IMG
|
||||
WIDTH="26" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
@@ -217,12 +223,6 @@ $cached_env_img{$key} = q|<IMG
|
||||
SRC="|."$dir".q|img17.png"
|
||||
ALT="$W_i^\delta$">|;
|
||||
|
||||
$key = q/A_i^delta;MSF=1.6;LFS=11;AAT/;
|
||||
$cached_env_img{$key} = q|<IMG
|
||||
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="|."$dir".q|img28.png"
|
||||
ALT="$A_i^\delta$">|;
|
||||
|
||||
$key = q/{displaymath}M_{AS}^{-1}=sum_{i=1}^m(R_i^{delta})^T(A_i^delta)^{-1}R_i^{delta},{displaymath};MSF=1.6;LFS=11;AAT/;
|
||||
$cached_env_img{$key} = q|<IMG
|
||||
WIDTH="206" HEIGHT="58" BORDER="0"
|
||||
@@ -323,9 +323,9 @@ $cached_env_img{$key} = q|<IMG
|
||||
|
||||
$key = q/>0;MSF=1.6;LFS=11;AAT/;
|
||||
$cached_env_img{$key} = q|<IMG
|
||||
WIDTH="32" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="|."$dir".q|img92.png"
|
||||
ALT="$>0$">|;
|
||||
WIDTH="31" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="|."$dir".q|img93.png"
|
||||
ALT="$> 0$">|;
|
||||
|
||||
$key = q/(tilde{R}_i^0)^TinRe^{n_i^deltatimesn};MSF=1.6;LFS=11;AAT/;
|
||||
$cached_env_img{$key} = q|<IMG
|
||||
@@ -347,8 +347,8 @@ $cached_env_img{$key} = q|<IMG
|
||||
|
||||
$key = q/x;MSF=1.6;LFS=11;AAT/;
|
||||
$cached_env_img{$key} = q|<IMG
|
||||
WIDTH="14" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="|."$dir".q|img95.png"
|
||||
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="|."$dir".q|img96.png"
|
||||
ALT="$x$">|;
|
||||
|
||||
$key = q/G=(W,E);MSF=1.6;LFS=11;AAT/;
|
||||
@@ -388,7 +388,7 @@ $cached_env_img{$key} = q|<IMG
|
||||
$key = q/M^{-1});MSF=1.6;LFS=11;AAT/;
|
||||
$cached_env_img{$key} = q|<IMG
|
||||
WIDTH="48" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="|."$dir".q|img99.png"
|
||||
SRC="|."$dir".q|img100.png"
|
||||
ALT="$M^{-1})$">|;
|
||||
|
||||
$key = q/{displaymath}M_{C}^{-1}=R_C^TA_C^{-1}R_C,{displaymath};MSF=1.6;LFS=11;AAT/;
|
||||
@@ -418,6 +418,12 @@ $cached_env_img{$key} = q|<IMG
|
||||
SRC="|."$dir".q|img90.png"
|
||||
ALT="$\in [0, 1]$">|;
|
||||
|
||||
$key = q/Re^n;MSF=1.6;LFS=11;AAT/;
|
||||
$cached_env_img{$key} = q|<IMG
|
||||
WIDTH="26" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="|."$dir".q|img24.png"
|
||||
ALT="$\Re^n$">|;
|
||||
|
||||
$key = q/{displaymath}array{{l}w=M_{C}^{-1}v,z=w+M_{1L}^{-1}(v-Aw),array{{displaymath};MSF=1.6;LFS=11;AAT/;
|
||||
$cached_env_img{$key} = q|<IMG
|
||||
WIDTH="177" HEIGHT="51" BORDER="0"
|
||||
@@ -434,12 +440,6 @@ $cached_env_img{$key} = q|<IMG
|
||||
SRC="|."$dir".q|img35.png"
|
||||
ALT="$p,t$">|;
|
||||
|
||||
$key = q/Re^n;MSF=1.6;LFS=11;AAT/;
|
||||
$cached_env_img{$key} = q|<IMG
|
||||
WIDTH="26" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="|."$dir".q|img24.png"
|
||||
ALT="$\Re^n$">|;
|
||||
|
||||
$key = q/t;MSF=1.6;LFS=11;AAT/;
|
||||
$cached_env_img{$key} = q|<IMG
|
||||
WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
@@ -504,8 +504,8 @@ M_{2LH-POST}^{-1} = M_{1L}^{-1} + \left( I - M_{1L}^{-1}A \right) M_{C}^{-1}.
|
||||
|
||||
$key = q/op;MSF=1.6;LFS=11;AAT/;
|
||||
$cached_env_img{$key} = q|<IMG
|
||||
WIDTH="22" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="|."$dir".q|img94.png"
|
||||
WIDTH="21" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="|."$dir".q|img95.png"
|
||||
ALT="$op$">|;
|
||||
|
||||
$key = q/G;MSF=1.6;LFS=11;AAT/;
|
||||
@@ -520,6 +520,12 @@ $cached_env_img{$key} = q|<IMG
|
||||
SRC="|."$dir".q|img84.png"
|
||||
ALT="$\omega$">|;
|
||||
|
||||
$key = q/4slash(3||D^{-1}A||_infty);MSF=1.6;LFS=11;AAT/;
|
||||
$cached_env_img{$key} = q|<IMG
|
||||
WIDTH="124" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="|."$dir".q|img92.png"
|
||||
ALT="$4/(3\vert\vert D^{-1}A\vert\vert _\infty)$">|;
|
||||
|
||||
$key = q/P_C;MSF=1.6;LFS=11;AAT/;
|
||||
$cached_env_img{$key} = q|<IMG
|
||||
WIDTH="27" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
@@ -541,7 +547,7 @@ $cached_env_img{$key} = q|<IMG
|
||||
$key = q/op(M^{-1})=M^{-1};MSF=1.6;LFS=11;AAT/;
|
||||
$cached_env_img{$key} = q|<IMG
|
||||
WIDTH="132" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="|."$dir".q|img97.png"
|
||||
SRC="|."$dir".q|img98.png"
|
||||
ALT="$op(M^{-1}) = M^{-1}$">|;
|
||||
|
||||
$key = q/W={1,2,ldots,n};MSF=1.6;LFS=11;AAT/;
|
||||
@@ -578,8 +584,8 @@ $cached_env_img{$key} = q|<IMG
|
||||
|
||||
$key = q/y=op(M^{-1}),x;MSF=1.6;LFS=11;AAT/;
|
||||
$cached_env_img{$key} = q|<IMG
|
||||
WIDTH="117" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="|."$dir".q|img93.png"
|
||||
WIDTH="118" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="|."$dir".q|img94.png"
|
||||
ALT="$y = op(M^{-1}) x$">|;
|
||||
|
||||
$key = q/{displaymath}S=I-omegaD^{-1}A,{displaymath};MSF=1.6;LFS=11;AAT/;
|
||||
@@ -613,7 +619,7 @@ $cached_env_img{$key} = q|<IMG
|
||||
$key = q/op(M^{-1})=M^{-C};MSF=1.6;LFS=11;AAT/;
|
||||
$cached_env_img{$key} = q|<IMG
|
||||
WIDTH="136" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="|."$dir".q|img100.png"
|
||||
SRC="|."$dir".q|img101.png"
|
||||
ALT="$op(M^{-1}) = M^{-C}$">|;
|
||||
|
||||
$key = q/W_i^deltasupsetW_i^{delta-1};MSF=1.6;LFS=11;AAT/;
|
||||
|
||||
+201
-95
@@ -1,4 +1,4 @@
|
||||
\batchmode
|
||||
\nonstopmode
|
||||
\documentclass[a4paper,twoside,11pt]{article}
|
||||
|
||||
\usepackage{pstricks}
|
||||
@@ -180,7 +180,8 @@
|
||||
\endgroup
|
||||
\stepcounter{section}
|
||||
{\newpage\clearpage
|
||||
\lthtmldisplayA{displaymath238}%
|
||||
% contents=begin{displaymath} Ax=b, end{displaymath}
|
||||
\lthtmldisplayA{displaymath242}%
|
||||
\begin{displaymath}
|
||||
Ax=b,
|
||||
\end{displaymath}%
|
||||
@@ -188,12 +189,12 @@ Ax=b,
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline265}%
|
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% contents=$A$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline268}%
|
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$A$%
|
||||
\lthtmlinlinemathZ
|
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\lthtmlcheckvsize\clearpage}
|
||||
|
||||
\stepcounter{section}
|
||||
\stepcounter{section}
|
||||
\stepcounter{section}
|
||||
\stepcounter{subsection}
|
||||
@@ -203,153 +204,178 @@ $A$%
|
||||
\stepcounter{section}
|
||||
\stepcounter{subsection}
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline586}%
|
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% contents=$A=(a_{ij}) \in \Re^{n \times n}$
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\lthtmlinlinemathA{tex2html_wrap_inline588}%
|
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$A=(a_{ij}) \in \Re^{n \times n}$%
|
||||
\lthtmlinlinemathZ
|
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\lthtmlcheckvsize\clearpage}
|
||||
|
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{\newpage\clearpage
|
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\lthtmlinlinemathA{tex2html_wrap_inline588}%
|
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% contents=$G=(W,E)$
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\lthtmlinlinemathA{tex2html_wrap_inline590}%
|
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$G=(W,E)$%
|
||||
\lthtmlinlinemathZ
|
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\lthtmlcheckvsize\clearpage}
|
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|
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{\newpage\clearpage
|
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\lthtmlinlinemathA{tex2html_wrap_inline592}%
|
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% contents=$W=\{1, 2, \ldots, n\}$
|
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\lthtmlinlinemathA{tex2html_wrap_inline594}%
|
||||
$W=\{1, 2, \ldots, n\}$%
|
||||
\lthtmlinlinemathZ
|
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\lthtmlcheckvsize\clearpage}
|
||||
|
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{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline594}%
|
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% contents=$E=\{(i,j) : a_{ij} \neq 0\}$
|
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\lthtmlinlinemathA{tex2html_wrap_inline596}%
|
||||
$E=\{(i,j) : a_{ij} \neq 0\}$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
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{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline596}%
|
||||
% contents=$G$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline598}%
|
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$G$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline598}%
|
||||
% contents=$\delta > 0$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline600}%
|
||||
$\delta > 0$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline600}%
|
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% contents=$\delta$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline602}%
|
||||
$\delta$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline602}%
|
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% contents=$W$
|
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\lthtmlinlinemathA{tex2html_wrap_inline604}%
|
||||
$W$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline606}%
|
||||
% contents=$m$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline608}%
|
||||
$m$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
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{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline608}%
|
||||
% contents=$W_i^0 \subset W$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline610}%
|
||||
$W_i^0 \subset W$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline610}%
|
||||
% contents=$\cup_{i=1}^m W_i^0 = W$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline612}%
|
||||
$\cup_{i=1}^m W_i^0 = W$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline616}%
|
||||
% contents=$W_i^\delta \supset W_i^{\delta-1}$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline618}%
|
||||
$W_i^\delta \supset W_i^{\delta-1}$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline618}%
|
||||
% contents=$W_i^{\delta-1}$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline620}%
|
||||
$W_i^{\delta-1}$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline620}%
|
||||
% contents=$n_i^\delta$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline622}%
|
||||
$n_i^\delta$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline622}%
|
||||
% contents=$W_i^\delta$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline624}%
|
||||
$W_i^\delta$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline624}%
|
||||
% contents=$R_i^{\delta} \in \Re^{n_i^\delta \times n}$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline626}%
|
||||
$R_i^{\delta} \in
|
||||
\Re^{n_i^\delta \times n}$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline626}%
|
||||
% contents=$v \in \Re^n$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline628}%
|
||||
$v \in \Re^n$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline628}%
|
||||
% contents=$v_i^{\delta} \in \Re^{n_i^\delta}$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline630}%
|
||||
$v_i^{\delta} \in \Re^{n_i^\delta}$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline630}%
|
||||
% contents=$v$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline632}%
|
||||
$v$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline634}%
|
||||
% contents=$R_i^{\delta}$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline636}%
|
||||
$R_i^{\delta}$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline636}%
|
||||
% contents=$\Re^{n_i^\delta}$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline638}%
|
||||
$\Re^{n_i^\delta}$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline638}%
|
||||
% contents=$\Re^n$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline640}%
|
||||
$\Re^n$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline640}%
|
||||
% contents=$A_i^\delta=R_i^\delta A (R_i^\delta)^T \in \Re^{n_i^\delta \times n_i^\delta}$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline642}%
|
||||
$A_i^\delta=R_i^\delta A (R_i^\delta)^T \in
|
||||
\Re^{n_i^\delta \times n_i^\delta}$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline644}%
|
||||
% contents=$W_i^{\delta}$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline646}%
|
||||
$W_i^{\delta}$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmldisplayA{displaymath564}%
|
||||
% contents=begin{displaymath} M_{AS}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T (A_i^\delta)^{-1} R_i^{\delta}, end{displaymath}
|
||||
\lthtmldisplayA{displaymath566}%
|
||||
\begin{displaymath}
|
||||
M_{AS}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
|
||||
(A_i^\delta)^{-1} R_i^{\delta},
|
||||
@@ -358,79 +384,92 @@ M_{AS}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline646}%
|
||||
% contents=$A_i^\delta$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline648}%
|
||||
$A_i^\delta$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline652}%
|
||||
% contents=$v_i = R_i^{\delta} v$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline654}%
|
||||
$v_i = R_i^{\delta} v$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline654}%
|
||||
% contents=$i=1,\ldots,m$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline656}%
|
||||
$i=1,\ldots,m$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline656}%
|
||||
% contents=$A_i^\delta w_i = v_i$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline658}%
|
||||
$A_i^\delta w_i = v_i$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline660}%
|
||||
% contents=$w_i$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline662}%
|
||||
$w_i$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline662}%
|
||||
% contents=$w = \sum_{i=1}^m (R_i^{\delta})^T w_i$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline664}%
|
||||
$w = \sum_{i=1}^m (R_i^{\delta})^T w_i$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline664}%
|
||||
% contents=$p$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline666}%
|
||||
$p$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline668}%
|
||||
% contents=$p,t$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline670}%
|
||||
$p,t$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline672}%
|
||||
% contents=$(\tilde{R}_i^0)^T \in \Re^{n_i^\delta \times n}$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline674}%
|
||||
$(\tilde{R}_i^0)^T \in \Re^{n_i^\delta \times n}$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline674}%
|
||||
% contents=$\tilde{R}_i^0$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline676}%
|
||||
$\tilde{R}_i^0$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline676}%
|
||||
% contents=$R_i^\delta$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline678}%
|
||||
$R_i^\delta$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline678}%
|
||||
% contents=$W_i^\delta \backslash W_i^0$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline680}%
|
||||
$W_i^\delta \backslash W_i^0$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmldisplayA{displaymath565}%
|
||||
% contents=begin{displaymath} M_{RAS}^{-1}= \sum_{i=1}^m (\tilde{R}_i^0)^T (A_i^\delta)^{-1} R_i^{\delta}. end{displaymath}
|
||||
\lthtmldisplayA{displaymath567}%
|
||||
\begin{displaymath}
|
||||
M_{RAS}^{-1}= \sum_{i=1}^m (\tilde{R}_i^0)^T
|
||||
(A_i^\delta)^{-1} R_i^{\delta}.
|
||||
@@ -439,7 +478,8 @@ M_{RAS}^{-1}= \sum_{i=1}^m (\tilde{R}_i^0)^T
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmldisplayA{displaymath566}%
|
||||
% contents=begin{displaymath} M_{ASH}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T (A_i^\delta)^{-1} \tilde{R}_i^0. end{displaymath}
|
||||
\lthtmldisplayA{displaymath568}%
|
||||
\begin{displaymath} M_{ASH}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
|
||||
(A_i^\delta)^{-1} \tilde{R}_i^0.
|
||||
\end{displaymath}%
|
||||
@@ -447,37 +487,43 @@ M_{RAS}^{-1}= \sum_{i=1}^m (\tilde{R}_i^0)^T
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline680}%
|
||||
% contents=$\delta=0$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline682}%
|
||||
$\delta=0$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline686}%
|
||||
% contents=$A_C$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline688}%
|
||||
$A_C$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline692}%
|
||||
% contents=$W_C$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline694}%
|
||||
$W_C$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline694}%
|
||||
% contents=$n_C$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline696}%
|
||||
$n_C$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline696}%
|
||||
% contents=$R_C \in \Re^{n_C \times n}$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline698}%
|
||||
$R_C \in \Re^{n_C \times n}$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmldisplayA{displaymath567}%
|
||||
% contents=begin{displaymath} A_C=R_C A R_C^T end{displaymath}
|
||||
\lthtmldisplayA{displaymath569}%
|
||||
\begin{displaymath}
|
||||
A_C=R_C A R_C^T
|
||||
\end{displaymath}%
|
||||
@@ -485,13 +531,15 @@ A_C=R_C A R_C^T
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline700}%
|
||||
% contents=$M_{1L}$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline702}%
|
||||
$M_{1L}$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmldisplayA{displaymath568}%
|
||||
% contents=begin{displaymath} M_{C}^{-1}= R_C^T A_C^{-1} R_C, end{displaymath}
|
||||
\lthtmldisplayA{displaymath570}%
|
||||
\begin{displaymath}
|
||||
M_{C}^{-1}= R_C^T A_C^{-1} R_C,
|
||||
\end{displaymath}%
|
||||
@@ -499,19 +547,22 @@ M_{C}^{-1}= R_C^T A_C^{-1} R_C,
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline704}%
|
||||
% contents=$M_{C}^{-1}$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline706}%
|
||||
$M_{C}^{-1}$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline710}%
|
||||
% contents=$M_{C}$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline712}%
|
||||
$M_{C}$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmldisplayA{displaymath569}%
|
||||
% contents=begin{displaymath} M_{2LA}^{-1} = M_{C}^{-1} + M_{1L}^{-1}. end{displaymath}
|
||||
\lthtmldisplayA{displaymath571}%
|
||||
\begin{displaymath}
|
||||
M_{2LA}^{-1} = M_{C}^{-1} + M_{1L}^{-1}.
|
||||
\end{displaymath}%
|
||||
@@ -519,19 +570,22 @@ M_{2LA}^{-1} = M_{C}^{-1} + M_{1L}^{-1}.
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline714}%
|
||||
% contents=$M_{2L-A}^{-1}$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline716}%
|
||||
$M_{2L-A}^{-1}$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline720}%
|
||||
% contents=$M_{1L}^{-1}$
|
||||
\lthtmlinlinemathA{tex2html_wrap_inline722}%
|
||||
$M_{1L}^{-1}$%
|
||||
\lthtmlinlinemathZ
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmldisplayA{displaymath570}%
|
||||
% contents=begin{displaymath} begin{array}{l} w = M_{1L}^{-1} v, \\z = w + M_{C}^{-1} (v-Aw); end{array} end{displaymath}
|
||||
\lthtmldisplayA{displaymath572}%
|
||||
\begin{displaymath}
|
||||
\begin{array}{l}
|
||||
w = M_{1L}^{-1} v, \\
|
||||
@@ -542,7 +596,8 @@ z = w + M_{C}^{-1} (v-Aw);
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmldisplayA{displaymath571}%
|
||||
% contents=begin{displaymath} M_{2LH-PRE}^{-1} = M_{C}^{-1} + \left( I - M_{C}^{-1}A \right) M_{1L}^{-1}. end{displaymath}
|
||||
\lthtmldisplayA{displaymath573}%
|
||||
\begin{displaymath}
|
||||
M_{2LH-PRE}^{-1} = M_{C}^{-1} + \left( I - M_{C}^{-1}A \right) M_{1L}^{-1}.
|
||||
\end{displaymath}%
|
||||
@@ -550,7 +605,8 @@ M_{2LH-PRE}^{-1} = M_{C}^{-1} + \left( I - M_{C}^{-1}A \right) M_{1L}^{-1}.
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmldisplayA{displaymath572}%
|
||||
% contents=begin{displaymath} begin{array}{l} w = M_{C}^{-1} v , \\z = w + M_{1L}^{-1} (v-Aw) , end{array} end{displaymath}
|
||||
\lthtmldisplayA{displaymath574}%
|
||||
\begin{displaymath}
|
||||
\begin{array}{l}
|
||||
w = M_{C}^{-1} v , \\
|
||||
@@ -561,7 +617,8 @@ z = w + M_{1L}^{-1} (v-Aw) ,
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
||||
{\newpage\clearpage
|
||||
\lthtmldisplayA{displaymath573}%
|
||||
% contents=begin{displaymath} M_{2LH-POST}^{-1} = M_{1L}^{-1} + \left( I - M_{1L}^{-1}A \right) M_{C}^{-1}. end{displaymath}
|
||||
\lthtmldisplayA{displaymath575}%
|
||||
\begin{displaymath}
|
||||
M_{2LH-POST}^{-1} = M_{1L}^{-1} + \left( I - M_{1L}^{-1}A \right) M_{C}^{-1}.
|
||||
\end{displaymath}%
|
||||
@@ -569,49 +626,57 @@ M_{2LH-POST}^{-1} = M_{1L}^{-1} + \left( I - M_{1L}^{-1}A \right) M_{C}^{-1}.
|
||||
\lthtmlcheckvsize\clearpage}
|
||||
|
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{\newpage\clearpage
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MLD2P4 User's and Reference Guidetitle
|
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<BR>
|
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<TABLE WIDTH="100%">
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|
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<FONT SIZE="+4"><B>MLD2P4
|
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<BIG CLASS="XLARGE"><B>MLD2P4
|
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<BR>
|
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User's and Reference Guide
|
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|
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<BR>
|
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<BR>
|
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<BR>
|
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<BR><I><FONT SIZE="+2">A guide for the Multi-Level Domain Decomposition <BR>
|
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User's and Reference Guide</B></BIG>
|
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<BR><SPAN CLASS="textit"><BIG CLASS="LARGE">A guide for the Multi-Level Domain Decomposition <BR>
|
||||
<BR>
|
||||
Parallel Preconditioners Package
|
||||
based on PSBLAS</FONT></I>
|
||||
|
||||
</DIV></TD></TR>
|
||||
</TABLE>
|
||||
|
||||
<P>
|
||||
based on PSBLAS</BIG></SPAN>
|
||||
<B>Pasqua D'Ambra</B>
|
||||
<BR>
|
||||
<TABLE WIDTH="100%">
|
||||
<TR><TD>
|
||||
<DIV ALIGN="RIGHT">
|
||||
<FONT SIZE="+1"><B>Pasqua D'Ambra</B></FONT>
|
||||
<BR><FONT SIZE="+1">ICAR-CNR, Naples, Italy
|
||||
ICAR-CNR, Naples, Italy
|
||||
<BR>
|
||||
<BR>
|
||||
<BR>
|
||||
<BR><FONT SIZE="+1"><B>Daniela di Serafino</B></FONT>
|
||||
<BR><FONT SIZE="+1">Second University of Naples, Italy
|
||||
<BR><B>Daniela di Serafino</B>
|
||||
<BR>
|
||||
Second University of Naples, Italy
|
||||
<BR>
|
||||
<BR>
|
||||
<BR>
|
||||
<BR><FONT SIZE="+1"><B>Salvatore Filippone</B></FONT>
|
||||
<BR><FONT SIZE="+1">University of Rome ``Tor Vergata'', Italy
|
||||
|
||||
</FONT></FONT></FONT></DIV></TD></TR>
|
||||
</TABLE>
|
||||
|
||||
<P>
|
||||
<BR><B>Salvatore Filippone</B>
|
||||
<BR>
|
||||
<TABLE WIDTH="100%">
|
||||
<TR><TD>
|
||||
<DIV ALIGN="RIGHT">
|
||||
<FONT SIZE="+1">Software version: 1.0
|
||||
<BR>
|
||||
July 22, 2008
|
||||
|
||||
</FONT></DIV></TD></TR>
|
||||
</TABLE>
|
||||
|
||||
<P>
|
||||
University of Rome ``Tor Vergata'', Italy
|
||||
Software version: 1.0
|
||||
<BR>
|
||||
July 24, 2008
|
||||
|
||||
<BR>
|
||||
|
||||
<P>
|
||||
|
||||
<P>
|
||||
|
||||
<H3>Abstract:</H3>
|
||||
<DIV>
|
||||
MLD2P4 (M<SMALL>ULTI-</SMALL>L<SMALL>EVEL </SMALL>D<SMALL>OMAIN </SMALL>D<SMALL>ECOMPOSITION </SMALL>P<SMALL>ARALLEL </SMALL>P<SMALL>RECONDITIONERS </SMALL>P<SMALL>ACKAGE BASED ON
|
||||
</SMALL>PSBLAS) is a package of parallel algebraic multi-level preconditioners.
|
||||
It implements various versions of one-level additive and of multi-level additive
|
||||
and hybrid Schwarz algorithms. In the multi-level case, a purely algebraic approach
|
||||
is applied to generate coarse-level corrections, so that no geometric background is needed
|
||||
concerning the matrix to be preconditioned. The matrix is required to be square, real
|
||||
or complex, with a symmetric sparsity pattern.
|
||||
|
||||
<P>
|
||||
MLD2P4 has been designed to provide scalable and easy-to-use preconditioners in the
|
||||
context of the PSBLAS (Parallel Sparse Basic Linear Algebra Subprograms)
|
||||
computational framework and can be used in conjuction with the Krylov solvers
|
||||
available in this framework. MLD2P4 enables the user to easily specify different aspects
|
||||
of a generic algebraic multilevel Schwarz preconditioner, thus allowing to search
|
||||
for the ``best'' preconditioner for the problem at hand.
|
||||
|
||||
<P>
|
||||
The package has been designed employing object-oriented techniques,
|
||||
using Fortran 95, with interfaces to additional third party libraries
|
||||
such as UMFPACK, SuperLU and SuperLU_Dist, that
|
||||
can be exploited in building multi-level preconditioners. The parallel
|
||||
implementation is based on a Single Program Multiple Data (SPMD)
|
||||
paradigm for distributed-memory architectures; the inter-process data
|
||||
communication is based on MPI and is managed mainly through PSBLAS.
|
||||
|
||||
<P>
|
||||
This guide provides a brief description of the functionalities and
|
||||
the user interface of MLD2P4.
|
||||
</DIV>
|
||||
<P>
|
||||
|
||||
<P>
|
||||
|
||||
<BR><HR>
|
||||
<BR><HR>
|
||||
<!--Table of Child-Links-->
|
||||
<A NAME="CHILD_LINKS"></A>
|
||||
|
||||
<UL>
|
||||
<UL CLASS="ChildLinks">
|
||||
<LI><A NAME="tex2html15"
|
||||
HREF="node1.html">Contents</A>
|
||||
HREF="node1.html">Abstract</A>
|
||||
<LI><A NAME="tex2html16"
|
||||
HREF="node2.html">General Overview</A>
|
||||
HREF="node2.html">Contents</A>
|
||||
<LI><A NAME="tex2html17"
|
||||
HREF="node3.html">Notational Conventions</A>
|
||||
HREF="node3.html">General Overview</A>
|
||||
<LI><A NAME="tex2html18"
|
||||
HREF="node4.html">Code Distribution</A>
|
||||
<LI><A NAME="tex2html19"
|
||||
@@ -238,29 +169,10 @@ the user interface of MLD2P4.
|
||||
HREF="node31.html">About this document ...</A>
|
||||
</UL>
|
||||
<!--End of Table of Child-Links-->
|
||||
<HR>
|
||||
<!--Navigation Panel-->
|
||||
<A NAME="tex2html13"
|
||||
HREF="node1.html">
|
||||
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
||||
SRC="file:/usr/share/latex2html/icons/next.png"></A>
|
||||
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
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SRC="file:/usr/share/latex2html/icons/up_g.png">
|
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<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
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SRC="file:/usr/share/latex2html/icons/prev_g.png">
|
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<A NAME="tex2html11"
|
||||
HREF="node1.html">
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||||
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
||||
SRC="file:/usr/share/latex2html/icons/contents.png"></A>
|
||||
<BR>
|
||||
<B> Next:</B> <A NAME="tex2html14"
|
||||
HREF="node1.html">Contents</A>
|
||||
<B> <A NAME="tex2html12"
|
||||
HREF="node1.html">Contents</A></B>
|
||||
<!--End of Navigation Panel-->
|
||||
<BR><HR>
|
||||
<ADDRESS>
|
||||
Salvatore Filippone
|
||||
2008-07-22
|
||||
2008-07-23
|
||||
</ADDRESS>
|
||||
</BODY>
|
||||
</HTML>
|
||||
|
||||
+35
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$ref_files{$key} = "$dir".q|node30.html|;
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$noresave{$key} = "$nosave";
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$ref_files{$key} = "$dir".q|node3.html|;
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$ref_files{$key} = "$dir".q|node30.html|;
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$noresave{$key} = "$nosave";
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|
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@@ -38,6 +42,10 @@ $key = q/cite_dd1_94/;
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$ref_files{$key} = "$dir".q|node30.html|;
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$noresave{$key} = "$nosave";
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$noresave{$key} = "$nosave";
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@@ -55,7 +63,7 @@ $ref_files{$key} = "$dir".q|node12.html|;
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$noresave{$key} = "$nosave";
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|
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$key = q/sec:overview/;
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$ref_files{$key} = "$dir".q|node3.html|;
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|
||||
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|
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|
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$noresave{$key} = "$nosave";
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|
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@@ -102,6 +114,10 @@ $key = q/sec:errors/;
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$noresave{$key} = "$nosave";
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$key = q/fig:ex_3lh/;
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$ref_files{$key} = "$dir".q|node30.html|;
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1;
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$external_labels{$key} = "$URL/" . q|node3.html|;
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$external_labels{$key} = "$URL/" . q|node30.html|;
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$noresave{$key} = "$nosave";
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$noresave{$key} = "$nosave";
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@@ -55,7 +63,7 @@ $external_labels{$key} = "$URL/" . q|node12.html|;
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$noresave{$key} = "$nosave";
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$key = q/sec:overview/;
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$external_labels{$key} = "$URL/" . q|node3.html|;
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$key = q/sec:multilevel/;
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@@ -87,7 +95,11 @@ $external_labels{$key} = "$URL/" . q|node30.html|;
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$noresave{$key} = "$nosave";
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$key = q/system1/;
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$noresave{$key} = "$nosave";
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$key = q/fig:ex_3la/;
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@@ -102,6 +114,10 @@ $key = q/sec:errors/;
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$external_labels{$key} = "$URL/" . q|node28.html|;
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@@ -122,14 +138,14 @@ $key = q/eq:jac_smoother/;
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$external_labels{$key} = "$URL/" . q|node12.html|;
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$noresave{$key} = "$nosave";
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$key = q/fig:ex_3lh/;
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$external_labels{$key} = "$URL/" . q|node14.html|;
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$noresave{$key} = "$nosave";
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$key = q/sec:precaply/;
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$external_labels{$key} = "$URL/" . q|node22.html|;
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$noresave{$key} = "$nosave";
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$key = q/fig:ex_3lh/;
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$external_labels{$key} = "$URL/" . q|node14.html|;
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$noresave{$key} = "$nosave";
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$key = q/cite_BREZINA_VANEK/;
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$external_labels{$key} = "$URL/" . q|node30.html|;
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$noresave{$key} = "$nosave";
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@@ -142,6 +158,10 @@ $key = q/cite_para_04/;
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$external_labels{$key} = "$URL/" . q|node30.html|;
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$noresave{$key} = "$nosave";
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$key = q/cite_BLACS/;
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$key = q/cite_EFSTATHIOU/;
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$external_labels{$key} = "$URL/" . q|node30.html|;
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$noresave{$key} = "$nosave";
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@@ -174,6 +194,10 @@ $key = q/eq:smoothed_prol/;
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$external_labels{$key} = "$URL/" . q|node12.html|;
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$noresave{$key} = "$nosave";
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$key = q/cite_blas1/;
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$external_labels{$key} = "$URL/" . q|node30.html|;
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$noresave{$key} = "$nosave";
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$key = q/fig:ex_1l/;
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$external_labels{$key} = "$URL/" . q|node14.html|;
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$noresave{$key} = "$nosave";
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@@ -186,6 +210,10 @@ $key = q/sec:precfree/;
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$external_labels{$key} = "$URL/" . q|node24.html|;
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$noresave{$key} = "$nosave";
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$noresave{$key} = "$nosave";
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1;
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# labels from external_latex_labels array.
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$external_latex_labels{$key} = q|4|;
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$noresave{$key} = "$nosave";
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$key = q/sec:errors/;
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$external_latex_labels{$key} = q|7|;
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$key = q/sec:ex_and_test/;
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1;
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|
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Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
|
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<HTML>
|
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<HEAD>
|
||||
<TITLE>Contents</TITLE>
|
||||
<META NAME="description" CONTENT="Contents">
|
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<TITLE>Abstract</TITLE>
|
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<META NAME="description" CONTENT="Abstract">
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<META NAME="resource-type" CONTENT="document">
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|
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<H2><A NAME="SECTION00010000000000000000">
|
||||
Contents</A>
|
||||
</H2>
|
||||
<!--Table of Contents-->
|
||||
<H1><A NAME="SECTION00010000000000000000">
|
||||
Abstract</A>
|
||||
</H1>
|
||||
|
||||
<UL>
|
||||
<LI><A NAME="tex2html56"
|
||||
HREF="node2.html">General Overview</A>
|
||||
<LI><A NAME="tex2html57"
|
||||
HREF="node3.html">Notational Conventions</A>
|
||||
<LI><A NAME="tex2html58"
|
||||
HREF="node4.html">Code Distribution</A>
|
||||
<LI><A NAME="tex2html59"
|
||||
HREF="node5.html">Configuring and Building MLD2P4</A>
|
||||
<UL>
|
||||
<LI><A NAME="tex2html60"
|
||||
HREF="node6.html">Prerequisites</A>
|
||||
<LI><A NAME="tex2html61"
|
||||
HREF="node7.html">Optional third party libraries</A>
|
||||
<LI><A NAME="tex2html62"
|
||||
HREF="node8.html">Configuration options</A>
|
||||
<LI><A NAME="tex2html63"
|
||||
HREF="node9.html">Example and test programs</A>
|
||||
</UL>
|
||||
<BR>
|
||||
<LI><A NAME="tex2html64"
|
||||
HREF="node10.html">Multi-level Domain Decomposition Background</A>
|
||||
<UL>
|
||||
<LI><A NAME="tex2html65"
|
||||
HREF="node11.html">Multi-level Schwarz Preconditioners</A>
|
||||
<LI><A NAME="tex2html66"
|
||||
HREF="node12.html">Smoothed Aggregation</A>
|
||||
</UL>
|
||||
<BR>
|
||||
<LI><A NAME="tex2html67"
|
||||
HREF="node13.html">Getting Started</A>
|
||||
<UL>
|
||||
<LI><A NAME="tex2html68"
|
||||
HREF="node14.html">Examples</A>
|
||||
</UL>
|
||||
<BR>
|
||||
<LI><A NAME="tex2html69"
|
||||
HREF="node15.html">User Interface</A>
|
||||
<UL>
|
||||
<LI><A NAME="tex2html70"
|
||||
HREF="node16.html">Subroutine mld_precinit</A>
|
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<LI><A NAME="tex2html71"
|
||||
HREF="node18.html">Subroutine mld_precset</A>
|
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<LI><A NAME="tex2html72"
|
||||
HREF="node20.html">Subroutine mld_precbld</A>
|
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<LI><A NAME="tex2html73"
|
||||
HREF="node22.html">Subroutine mld_precaply</A>
|
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<LI><A NAME="tex2html74"
|
||||
HREF="node24.html">Subroutine mld_precfree</A>
|
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<LI><A NAME="tex2html75"
|
||||
HREF="node26.html">Subroutine mld_precdescr</A>
|
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</UL>
|
||||
<BR>
|
||||
<LI><A NAME="tex2html76"
|
||||
HREF="node28.html">Error Handling</A>
|
||||
<LI><A NAME="tex2html77"
|
||||
HREF="node29.html">License</A>
|
||||
<LI><A NAME="tex2html78"
|
||||
HREF="node30.html">Bibliography</A>
|
||||
</UL>
|
||||
<!--End of Table of Contents-->
|
||||
MLD2P4 (M<SMALL>ULTI-</SMALL>L<SMALL>EVEL </SMALL>D<SMALL>OMAIN </SMALL>D<SMALL>ECOMPOSITION </SMALL>P<SMALL>ARALLEL </SMALL>P<SMALL>RECONDITIONERS </SMALL>P<SMALL>ACKAGE BASED ON
|
||||
</SMALL>PSBLAS) is a package of parallel algebraic multi-level preconditioners.
|
||||
It implements various versions of one-level additive and of multi-level additive
|
||||
and hybrid Schwarz algorithms. In the multi-level case, a purely algebraic approach
|
||||
is applied to generate coarse-level corrections, so that no geometric background is needed
|
||||
concerning the matrix to be preconditioned. The matrix is required to be square, real
|
||||
or complex, with a symmetric sparsity pattern.
|
||||
|
||||
<P>
|
||||
MLD2P4 has been designed to provide scalable and easy-to-use preconditioners in the
|
||||
context of the PSBLAS (Parallel Sparse Basic Linear Algebra Subprograms)
|
||||
computational framework and can be used in conjuction with the Krylov solvers
|
||||
available in this framework. MLD2P4 enables the user to easily specify different aspects
|
||||
of a generic algebraic multilevel Schwarz preconditioner, thus allowing to search
|
||||
for the ``best'' preconditioner for the problem at hand.
|
||||
|
||||
<P>
|
||||
The package has been designed employing object-oriented techniques,
|
||||
using Fortran 95, with interfaces to additional third party libraries
|
||||
such as UMFPACK, SuperLU and SuperLU_Dist, that
|
||||
can be exploited in building multi-level preconditioners. The parallel
|
||||
implementation is based on a Single Program Multiple Data (SPMD)
|
||||
paradigm for distributed-memory architectures; the inter-process data
|
||||
communication is based on MPI and is managed mainly through PSBLAS.
|
||||
|
||||
<P>
|
||||
This guide provides a brief description of the functionalities and
|
||||
the user interface of MLD2P4.
|
||||
|
||||
<P>
|
||||
|
||||
<P>
|
||||
<BR><HR>
|
||||
|
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<DIV CLASS="navigation"><HR>
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HREF="node2.html">Contents</A></B> </DIV>
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<!--End of Navigation Panel-->
|
||||
<ADDRESS>
|
||||
Salvatore Filippone
|
||||
2008-07-22
|
||||
2008-07-23
|
||||
</ADDRESS>
|
||||
</BODY>
|
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</HTML>
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|
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+43
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
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<!--Converted with LaTeX2HTML 2002-2-1 (1.71)
|
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original version by: Nikos Drakos, CBLU, University of Leeds
|
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@@ -25,34 +25,35 @@ original version by: Nikos Drakos, CBLU, University of Leeds
|
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HREF="node1.html">Contents</A></B>
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|
||||
<H1><A NAME="SECTION00060000000000000000"></A><A NAME="sec:background"></A>
|
||||
@@ -61,7 +62,7 @@ Multi-level Domain Decomposition Background
|
||||
</H1>
|
||||
|
||||
<P>
|
||||
<I>Domain Decomposition</I> (DD) preconditioners, coupled with Krylov iterative
|
||||
<SPAN CLASS="textit">Domain Decomposition</SPAN> (DD) preconditioners, coupled with Krylov iterative
|
||||
solvers, are widely used in the parallel solution of large and sparse linear systems.
|
||||
These preconditioners are based on the divide and conquer technique: the matrix
|
||||
to be preconditioned is divided into submatrices, a ``local'' linear system
|
||||
@@ -74,13 +75,13 @@ solution of the original problem from the local solutions
|
||||
[<A
|
||||
HREF="node30.html#Cai_Widlund_92">6</A>,<A
|
||||
HREF="node30.html#dd1_94">7</A>,<A
|
||||
HREF="node30.html#dd2_96">15</A>].
|
||||
HREF="node30.html#dd2_96">20</A>].
|
||||
|
||||
<P>
|
||||
<I>Additive Schwarz</I> preconditioners are DD preconditioners using overlapping
|
||||
<SPAN CLASS="textit">Additive Schwarz</SPAN> preconditioners are DD preconditioners using overlapping
|
||||
submatrices, i.e. with some common rows, to couple the local information
|
||||
related to the submatrices (see, e.g., [<A
|
||||
HREF="node30.html#dd2_96">15</A>]).
|
||||
HREF="node30.html#dd2_96">20</A>]).
|
||||
The main motivation for choosing Additive Schwarz preconditioners is their
|
||||
intrinsic parallelism. A drawback of these
|
||||
preconditioners is that the number of iterations of the preconditioned solvers
|
||||
@@ -93,15 +94,15 @@ in a coarse space, which globally couples the information related to the single
|
||||
submatrices.
|
||||
|
||||
<P>
|
||||
<I>Two-level Schwarz</I> preconditioners are obtained
|
||||
<SPAN CLASS="textit">Two-level Schwarz</SPAN> preconditioners are obtained
|
||||
by combining basic (one-level) Schwarz preconditioners with a coarse-level
|
||||
correction. In this context, the one-level preconditioner is often
|
||||
called `smoother'. Different two-level preconditioners are obtained by varying the
|
||||
choice of the smoother and of the coarse-level correction, and the
|
||||
way they are combined [<A
|
||||
HREF="node30.html#dd2_96">15</A>]. The same reasoning can be applied starting
|
||||
HREF="node30.html#dd2_96">20</A>]. The same reasoning can be applied starting
|
||||
from the coarse-level system, i.e. a coarse-space correction can be built
|
||||
from this system, thus obtaining <I>multi-level</I> preconditioners.
|
||||
from this system, thus obtaining <SPAN CLASS="textit">multi-level</SPAN> preconditioners.
|
||||
|
||||
<P>
|
||||
It is worth noting that optimal preconditioners do not necessarily correspond
|
||||
@@ -123,68 +124,69 @@ are considered. The algebraic approach builds coarse-space corrections using onl
|
||||
information. It performs a fully automatic coarsening and enforces the interplay between
|
||||
the fine and coarse levels by suitably choosing the coarse space and the coarse-to-fine
|
||||
interpolation [<A
|
||||
HREF="node30.html#StubenGMD69_99">17</A>].
|
||||
HREF="node30.html#StubenGMD69_99">22</A>].
|
||||
|
||||
<P>
|
||||
MLD2P4 uses a pure algebraic approach for building the sequence of coarse matrices
|
||||
starting from the original matrix. The algebraic approach is based on the <I>smoothed
|
||||
aggregation</I> algorithm [<A
|
||||
starting from the original matrix. The algebraic approach is based on the <SPAN CLASS="textit">smoothed
|
||||
aggregation</SPAN> algorithm [<A
|
||||
HREF="node30.html#BREZINA_VANEK">1</A>,<A
|
||||
HREF="node30.html#VANEK_MANDEL_BREZINA">19</A>]. A decoupled version
|
||||
HREF="node30.html#VANEK_MANDEL_BREZINA">24</A>]. A decoupled version
|
||||
of this algorithm is implemented, where the smoothed aggregation is applied locally
|
||||
to each submatrix [<A
|
||||
HREF="node30.html#TUMINARO_TONG">18</A>]. In the next two subsections we provide
|
||||
HREF="node30.html#TUMINARO_TONG">23</A>]. In the next two subsections we provide
|
||||
a brief description of the multi-level Schwarz preconditioners and of the smoothed
|
||||
aggregation technique as implemented in MLD2P4. For further details the user
|
||||
is referred to [<A
|
||||
HREF="node30.html#para_04">2</A>,<A
|
||||
HREF="node30.html#aaecc_07">3</A>,<A
|
||||
HREF="node30.html#apnum_07">4</A>,<A
|
||||
HREF="node30.html#dd2_96">15</A>].
|
||||
HREF="node30.html#dd2_96">20</A>].
|
||||
|
||||
<P>
|
||||
<BR><HR>
|
||||
<!--Table of Child-Links-->
|
||||
<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
|
||||
|
||||
<UL>
|
||||
<LI><A NAME="tex2html189"
|
||||
<UL CLASS="ChildLinks">
|
||||
<LI><A NAME="tex2html188"
|
||||
HREF="node11.html">Multi-level Schwarz Preconditioners</A>
|
||||
<LI><A NAME="tex2html190"
|
||||
<LI><A NAME="tex2html189"
|
||||
HREF="node12.html">Smoothed Aggregation</A>
|
||||
</UL>
|
||||
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|
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|
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|
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<DIV CLASS="navigation"><HR>
|
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|
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<B> Previous:</B> <A NAME="tex2html178"
|
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<B> Previous:</B> <A NAME="tex2html177"
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HREF="node9.html">Example and test programs</A>
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<B> <A NAME="tex2html186"
|
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HREF="node1.html">Contents</A></B>
|
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<B> <A NAME="tex2html185"
|
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HREF="node2.html">Contents</A></B> </DIV>
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<!--End of Navigation Panel-->
|
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<ADDRESS>
|
||||
Salvatore Filippone
|
||||
2008-07-22
|
||||
2008-07-23
|
||||
</ADDRESS>
|
||||
</BODY>
|
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</HTML>
|
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|
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+196
-195
@@ -1,4 +1,4 @@
|
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
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|
||||
original version by: Nikos Drakos, CBLU, University of Leeds
|
||||
@@ -25,34 +25,35 @@ original version by: Nikos Drakos, CBLU, University of Leeds
|
||||
</HEAD>
|
||||
|
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<BODY >
|
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<A NAME="tex2html201"
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<DIV CLASS="navigation"><!--Navigation Panel-->
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HREF="node12.html">
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SRC="file:/usr/share/latex2html/icons/next.png"></A>
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<B> Up:</B> <A NAME="tex2html197"
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HREF="node1.html">Contents</A></B>
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<BR>
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<B> <A NAME="tex2html199"
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HREF="node2.html">Contents</A></B>
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<!--End of Navigation Panel-->
|
||||
|
||||
<H2><A NAME="SECTION00061000000000000000"></A><A NAME="sec:multilevel"></A>
|
||||
@@ -70,157 +71,157 @@ Given the linear system ,
|
||||
where <!-- MATH
|
||||
$A=(a_{ij}) \in \Re^{n \times n}$
|
||||
-->
|
||||
<IMG
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="137" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img3.png"
|
||||
ALT="$A=(a_{ij}) \in \Re^{n \times n}$"> is a
|
||||
ALT="$A=(a_{ij}) \in \Re^{n \times n}$"></SPAN> is a
|
||||
nonsingular sparse matrix with a symmetric nonzero pattern,
|
||||
let <IMG
|
||||
let <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="93" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img4.png"
|
||||
ALT="$G=(W,E)$"> be the adjacency graph of <IMG
|
||||
ALT="$G=(W,E)$"></SPAN> be the adjacency graph of <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img2.png"
|
||||
ALT="$A$">, where <!-- MATH
|
||||
ALT="$A$"></SPAN>, where <!-- MATH
|
||||
$W=\{1, 2, \ldots, n\}$
|
||||
-->
|
||||
<IMG
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="138" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img5.png"
|
||||
ALT="$W=\{1, 2, \ldots, n\}$">
|
||||
ALT="$W=\{1, 2, \ldots, n\}$"></SPAN>
|
||||
and <!-- MATH
|
||||
$E=\{(i,j) : a_{ij} \neq 0\}$
|
||||
-->
|
||||
<IMG
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="162" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img6.png"
|
||||
ALT="$E=\{(i,j) : a_{ij} \neq 0\}$"> are the vertex set and the edge set of <IMG
|
||||
ALT="$E=\{(i,j) : a_{ij} \neq 0\}$"></SPAN> are the vertex set and the edge set of <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img7.png"
|
||||
ALT="$G$">,
|
||||
ALT="$G$"></SPAN>,
|
||||
respectively. Two vertices are called adjacent if there is an edge connecting
|
||||
them. For any integer <IMG
|
||||
them. For any integer <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="45" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img8.png"
|
||||
ALT="$\delta > 0$">, a <IMG
|
||||
ALT="$\delta > 0$"></SPAN>, a <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="13" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img9.png"
|
||||
ALT="$\delta$">-overlap
|
||||
partition of <IMG
|
||||
ALT="$\delta$"></SPAN>-overlap
|
||||
partition of <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="24" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img10.png"
|
||||
ALT="$W$"> can be defined recursively as follows.
|
||||
Given a 0-overlap (or non-overlapping) partition of <IMG
|
||||
ALT="$W$"></SPAN> can be defined recursively as follows.
|
||||
Given a 0-overlap (or non-overlapping) partition of <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="24" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img10.png"
|
||||
ALT="$W$">,
|
||||
i.e. a set of <IMG
|
||||
ALT="$W$"></SPAN>,
|
||||
i.e. a set of <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="20" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img11.png"
|
||||
ALT="$m$"> disjoint nonempty sets <!-- MATH
|
||||
ALT="$m$"></SPAN> disjoint nonempty sets <!-- MATH
|
||||
$W_i^0 \subset W$
|
||||
-->
|
||||
<IMG
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="73" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img12.png"
|
||||
ALT="$W_i^0 \subset W$"> such that
|
||||
ALT="$W_i^0 \subset W$"></SPAN> such that
|
||||
<!-- MATH
|
||||
$\cup_{i=1}^m W_i^0 = W$
|
||||
-->
|
||||
<IMG
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="107" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img13.png"
|
||||
ALT="$\cup_{i=1}^m W_i^0 = W$">, a <IMG
|
||||
ALT="$\cup_{i=1}^m W_i^0 = W$"></SPAN>, a <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="13" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img9.png"
|
||||
ALT="$\delta$">-overlap
|
||||
partition of <IMG
|
||||
ALT="$\delta$"></SPAN>-overlap
|
||||
partition of <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="24" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img10.png"
|
||||
ALT="$W$"> is obtained by considering the sets
|
||||
ALT="$W$"></SPAN> is obtained by considering the sets
|
||||
<!-- MATH
|
||||
$W_i^\delta \supset W_i^{\delta-1}$
|
||||
-->
|
||||
<IMG
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="97" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img14.png"
|
||||
ALT="$W_i^\delta \supset W_i^{\delta-1}$"> obtained by including the vertices that
|
||||
ALT="$W_i^\delta \supset W_i^{\delta-1}$"></SPAN> obtained by including the vertices that
|
||||
are adjacent to any vertex in <!-- MATH
|
||||
$W_i^{\delta-1}$
|
||||
-->
|
||||
<IMG
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="48" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img15.png"
|
||||
ALT="$W_i^{\delta-1}$">.
|
||||
ALT="$W_i^{\delta-1}$"></SPAN>.
|
||||
|
||||
<P>
|
||||
Let <IMG
|
||||
Let <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img16.png"
|
||||
ALT="$n_i^\delta$"> be the size of <IMG
|
||||
ALT="$n_i^\delta$"></SPAN> be the size of <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="30" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img17.png"
|
||||
ALT="$W_i^\delta$"> and <!-- MATH
|
||||
ALT="$W_i^\delta$"></SPAN> and <!-- MATH
|
||||
$R_i^{\delta} \in
|
||||
\Re^{n_i^\delta \times n}$
|
||||
-->
|
||||
<IMG
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="93" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img18.png"
|
||||
ALT="$R_i^{\delta} \in
|
||||
\Re^{n_i^\delta \times n}$"> the restriction operator that maps
|
||||
a vector <IMG
|
||||
\Re^{n_i^\delta \times n}$"></SPAN> the restriction operator that maps
|
||||
a vector <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="56" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img19.png"
|
||||
ALT="$v \in \Re^n$"> onto the vector <!-- MATH
|
||||
ALT="$v \in \Re^n$"></SPAN> onto the vector <!-- MATH
|
||||
$v_i^{\delta} \in \Re^{n_i^\delta}$
|
||||
-->
|
||||
<IMG
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="70" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img20.png"
|
||||
ALT="$v_i^{\delta} \in \Re^{n_i^\delta}$">
|
||||
containing the components of <IMG
|
||||
ALT="$v_i^{\delta} \in \Re^{n_i^\delta}$"></SPAN>
|
||||
containing the components of <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img21.png"
|
||||
ALT="$v$"> corresponding to the vertices in
|
||||
<IMG
|
||||
ALT="$v$"></SPAN> corresponding to the vertices in
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="30" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img17.png"
|
||||
ALT="$W_i^\delta$">. The transpose of <IMG
|
||||
ALT="$W_i^\delta$"></SPAN>. The transpose of <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img22.png"
|
||||
ALT="$R_i^{\delta}$"> is a
|
||||
ALT="$R_i^{\delta}$"></SPAN> is a
|
||||
prolongation operator from <!-- MATH
|
||||
$\Re^{n_i^\delta}$
|
||||
-->
|
||||
<IMG
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="32" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img23.png"
|
||||
ALT="$\Re^{n_i^\delta}$"> to <IMG
|
||||
ALT="$\Re^{n_i^\delta}$"></SPAN> to <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="26" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img24.png"
|
||||
ALT="$\Re^n$">.
|
||||
ALT="$\Re^n$"></SPAN>.
|
||||
The matrix <!-- MATH
|
||||
$A_i^\delta=R_i^\delta A (R_i^\delta)^T \in
|
||||
\Re^{n_i^\delta \times n_i^\delta}$
|
||||
-->
|
||||
<IMG
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="201" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img25.png"
|
||||
ALT="$A_i^\delta=R_i^\delta A (R_i^\delta)^T \in
|
||||
\Re^{n_i^\delta \times n_i^\delta}$"> can be considered
|
||||
as a restriction of <IMG
|
||||
\Re^{n_i^\delta \times n_i^\delta}$"></SPAN> can be considered
|
||||
as a restriction of <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img2.png"
|
||||
ALT="$A$"> corresponding to the set <IMG
|
||||
ALT="$A$"></SPAN> corresponding to the set <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="31" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img26.png"
|
||||
ALT="$W_i^{\delta}$">.
|
||||
ALT="$W_i^{\delta}$"></SPAN>.
|
||||
|
||||
<P>
|
||||
The <I>classical one-level AS</I> preconditioner is defined by
|
||||
The <SPAN CLASS="textit">classical one-level AS</SPAN> preconditioner is defined by
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<DIV ALIGN="CENTER" CLASS="mathdisplay">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
M_{AS}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
|
||||
@@ -238,105 +239,105 @@ M_{AS}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
where <IMG
|
||||
where <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img28.png"
|
||||
ALT="$A_i^\delta$"> is assumed to be nonsingular. Its application
|
||||
to a vector <IMG
|
||||
ALT="$A_i^\delta$"></SPAN> is assumed to be nonsingular. Its application
|
||||
to a vector <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="56" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img19.png"
|
||||
ALT="$v \in \Re^n$"> within a Krylov solver requires the following
|
||||
ALT="$v \in \Re^n$"></SPAN> within a Krylov solver requires the following
|
||||
three steps:
|
||||
|
||||
<OL>
|
||||
<LI>restriction of <IMG
|
||||
<LI>restriction of <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img21.png"
|
||||
ALT="$v$"> as <!-- MATH
|
||||
ALT="$v$"></SPAN> as <!-- MATH
|
||||
$v_i = R_i^{\delta} v$
|
||||
-->
|
||||
<IMG
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="71" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img29.png"
|
||||
ALT="$v_i = R_i^{\delta} v$">, <IMG
|
||||
ALT="$v_i = R_i^{\delta} v$"></SPAN>, <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="97" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img30.png"
|
||||
ALT="$i=1,\ldots,m$">;
|
||||
ALT="$i=1,\ldots,m$"></SPAN>;
|
||||
</LI>
|
||||
<LI>solution of the linear systems <!-- MATH
|
||||
$A_i^\delta w_i = v_i$
|
||||
-->
|
||||
<IMG
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="80" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img31.png"
|
||||
ALT="$A_i^\delta w_i = v_i$">,
|
||||
<IMG
|
||||
ALT="$A_i^\delta w_i = v_i$"></SPAN>,
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="97" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img30.png"
|
||||
ALT="$i=1,\ldots,m$">;
|
||||
ALT="$i=1,\ldots,m$"></SPAN>;
|
||||
</LI>
|
||||
<LI>prolongation and sum of the <IMG
|
||||
<LI>prolongation and sum of the <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="23" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img32.png"
|
||||
ALT="$w_i$">'s, i.e. <!-- MATH
|
||||
ALT="$w_i$"></SPAN>'s, i.e. <!-- MATH
|
||||
$w = \sum_{i=1}^m (R_i^{\delta})^T w_i$
|
||||
-->
|
||||
<IMG
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="144" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img33.png"
|
||||
ALT="$w = \sum_{i=1}^m (R_i^{\delta})^T w_i$">.
|
||||
ALT="$w = \sum_{i=1}^m (R_i^{\delta})^T w_i$"></SPAN>.
|
||||
</LI>
|
||||
</OL>
|
||||
Note that the linear systems at step 2 are usually solved approximately,
|
||||
e.g. using incomplete LU factorizations such as ILU(<IMG
|
||||
e.g. using incomplete LU factorizations such as ILU(<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img34.png"
|
||||
ALT="$p$">), MILU(<IMG
|
||||
ALT="$p$"></SPAN>), MILU(<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img34.png"
|
||||
ALT="$p$">) and
|
||||
ILU(<IMG
|
||||
ALT="$p$"></SPAN>) and
|
||||
ILU(<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="27" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img35.png"
|
||||
ALT="$p,t$">) [<A
|
||||
HREF="node30.html#Saad_book">14</A>, Chapter 10].
|
||||
ALT="$p,t$"></SPAN>) [<A
|
||||
HREF="node30.html#Saad_book">19</A>, Chapter 10].
|
||||
|
||||
<P>
|
||||
A variant of the classical AS preconditioner that outperforms it
|
||||
in terms of convergence rate and of computation and communication
|
||||
time on parallel distributed-memory computers is the so-called <I>Restricted AS
|
||||
(RAS)</I> preconditioner [<A
|
||||
time on parallel distributed-memory computers is the so-called <SPAN CLASS="textit">Restricted AS
|
||||
(RAS)</SPAN> preconditioner [<A
|
||||
HREF="node30.html#CAI_SARKIS">5</A>,<A
|
||||
HREF="node30.html#EFSTATHIOU">10</A>]. It
|
||||
is obtained by zeroing the components of <IMG
|
||||
HREF="node30.html#EFSTATHIOU">13</A>]. It
|
||||
is obtained by zeroing the components of <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="23" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img32.png"
|
||||
ALT="$w_i$"> corresponding to the
|
||||
ALT="$w_i$"></SPAN> corresponding to the
|
||||
overlapping vertices when applying the prolongation. Therefore,
|
||||
RAS differs from classical AS by the prolongation operators,
|
||||
which are substituted by <!-- MATH
|
||||
$(\tilde{R}_i^0)^T \in \Re^{n_i^\delta \times n}$
|
||||
-->
|
||||
<IMG
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="118" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img36.png"
|
||||
ALT="$(\tilde{R}_i^0)^T \in \Re^{n_i^\delta \times n}$">,
|
||||
where <IMG
|
||||
ALT="$(\tilde{R}_i^0)^T \in \Re^{n_i^\delta \times n}$"></SPAN>,
|
||||
where <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="25" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img37.png"
|
||||
ALT="$\tilde{R}_i^0$"> is obtained by zeroing the rows of <IMG
|
||||
ALT="$\tilde{R}_i^0$"></SPAN> is obtained by zeroing the rows of <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img38.png"
|
||||
ALT="$R_i^\delta$">
|
||||
ALT="$R_i^\delta$"></SPAN>
|
||||
corresponding to the vertices in <!-- MATH
|
||||
$W_i^\delta \backslash W_i^0$
|
||||
-->
|
||||
<IMG
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="66" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img39.png"
|
||||
ALT="$W_i^\delta \backslash W_i^0$">:
|
||||
ALT="$W_i^\delta \backslash W_i^0$"></SPAN>:
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<DIV ALIGN="CENTER" CLASS="mathdisplay">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
M_{RAS}^{-1}= \sum_{i=1}^m (\tilde{R}_i^0)^T
|
||||
@@ -354,10 +355,10 @@ M_{RAS}^{-1}= \sum_{i=1}^m (\tilde{R}_i^0)^T
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
Analogously, the AS variant called <I>AS with Harmonic extension (ASH)</I>
|
||||
Analogously, the AS variant called <SPAN CLASS="textit">AS with Harmonic extension (ASH)</SPAN>
|
||||
is defined by
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<DIV ALIGN="CENTER" CLASS="mathdisplay">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
M_{ASH}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
|
||||
@@ -374,57 +375,57 @@ M_{ASH}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
We note that for <IMG
|
||||
We note that for <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="45" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img42.png"
|
||||
ALT="$\delta=0$"> the three variants of the AS preconditioner are
|
||||
ALT="$\delta=0$"></SPAN> the three variants of the AS preconditioner are
|
||||
all equal to the block-Jacobi preconditioner.
|
||||
|
||||
<P>
|
||||
As already observed, the convergence rate of the one-level Schwarz
|
||||
preconditioned iterative solvers deteriorates as the number <IMG
|
||||
preconditioned iterative solvers deteriorates as the number <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="20" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img11.png"
|
||||
ALT="$m$"> of partitions
|
||||
of <IMG
|
||||
ALT="$m$"></SPAN> of partitions
|
||||
of <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="24" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img10.png"
|
||||
ALT="$W$"> increases [<A
|
||||
ALT="$W$"></SPAN> increases [<A
|
||||
HREF="node30.html#dd1_94">7</A>,<A
|
||||
HREF="node30.html#dd2_96">15</A>]. To reduce the dependency
|
||||
HREF="node30.html#dd2_96">20</A>]. To reduce the dependency
|
||||
of the number of iterations on the degree of parallelism we may
|
||||
introduce a global coupling among the overlapping partitions by defining
|
||||
a coarse-space approximation <IMG
|
||||
a coarse-space approximation <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img43.png"
|
||||
ALT="$A_C$"> of the matrix <IMG
|
||||
ALT="$A_C$"></SPAN> of the matrix <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img2.png"
|
||||
ALT="$A$">.
|
||||
In a pure algebraic setting, <IMG
|
||||
ALT="$A$"></SPAN>.
|
||||
In a pure algebraic setting, <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img43.png"
|
||||
ALT="$A_C$"> is usually built with
|
||||
a Galerkin approach. Given a set <IMG
|
||||
ALT="$A_C$"></SPAN> is usually built with
|
||||
a Galerkin approach. Given a set <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="32" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img44.png"
|
||||
ALT="$W_C$"> of <I>coarse vertices</I>,
|
||||
with size <IMG
|
||||
ALT="$W_C$"></SPAN> of <SPAN CLASS="textit">coarse vertices</SPAN>,
|
||||
with size <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="26" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img45.png"
|
||||
ALT="$n_C$">, and a suitable restriction operator
|
||||
ALT="$n_C$"></SPAN>, and a suitable restriction operator
|
||||
<!-- MATH
|
||||
$R_C \in \Re^{n_C \times n}$
|
||||
-->
|
||||
<IMG
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="101" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img46.png"
|
||||
ALT="$R_C \in \Re^{n_C \times n}$">, <IMG
|
||||
ALT="$R_C \in \Re^{n_C \times n}$"></SPAN>, <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img43.png"
|
||||
ALT="$A_C$"> is defined as
|
||||
ALT="$A_C$"></SPAN> is defined as
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<DIV ALIGN="CENTER" CLASS="mathdisplay">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
A_C=R_C A R_C^T
|
||||
@@ -441,12 +442,12 @@ A_C=R_C A R_C^T
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
and the coarse-level correction matrix to be combined with a generic
|
||||
one-level AS preconditioner <IMG
|
||||
one-level AS preconditioner <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img48.png"
|
||||
ALT="$M_{1L}$"> is obtained as
|
||||
ALT="$M_{1L}$"></SPAN> is obtained as
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<DIV ALIGN="CENTER" CLASS="mathdisplay">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
M_{C}^{-1}= R_C^T A_C^{-1} R_C,
|
||||
@@ -462,36 +463,36 @@ M_{C}^{-1}= R_C^T A_C^{-1} R_C,
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
where <IMG
|
||||
where <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img43.png"
|
||||
ALT="$A_C$"> is assumed to be nonsingular. The application of <IMG
|
||||
ALT="$A_C$"></SPAN> is assumed to be nonsingular. The application of <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="42" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img50.png"
|
||||
ALT="$M_{C}^{-1}$">
|
||||
to a vector <IMG
|
||||
ALT="$M_{C}^{-1}$"></SPAN>
|
||||
to a vector <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img21.png"
|
||||
ALT="$v$"> corresponds to a restriction, a solution and
|
||||
a prolongation step; the solution step, involving the matrix <IMG
|
||||
ALT="$v$"></SPAN> corresponds to a restriction, a solution and
|
||||
a prolongation step; the solution step, involving the matrix <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img43.png"
|
||||
ALT="$A_C$">,
|
||||
ALT="$A_C$"></SPAN>,
|
||||
may be carried out also approximately.
|
||||
|
||||
<P>
|
||||
The combination of <IMG
|
||||
The combination of <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="32" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img51.png"
|
||||
ALT="$M_{C}$"> and <IMG
|
||||
ALT="$M_{C}$"></SPAN> and <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img48.png"
|
||||
ALT="$M_{1L}$"> may be
|
||||
ALT="$M_{1L}$"></SPAN> may be
|
||||
performed in either an additive or a multiplicative framework.
|
||||
In the former case, the <I>two-level additive</I> Schwarz preconditioner
|
||||
In the former case, the <SPAN CLASS="textit">two-level additive</SPAN> Schwarz preconditioner
|
||||
is obtained:
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<DIV ALIGN="CENTER" CLASS="mathdisplay">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
M_{2LA}^{-1} = M_{C}^{-1} + M_{1L}^{-1}.
|
||||
@@ -507,38 +508,38 @@ M_{2LA}^{-1} = M_{C}^{-1} + M_{1L}^{-1}.
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
Applying <IMG
|
||||
Applying <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="59" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img53.png"
|
||||
ALT="$M_{2L-A}^{-1}$"> to a vector <IMG
|
||||
ALT="$M_{2L-A}^{-1}$"></SPAN> to a vector <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img21.png"
|
||||
ALT="$v$"> within a Krylov solver
|
||||
corresponds to applying <IMG
|
||||
ALT="$v$"></SPAN> within a Krylov solver
|
||||
corresponds to applying <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="42" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img50.png"
|
||||
ALT="$M_{C}^{-1}$">
|
||||
and <IMG
|
||||
ALT="$M_{C}^{-1}$"></SPAN>
|
||||
and <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="42" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img54.png"
|
||||
ALT="$M_{1L}^{-1}$"> to <IMG
|
||||
ALT="$M_{1L}^{-1}$"></SPAN> to <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img21.png"
|
||||
ALT="$v$"> independently and then summing up
|
||||
ALT="$v$"></SPAN> independently and then summing up
|
||||
the results.
|
||||
|
||||
<P>
|
||||
In the multiplicative case, the combination can be
|
||||
performed by first applying the smoother <IMG
|
||||
performed by first applying the smoother <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="42" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img54.png"
|
||||
ALT="$M_{1L}^{-1}$"> and then
|
||||
the coarse-level correction operator <IMG
|
||||
ALT="$M_{1L}^{-1}$"></SPAN> and then
|
||||
the coarse-level correction operator <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="42" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img50.png"
|
||||
ALT="$M_{C}^{-1}$">:
|
||||
ALT="$M_{C}^{-1}$"></SPAN>:
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<DIV ALIGN="CENTER" CLASS="mathdisplay">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
\begin{array}{l}
|
||||
@@ -559,10 +560,10 @@ z = w + M_{C}^{-1} (v-Aw);
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
this corresponds to the following <I>two-level hybrid pre-smoothed</I>
|
||||
this corresponds to the following <SPAN CLASS="textit">two-level hybrid pre-smoothed</SPAN>
|
||||
Schwarz preconditioner:
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<DIV ALIGN="CENTER" CLASS="mathdisplay">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
M_{2LH-PRE}^{-1} = M_{C}^{-1} + \left( I - M_{C}^{-1}A \right) M_{1L}^{-1}.
|
||||
@@ -581,7 +582,7 @@ M_{2LH-PRE}^{-1} = M_{C}^{-1} + \left( I - M_{C}^{-1}A \right) M_{1L}^{-1}.
|
||||
On the other hand, by applying the smoother after the coarse-level correction,
|
||||
i.e. by computing
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<DIV ALIGN="CENTER" CLASS="mathdisplay">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
\begin{array}{l}
|
||||
@@ -602,10 +603,10 @@ z = w + M_{1L}^{-1} (v-Aw) ,
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
the <I>two-level hybrid post-smoothed</I>
|
||||
the <SPAN CLASS="textit">two-level hybrid post-smoothed</SPAN>
|
||||
Schwarz preconditioner is obtained:
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<DIV ALIGN="CENTER" CLASS="mathdisplay">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
M_{2LH-POST}^{-1} = M_{1L}^{-1} + \left( I - M_{1L}^{-1}A \right) M_{C}^{-1}.
|
||||
@@ -623,16 +624,16 @@ M_{2LH-POST}^{-1} = M_{1L}^{-1} + \left( I - M_{1L}^{-1}A \right) M_{C}^{-1}.
|
||||
<P></P>
|
||||
One more variant of two-level hybrid preconditioner is obtained by applying
|
||||
the smoother before and after the coarse-level correction. In this case, the
|
||||
preconditioner is symmetric if <IMG
|
||||
preconditioner is symmetric if <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img2.png"
|
||||
ALT="$A$">, <IMG
|
||||
ALT="$A$"></SPAN>, <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img48.png"
|
||||
ALT="$M_{1L}$"> and <IMG
|
||||
ALT="$M_{1L}$"></SPAN> and <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="32" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img51.png"
|
||||
ALT="$M_{C}$"> are symmetric.
|
||||
ALT="$M_{C}$"></SPAN> are symmetric.
|
||||
|
||||
<P>
|
||||
As previously noted, on parallel computers the number of submatrices usually matches
|
||||
@@ -643,52 +644,51 @@ On the other hand, the use of few processors often leads to local sumatrices tha
|
||||
are too expensive to be processed on single processors, because of memory and/or
|
||||
computing requirements. Therefore, it seems natural to use a recursive approach,
|
||||
in which the coarse-level correction is re-applied starting from the current
|
||||
coarse-level system. The corresponding preconditioners, called <I>multi-level</I>
|
||||
coarse-level system. The corresponding preconditioners, called <SPAN CLASS="textit">multi-level</SPAN>
|
||||
preconditioners, can significantly reduce the computational cost of preconditioning
|
||||
with respect to the two-level case (see [<A
|
||||
HREF="node30.html#dd2_96">15</A>, Chapter 3]).
|
||||
HREF="node30.html#dd2_96">20</A>, Chapter 3]).
|
||||
Additive and hybrid multilevel preconditioners
|
||||
are obtained as direct extensions of the two-level counterparts.
|
||||
For a detailed descrition of them, the reader is
|
||||
referred to [<A
|
||||
HREF="node30.html#dd2_96">15</A>, Chapter 3].
|
||||
HREF="node30.html#dd2_96">20</A>, Chapter 3].
|
||||
The algorithm for the application of a multi-level hybrid
|
||||
post-smoothed preconditioner <IMG
|
||||
post-smoothed preconditioner <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="23" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img59.png"
|
||||
ALT="$M$"> to a vector <IMG
|
||||
ALT="$M$"></SPAN> to a vector <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img21.png"
|
||||
ALT="$v$">, i.e. for the
|
||||
computation of <IMG
|
||||
ALT="$v$"></SPAN>, i.e. for the
|
||||
computation of <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="87" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img60.png"
|
||||
ALT="$w=M^{-1}v$">, is reported, for
|
||||
example, in Figure <A HREF="#fig:mlhpost_alg"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A>. Here the number of levels
|
||||
is denoted by <IMG
|
||||
ALT="$w=M^{-1}v$"></SPAN>, is reported, for
|
||||
example, in Figure <A HREF="#fig:mlhpost_alg">1</A>. Here the number of levels
|
||||
is denoted by <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="37" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img61.png"
|
||||
ALT="$nlev$"> and the levels are numbered in increasing order starting
|
||||
ALT="$nlev$"></SPAN> and the levels are numbered in increasing order starting
|
||||
from the finest one, i.e. the finest level is level 1; the coarse matrix
|
||||
and the corresponding basic preconditioner at each level <IMG
|
||||
and the corresponding basic preconditioner at each level <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="10" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img62.png"
|
||||
ALT="$l$"> are denoted by <IMG
|
||||
ALT="$l$"></SPAN> are denoted by <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="22" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img63.png"
|
||||
ALT="$A_l$"> and
|
||||
<IMG
|
||||
ALT="$A_l$"></SPAN> and
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="27" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img64.png"
|
||||
ALT="$M_l$">, respectively, with <IMG
|
||||
ALT="$M_l$"></SPAN>, respectively, with <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="61" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img65.png"
|
||||
ALT="$A_1=A$">.
|
||||
ALT="$A_1=A$"></SPAN>.
|
||||
|
||||
<DIV ALIGN="CENTER"><A NAME="fig:mlhpost_alg"></A><A NAME="557"></A>
|
||||
<DIV ALIGN="CENTER"><A NAME="fig:mlhpost_alg"></A><A NAME="508"></A>
|
||||
<TABLE>
|
||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure:</STRONG>
|
||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1:</STRONG>
|
||||
Application of the multi-level hybrid post-smoothed preconditioner.</CAPTION>
|
||||
<TR><TD>
|
||||
<DIV ALIGN="CENTER">
|
||||
@@ -728,7 +728,7 @@ $w = y_1$;
|
||||
\begin{tabbing}
|
||||
\quad \=\quad...
|
||||
...= y_l+r_l$\\
|
||||
\textbf{endfor} \ [1mm]
|
||||
\textbf{endfor} [1mm]
|
||||
$w = y_1$;
|
||||
\end{tabbing}}
|
||||
\end{minipage}}">
|
||||
@@ -738,37 +738,38 @@ $w = y_1$;
|
||||
</DIV>
|
||||
|
||||
<P>
|
||||
<HR>
|
||||
|
||||
<DIV CLASS="navigation"><HR>
|
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|
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<A NAME="tex2html201"
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<B> Up:</B> <A NAME="tex2html198"
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<B> Up:</B> <A NAME="tex2html197"
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<B> Previous:</B> <A NAME="tex2html192"
|
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<B> <A NAME="tex2html200"
|
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HREF="node1.html">Contents</A></B>
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<B> <A NAME="tex2html199"
|
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HREF="node2.html">Contents</A></B> </DIV>
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<ADDRESS>
|
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Salvatore Filippone
|
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2008-07-22
|
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2008-07-23
|
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</ADDRESS>
|
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|
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|
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|
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+92
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@@ -1,4 +1,4 @@
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
|
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
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|
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|
||||
original version by: Nikos Drakos, CBLU, University of Leeds
|
||||
@@ -24,34 +24,35 @@ original version by: Nikos Drakos, CBLU, University of Leeds
|
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</HEAD>
|
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|
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<BODY >
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<A NAME="tex2html211"
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<DIV CLASS="navigation"><!--Navigation Panel-->
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<B> Next:</B> <A NAME="tex2html212"
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HREF="node1.html">Contents</A></B>
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<BR>
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<B> <A NAME="tex2html209"
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|
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|
||||
<H2><A NAME="SECTION00062000000000000000"></A><A NAME="sec:aggregation"></A>
|
||||
@@ -60,29 +61,29 @@ Smoothed Aggregation
|
||||
</H2>
|
||||
|
||||
<P>
|
||||
In order to define the restriction operator <IMG
|
||||
In order to define the restriction operator <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img67.png"
|
||||
ALT="$R_C$">, which is used to compute
|
||||
the coarse-level matrix <IMG
|
||||
ALT="$R_C$"></SPAN>, which is used to compute
|
||||
the coarse-level matrix <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img43.png"
|
||||
ALT="$A_C$">, MLD2P4 uses the <I>smoothed aggregation</I>
|
||||
ALT="$A_C$"></SPAN>, MLD2P4 uses the <SPAN CLASS="textit">smoothed aggregation</SPAN>
|
||||
algorithm described in [<A
|
||||
HREF="node30.html#BREZINA_VANEK">1</A>,<A
|
||||
HREF="node30.html#VANEK_MANDEL_BREZINA">19</A>].
|
||||
HREF="node30.html#VANEK_MANDEL_BREZINA">24</A>].
|
||||
The basic idea of this algorithm is to build a coarse set of vertices
|
||||
<IMG
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="32" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img44.png"
|
||||
ALT="$W_C$"> by suitably grouping the vertices of <IMG
|
||||
ALT="$W_C$"></SPAN> by suitably grouping the vertices of <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="24" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img10.png"
|
||||
ALT="$W$"> into disjoint subsets
|
||||
(aggregates), and to define the coarse-to-fine space transfer operator <IMG
|
||||
ALT="$W$"></SPAN> into disjoint subsets
|
||||
(aggregates), and to define the coarse-to-fine space transfer operator <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="29" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img68.png"
|
||||
ALT="$R_C^T$"> by
|
||||
ALT="$R_C^T$"></SPAN> by
|
||||
applying a suitable smoother to a simple piecewise constant
|
||||
prolongation operator, to improve the quality of the coarse-space correction.
|
||||
|
||||
@@ -90,29 +91,29 @@ prolongation operator, to improve the quality of the coarse-space correction.
|
||||
Three main steps can be identified in the smoothed aggregation procedure:
|
||||
|
||||
<OL>
|
||||
<LI>coarsening of the vertex set <IMG
|
||||
<LI>coarsening of the vertex set <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="24" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img10.png"
|
||||
ALT="$W$">, to obtain <IMG
|
||||
ALT="$W$"></SPAN>, to obtain <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="32" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img44.png"
|
||||
ALT="$W_C$">;
|
||||
ALT="$W_C$"></SPAN>;
|
||||
</LI>
|
||||
<LI>construction of the prolongator <IMG
|
||||
<LI>construction of the prolongator <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="29" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img68.png"
|
||||
ALT="$R_C^T$">;
|
||||
ALT="$R_C^T$"></SPAN>;
|
||||
</LI>
|
||||
<LI>application of <IMG
|
||||
<LI>application of <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img67.png"
|
||||
ALT="$R_C$"> and <IMG
|
||||
ALT="$R_C$"></SPAN> and <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="29" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img68.png"
|
||||
ALT="$R_C^T$"> to build <IMG
|
||||
ALT="$R_C^T$"></SPAN> to build <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img43.png"
|
||||
ALT="$A_C$">.
|
||||
ALT="$A_C$"></SPAN>.
|
||||
</LI>
|
||||
</OL>
|
||||
|
||||
@@ -120,20 +121,20 @@ Three main steps can be identified in the smoothed aggregation procedure:
|
||||
To perform the coarsening step, we have implemented the aggregation algorithm sketched
|
||||
in [<A
|
||||
HREF="node30.html#apnum_07">4</A>]. According to [<A
|
||||
HREF="node30.html#VANEK_MANDEL_BREZINA">19</A>], a modification of
|
||||
HREF="node30.html#VANEK_MANDEL_BREZINA">24</A>], a modification of
|
||||
this algorithm has been actually considered,
|
||||
in which each aggregate <IMG
|
||||
in which each aggregate <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="26" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img69.png"
|
||||
ALT="$N_r$"> is made of vertices of <IMG
|
||||
ALT="$N_r$"></SPAN> is made of vertices of <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="24" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img10.png"
|
||||
ALT="$W$"> that are <I>strongly coupled</I>
|
||||
to a certain root vertex <IMG
|
||||
ALT="$W$"></SPAN> that are <SPAN CLASS="textit">strongly coupled</SPAN>
|
||||
to a certain root vertex <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="53" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img70.png"
|
||||
ALT="$r \in W$">, i.e. <BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
ALT="$r \in W$"></SPAN>, i.e. <BR><P></P>
|
||||
<DIV ALIGN="CENTER" CLASS="mathdisplay">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
N_r = \left\{s \in W: |a_{rs}| > \theta \sqrt{|a_{rr}a_{ss}|} \right\}
|
||||
@@ -153,47 +154,47 @@ N_r = \left\{s \in W: |a_{rs}| > \theta \sqrt{|a_{rr}a_{ss}|} \right\}
|
||||
for a given <!-- MATH
|
||||
$\theta \in [0,1]$
|
||||
-->
|
||||
<IMG
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="69" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img72.png"
|
||||
ALT="$\theta \in [0,1]$">.
|
||||
Since this algorithm has a sequential nature, a <I>decoupled</I> version of
|
||||
it has been chosen, where each processor <IMG
|
||||
ALT="$\theta \in [0,1]$"></SPAN>.
|
||||
Since this algorithm has a sequential nature, a <SPAN CLASS="textit">decoupled</SPAN> version of
|
||||
it has been chosen, where each processor <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="10" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img73.png"
|
||||
ALT="$i$"> independently applies the algorithm to
|
||||
the set of vertices <IMG
|
||||
ALT="$i$"></SPAN> independently applies the algorithm to
|
||||
the set of vertices <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="31" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img74.png"
|
||||
ALT="$W_i^0$"> assigned to it in the initial data distribution. This
|
||||
ALT="$W_i^0$"></SPAN> assigned to it in the initial data distribution. This
|
||||
version is embarrassingly parallel, since it does not require any data communication.
|
||||
On the other hand, it may produce non-uniform aggregates near boundary vertices,
|
||||
i.e. near vertices adjacent to vertices in other processors, and is strongly
|
||||
dependent on the number of processors and on the initial partitioning of the matrix <IMG
|
||||
dependent on the number of processors and on the initial partitioning of the matrix <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img2.png"
|
||||
ALT="$A$">.
|
||||
ALT="$A$"></SPAN>.
|
||||
Nevertheless, this algorithm has been chosen for the implementation in MLD2P4,
|
||||
since it has been shown to produce good results in practice
|
||||
[<A
|
||||
HREF="node30.html#aaecc_07">3</A>,<A
|
||||
HREF="node30.html#apnum_07">4</A>,<A
|
||||
HREF="node30.html#TUMINARO_TONG">18</A>].
|
||||
HREF="node30.html#TUMINARO_TONG">23</A>].
|
||||
|
||||
<P>
|
||||
The prolongator <IMG
|
||||
The prolongator <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="75" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img75.png"
|
||||
ALT="$P_C=R_C^T$"> is built starting from a <I>tentative prolongator</I>
|
||||
ALT="$P_C=R_C^T$"></SPAN> is built starting from a <SPAN CLASS="textit">tentative prolongator</SPAN>
|
||||
<!-- MATH
|
||||
$P \in \Re^{n \times n_C}$
|
||||
-->
|
||||
<IMG
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="90" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img76.png"
|
||||
ALT="$P \in \Re^{n \times n_C}$">, defined as
|
||||
ALT="$P \in \Re^{n \times n_C}$"></SPAN>, defined as
|
||||
<BR>
|
||||
<DIV ALIGN="RIGHT">
|
||||
<DIV ALIGN="RIGHT" CLASS="mathdisplay">
|
||||
|
||||
<!-- MATH
|
||||
\begin{equation}
|
||||
@@ -204,6 +205,7 @@ P=(p_{ij}), \quad p_{ij}=
|
||||
\end{array} \right. .
|
||||
\end{equation}
|
||||
-->
|
||||
<A NAME="eq:tent_prol"></A>
|
||||
<TABLE WIDTH="100%" ALIGN="CENTER">
|
||||
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:tent_prol"></A><IMG
|
||||
WIDTH="290" HEIGHT="52" BORDER="0"
|
||||
@@ -216,32 +218,33 @@ P=(p_{ij}), \quad p_{ij}=
|
||||
0 & \quad \mbox{otherwise}
|
||||
\end{array} \right. .
|
||||
\end{displaymath}"></TD>
|
||||
<TD WIDTH=10 ALIGN="RIGHT">
|
||||
(2)</TD></TR>
|
||||
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
|
||||
(<SPAN CLASS="arabic">2</SPAN>)</TD></TR>
|
||||
</TABLE>
|
||||
<BR CLEAR="ALL"></DIV><P></P>
|
||||
<IMG
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="27" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img78.png"
|
||||
ALT="$P_C$"> is obtained by
|
||||
applying to <IMG
|
||||
ALT="$P_C$"></SPAN> is obtained by
|
||||
applying to <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img79.png"
|
||||
ALT="$P$"> a smoother <!-- MATH
|
||||
ALT="$P$"></SPAN> a smoother <!-- MATH
|
||||
$S \in \Re^{n \times n}$
|
||||
-->
|
||||
<IMG
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="78" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img80.png"
|
||||
ALT="$S \in \Re^{n \times n}$">:
|
||||
ALT="$S \in \Re^{n \times n}$"></SPAN>:
|
||||
<BR>
|
||||
<DIV ALIGN="RIGHT">
|
||||
<DIV ALIGN="RIGHT" CLASS="mathdisplay">
|
||||
|
||||
<!-- MATH
|
||||
\begin{equation}
|
||||
P_C = S P,
|
||||
\end{equation}
|
||||
-->
|
||||
<A NAME="eq:smoothed_prol"></A>
|
||||
<TABLE WIDTH="100%" ALIGN="CENTER">
|
||||
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:smoothed_prol"></A><IMG
|
||||
WIDTH="73" HEIGHT="30" BORDER="0"
|
||||
@@ -249,27 +252,28 @@ P_C = S P,
|
||||
ALT="\begin{displaymath}
|
||||
P_C = S P,
|
||||
\end{displaymath}"></TD>
|
||||
<TD WIDTH=10 ALIGN="RIGHT">
|
||||
(3)</TD></TR>
|
||||
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
|
||||
(<SPAN CLASS="arabic">3</SPAN>)</TD></TR>
|
||||
</TABLE>
|
||||
<BR CLEAR="ALL"></DIV><P></P>
|
||||
in order to remove oscillatory components from the range of the prolongator
|
||||
and hence to improve the convergence properties of the multi-level
|
||||
Schwarz method [<A
|
||||
HREF="node30.html#BREZINA_VANEK">1</A>,<A
|
||||
HREF="node30.html#StubenGMD69_99">17</A>].
|
||||
A simple choice for <IMG
|
||||
HREF="node30.html#StubenGMD69_99">22</A>].
|
||||
A simple choice for <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="16" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img82.png"
|
||||
ALT="$S$"> is the damped Jacobi smoother:
|
||||
ALT="$S$"></SPAN> is the damped Jacobi smoother:
|
||||
<BR>
|
||||
<DIV ALIGN="RIGHT">
|
||||
<DIV ALIGN="RIGHT" CLASS="mathdisplay">
|
||||
|
||||
<!-- MATH
|
||||
\begin{equation}
|
||||
S = I - \omega D^{-1} A ,
|
||||
\end{equation}
|
||||
-->
|
||||
<A NAME="eq:jac_smoother"></A>
|
||||
<TABLE WIDTH="100%" ALIGN="CENTER">
|
||||
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:jac_smoother"></A><IMG
|
||||
WIDTH="126" HEIGHT="30" BORDER="0"
|
||||
@@ -277,52 +281,53 @@ S = I - \omega D^{-1} A ,
|
||||
ALT="\begin{displaymath}
|
||||
S = I - \omega D^{-1} A ,
|
||||
\end{displaymath}"></TD>
|
||||
<TD WIDTH=10 ALIGN="RIGHT">
|
||||
(4)</TD></TR>
|
||||
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
|
||||
(<SPAN CLASS="arabic">4</SPAN>)</TD></TR>
|
||||
</TABLE>
|
||||
<BR CLEAR="ALL"></DIV><P></P>
|
||||
where the value of <IMG
|
||||
where the value of <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img84.png"
|
||||
ALT="$\omega$"> can be chosen
|
||||
using some estimate of the spectral radius of <IMG
|
||||
ALT="$\omega$"></SPAN> can be chosen
|
||||
using some estimate of the spectral radius of <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="50" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img85.png"
|
||||
ALT="$D^{-1}A$"> [<A
|
||||
ALT="$D^{-1}A$"></SPAN> [<A
|
||||
HREF="node30.html#BREZINA_VANEK">1</A>].
|
||||
|
||||
<P>
|
||||
<HR>
|
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|
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<DIV CLASS="navigation"><HR>
|
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<!--Navigation Panel-->
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<B> Up:</B> <A NAME="tex2html208"
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<B> Up:</B> <A NAME="tex2html207"
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HREF="node10.html">Multi-level Domain Decomposition Background</A>
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<B> Previous:</B> <A NAME="tex2html203"
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<B> <A NAME="tex2html210"
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HREF="node1.html">Contents</A></B>
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<B> <A NAME="tex2html209"
|
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HREF="node2.html">Contents</A></B> </DIV>
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<!--End of Navigation Panel-->
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<ADDRESS>
|
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Salvatore Filippone
|
||||
2008-07-22
|
||||
2008-07-23
|
||||
</ADDRESS>
|
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</BODY>
|
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|
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+56
-62
@@ -1,4 +1,4 @@
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
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|
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original version by: Nikos Drakos, CBLU, University of Leeds
|
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@@ -25,34 +25,35 @@ original version by: Nikos Drakos, CBLU, University of Leeds
|
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|
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<BODY >
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<B> Previous:</B> <A NAME="tex2html213"
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<B> <A NAME="tex2html222"
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HREF="node1.html">Contents</A></B>
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<BR>
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<B> <A NAME="tex2html221"
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<!--End of Navigation Panel-->
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|
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<H1><A NAME="SECTION00070000000000000000"></A><A NAME="sec:started"></A>
|
||||
@@ -63,58 +64,51 @@ Getting Started
|
||||
<P>
|
||||
We describe the basics for building and applying MLD2P4 one-level and multi-level
|
||||
Schwarz preconditioners with the Krylov solvers included in PSBLAS [<A
|
||||
HREF="node30.html#PSBLASGUIDE">11</A>].
|
||||
HREF="node30.html#PSBLASGUIDE">14</A>].
|
||||
The following steps are required:
|
||||
|
||||
<OL>
|
||||
<LI><I>Declare the preconditioner data structure</I>. It is a derived data type,
|
||||
<code>mld_</code><I>x</I><code>prec_type</code>, where <I>x</I> may be <code>s</code>, <code>d</code>, <code>c</code>
|
||||
<LI><SPAN CLASS="textit">Declare the preconditioner data structure</SPAN>. It is a derived data type,
|
||||
<code>mld_</code><SPAN CLASS="textit">x</SPAN><code>prec_</code> <code>type</code>, where <SPAN CLASS="textit">x</SPAN> may be <code>s</code>, <code>d</code>, <code>c</code>
|
||||
or <code>z</code>, according to the basic data type of the sparse matrix
|
||||
(<code>s</code> = real single precision; <code>d</code> = real double precision;
|
||||
<code>c</code> = complex single precision; <code>z</code> = complex double precision).
|
||||
This data structure is accessed by the user only through the MLD2P4 routines,
|
||||
following an object-oriented approach.
|
||||
</LI>
|
||||
<LI><I>Allocate and initialize the preconditioner data structure, according to
|
||||
a preconditioner type chosen by the user</I>. This is performed by the routine
|
||||
<LI><SPAN CLASS="textit">Allocate and initialize the preconditioner data structure, according to
|
||||
a preconditioner type chosen by the user</SPAN>. This is performed by the routine
|
||||
<code>mld_precinit</code>, which also sets defaults for each preconditioner
|
||||
type selected by the user. The defaults associated to each preconditioner
|
||||
type are given in Table <A HREF="#tab:precinit"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A>, where the strings used by
|
||||
type are given in Table <A HREF="#tab:precinit">1</A>, where the strings used by
|
||||
<code>mld_precinit</code> to identify the preconditioner types are also given.
|
||||
Note that these strings are valid also if uppercase letters are substituted by
|
||||
corresponding lowercase ones.
|
||||
</LI>
|
||||
<LI><I>Modify the selected preconditioner type, by properly setting
|
||||
preconditioner parameters.</I> This is performed by the routine <code>mld_precset</code>.
|
||||
<LI><SPAN CLASS="textit">Modify the selected preconditioner type, by properly setting
|
||||
preconditioner parameters.</SPAN> This is performed by the routine <code>mld_precset</code>.
|
||||
This routine must be called only if the user wants to modify the default values
|
||||
of the parameters associated to the selected preconditioner type, to obtain a variant
|
||||
of the preconditioner. Examples of use of <code>mld_precset</code> are given in
|
||||
Section <A HREF="node14.html#sec:examples"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A>; a complete list of all the
|
||||
Section <A HREF="node14.html#sec:examples">5.1</A>; a complete list of all the
|
||||
preconditioner parameters and their allowed and default values is provided in
|
||||
Section <A HREF="node15.html#sec:userinterface"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A>, Tables <A HREF="#tab:p_type"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A>-<A HREF="#tab:p_coarse"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A>.
|
||||
Section <A HREF="node15.html#sec:userinterface">6</A>, Tables <A HREF="#tab:p_type">2</A>-<A HREF="#tab:p_coarse">5</A>.
|
||||
</LI>
|
||||
<LI><I>Build the preconditioner for a given matrix.</I> This is performed by
|
||||
<LI><SPAN CLASS="textit">Build the preconditioner for a given matrix.</SPAN> This is performed by
|
||||
the routine <code>mld_precbld</code>.
|
||||
</LI>
|
||||
<LI><I>Apply the preconditioner at each iteration of a Krylov solver.</I>
|
||||
<LI><SPAN CLASS="textit">Apply the preconditioner at each iteration of a Krylov solver.</SPAN>
|
||||
This is performed by the routine <code>mld_precaply</code>. When using the PSBLAS Krylov solvers,
|
||||
this step is completely transparent to the user, since <code>mld_precaply</code> is called
|
||||
by the PSBLAS routine implementing the Krylov solver (<code>psb_krylov</code>).
|
||||
</LI>
|
||||
<LI><I>Free the preconditioner data structure</I>. This is performed by
|
||||
the routine <code>mld_precfree</code>. This step is complementary to step 1 and should
|
||||
<LI><SPAN CLASS="textit">Free the preconditioner data structure</SPAN>. This is performed by
|
||||
the routine <code>mld_</code> <code>precfree</code>. This step is complementary to step 1 and should
|
||||
be performed when the preconditioner is no more used.
|
||||
</LI>
|
||||
</OL>
|
||||
A detailed description of the above routines is given in Section <A HREF="node15.html#sec:userinterface"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A>.
|
||||
Examples showing the basic use of MLD2P4 are reported in Section <A HREF="node14.html#sec:examples"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A>.
|
||||
A detailed description of the above routines is given in Section <A HREF="node15.html#sec:userinterface">6</A>.
|
||||
Examples showing the basic use of MLD2P4 are reported in Section <A HREF="node14.html#sec:examples">5.1</A>.
|
||||
|
||||
<P>
|
||||
Note that the Fortran 95 module <code>mld_prec_mod</code>, containing the definition of the
|
||||
@@ -122,12 +116,11 @@ preconditioner data type and the interfaces to the routines of MLD2P4,
|
||||
must be used in any program calling such routines.
|
||||
The modules <code>psb_base_mod</code>, for the sparse matrix and communication descriptor
|
||||
data types, and <code>psb_krylov_mod</code>, for interfacing with the
|
||||
Krylov solvers, must be also used (see Section <A HREF="node14.html#sec:examples"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A>).
|
||||
Krylov solvers, must be also used (see Section <A HREF="node14.html#sec:examples">5.1</A>).
|
||||
|
||||
<P>
|
||||
|
||||
<BR><B>Remark 1.</B> The coarsest-level solver used by the default two-level
|
||||
<BR><SPAN CLASS="textbf">Remark 1.</SPAN> The coarsest-level solver used by the default two-level
|
||||
preconditioner has been chosen by taking into account that, on parallel
|
||||
machines, it often leads to the smallest execution time when applied to
|
||||
linear systems coming from finite-difference discretizations of basic
|
||||
@@ -138,13 +131,12 @@ preconditioners [<A
|
||||
not necessarily correspond to the smallest number of iterations of the
|
||||
preconditioned Krylov method, which is usually obtained by applying
|
||||
a direct solver to the coarsest-level system, e.g. based on the LU
|
||||
factorization (see Section <A HREF="node15.html#sec:userinterface"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A>
|
||||
factorization (see Section <A HREF="node15.html#sec:userinterface">6</A>
|
||||
for the coarsest-level solvers available in MLD2P4).
|
||||
|
||||
<P>
|
||||
|
||||
<BR><B>Remark 2.</B> The include path for MLD2P4 must override
|
||||
<BR><SPAN CLASS="textbf">Remark 2.</SPAN> The include path for MLD2P4 must override
|
||||
those for PSBLAS, e.g. the latter must come first in the sequence
|
||||
passed to the compiler, as the MLD2P4 version of the Krylov solver
|
||||
interfaces must override that of PSBLAS. This will change in the future
|
||||
@@ -153,9 +145,9 @@ compilers.
|
||||
|
||||
<P>
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER"><A NAME="972"></A>
|
||||
<DIV ALIGN="CENTER"><A NAME="923"></A>
|
||||
<TABLE>
|
||||
<CAPTION><STRONG>Table:</STRONG>
|
||||
<CAPTION><STRONG>Table 1:</STRONG>
|
||||
Preconditioner types, corresponding strings and default choices.
|
||||
</CAPTION>
|
||||
<TR><TD>
|
||||
@@ -192,16 +184,17 @@ Preconditioner types, corresponding strings and default choices.
|
||||
Post-smoother: RAS with overlap 1 and ILU(0)
|
||||
on the local blocks.
|
||||
Aggregation: smoothed aggregation with
|
||||
threshold <IMG
|
||||
threshold <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="45" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img86.png"
|
||||
ALT="$\theta = 0$">.
|
||||
ALT="$\theta = 0$"></SPAN>.
|
||||
Coarsest matrix: distributed among the processors.
|
||||
Coarsest-level solver:
|
||||
4 sweeps of the block-Jacobi solver,
|
||||
with LU factorization of the blocks
|
||||
with LU (or ILU) factorization of the blocks
|
||||
(UMFPACK for the double precision versions and
|
||||
SuperLU for the single precision ones)</TD>
|
||||
SuperLU for the single precision ones, if they have been
|
||||
installed; ILU(0), otherwise).</TD>
|
||||
</TR>
|
||||
</TABLE>
|
||||
</DIV>
|
||||
@@ -217,42 +210,43 @@ Preconditioner types, corresponding strings and default choices.
|
||||
<!--Table of Child-Links-->
|
||||
<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
|
||||
|
||||
<UL>
|
||||
<LI><A NAME="tex2html225"
|
||||
<UL CLASS="ChildLinks">
|
||||
<LI><A NAME="tex2html224"
|
||||
HREF="node14.html">Examples</A>
|
||||
</UL>
|
||||
<!--End of Table of Child-Links-->
|
||||
<HR>
|
||||
|
||||
<DIV CLASS="navigation"><HR>
|
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<!--Navigation Panel-->
|
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<A NAME="tex2html223"
|
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<A NAME="tex2html222"
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HREF="node14.html">
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2008-07-22
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2008-07-23
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<H2><A NAME="SECTION00071000000000000000"></A><A NAME="sec:examples"></A>
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@@ -60,11 +61,9 @@ Examples
|
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</H2>
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<P>
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The code reported in Figure <A HREF="#fig:ex_default"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
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SRC="file:/usr/share/latex2html/icons/crossref.png"></A> shows how to set and apply the default
|
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The code reported in Figure <A HREF="#fig:ex_default">2</A> shows how to set and apply the default
|
||||
multi-level preconditioner available in the real double precision version
|
||||
of MLD2P4 (see Table <A HREF="#tab:precinit"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
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SRC="file:/usr/share/latex2html/icons/crossref.png"></A>). This preconditioner is chosen
|
||||
of MLD2P4 (see Table <A HREF="node13.html#tab:precinit">1</A>). This preconditioner is chosen
|
||||
by simply specifying <code>'ML'</code> as second argument of <code>mld_precinit</code>
|
||||
(a call to <code>mld_precset</code> is not needed) and is applied with the BiCGSTAB
|
||||
solver provided by PSBLAS. As previously observed, the modules <code>psb_base_mod</code>,
|
||||
@@ -78,25 +77,23 @@ here for brevity; the statements concerning the deallocation of the PSBLAS
|
||||
data structure are neglected too.
|
||||
The complete code can be found in the example program file <code>mld_dexample_ml.f90</code>,
|
||||
in the directory <code>examples/fileread</code> of the MLD2P4 tree (see
|
||||
Section <A HREF="node5.html#sec:building"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A>).
|
||||
Section <A HREF="node9.html#sec:ex_and_test">3.4</A>).
|
||||
For details on the use of the PSBLAS routines, see the PSBLAS User's
|
||||
Guide [<A
|
||||
HREF="node30.html#PSBLASGUIDE">11</A>].
|
||||
HREF="node30.html#PSBLASGUIDE">14</A>].
|
||||
|
||||
<P>
|
||||
The setup and application of the default multi-level
|
||||
preconditioners for the real single precision and the complex, single and double
|
||||
precision, versions are obtained with straightforward modifications of the previous
|
||||
example (see Section <A HREF="node15.html#sec:userinterface"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A> for details). If these versions are installed,
|
||||
example (see Section <A HREF="node15.html#sec:userinterface">6</A> for details). If these versions are installed,
|
||||
the corresponding Fortran 95 codes are available in <code>examples/fileread/</code>.
|
||||
|
||||
<P>
|
||||
|
||||
<DIV ALIGN="CENTER"><A NAME="fig:ex_default"></A><A NAME="975"></A>
|
||||
<DIV ALIGN="CENTER"><A NAME="fig:ex_default"></A><A NAME="926"></A>
|
||||
<TABLE>
|
||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure:</STRONG>
|
||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 2:</STRONG>
|
||||
Setup and application of the default multi-level Schwarz preconditioner.
|
||||
</CAPTION>
|
||||
<TR><TD>
|
||||
@@ -165,8 +162,7 @@ Setup and application of the default multi-level Schwarz preconditioner.
|
||||
<P>
|
||||
Different versions of multi-level preconditioners can be obtained by changing
|
||||
the default values of the preconditioner parameters. The code reported in
|
||||
Figure <A HREF="#fig:ex_3lh"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A> shows how to set a three-level hybrid Schwarz
|
||||
Figure <A HREF="#fig:ex_3lh">3</A> shows how to set a three-level hybrid Schwarz
|
||||
preconditioner, which uses block Jacobi with ILU(0) on the
|
||||
local blocks as post-smoother, has a coarsest matrix replicated on the processors,
|
||||
and solves the coarsest-level system with the LU factorization from UMFPACK [<A
|
||||
@@ -178,28 +174,22 @@ with post-smoothing only) is not specified since it is the default
|
||||
set by <code>mld_precinit</code>.
|
||||
|
||||
<P>
|
||||
Figure <A HREF="#fig:ex_3la"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A> shows how to
|
||||
Figure <A HREF="#fig:ex_3la">4</A> shows how to
|
||||
set a three-level additive Schwarz preconditioner,
|
||||
which uses RAS, with overlap 1 and ILU(0) on the blocks,
|
||||
as pre- and post-smoother, and applies five block-Jacobi sweeps, with
|
||||
the UMFPACK LU factorization on the blocks, as distributed coarsest-level
|
||||
solver. Again, <code>mld_precset</code> is used only to set
|
||||
non-default values of the parameters (see Tables <A HREF="#tab:p_type"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A>-<A HREF="#tab:p_coarse"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A>).
|
||||
non-default values of the parameters (see Tables <A HREF="#tab:p_type">2</A>-<A HREF="#tab:p_coarse">5</A>).
|
||||
In both cases, the construction and the application of the preconditioner
|
||||
are carried out as for the default multi-level preconditioner.
|
||||
The code fragments shown in in Figures <A HREF="#fig:ex_3lh"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A>-<A HREF="#fig:ex_3la"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A> are
|
||||
The code fragments shown in in Figures <A HREF="#fig:ex_3lh">3</A>-<A HREF="#fig:ex_3la">4</A> are
|
||||
included in the example program file <code>mld_dexample_ml.f90</code> too.
|
||||
|
||||
<P>
|
||||
Finally, Figure <A HREF="#fig:ex_1l"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A> shows the setup of a one-level
|
||||
Finally, Figure <A HREF="#fig:ex_1l">5</A> shows the setup of a one-level
|
||||
additive Schwarz preconditioner, i.e. RAS with overlap 2. The corresponding
|
||||
example program is available in <code>mld_dexample_1lev.f90</code>.
|
||||
example program is available in <code>mld_dexample_</code> <code>1lev.f90</code>.
|
||||
|
||||
<P>
|
||||
For all the previous preconditioners, example programs where the sparse matrix and
|
||||
@@ -208,9 +198,19 @@ boundary conditions are also available in the directory <code>examples/pdegen</c
|
||||
|
||||
<P>
|
||||
|
||||
<DIV ALIGN="CENTER"><A NAME="fig:ex_3lh"></A><A NAME="977"></A>
|
||||
<BR><SPAN CLASS="textbf">Remark 3.</SPAN> Any PSBLAS-based program using the basic preconditioners
|
||||
implemented in PSBLAS 2.0, i.e. the diagonal and block-Jacobi ones,
|
||||
can use the diagonal and block-Jacobi preconditioners
|
||||
implemented in MLD2P4 without any change in the code.
|
||||
The PSBLAS-based program must be only recompiled
|
||||
and linked to the MLD2P4 library.
|
||||
|
||||
<BR>
|
||||
<P>
|
||||
|
||||
<DIV ALIGN="CENTER"><A NAME="fig:ex_3lh"></A><A NAME="928"></A>
|
||||
<TABLE>
|
||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure:</STRONG>
|
||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 3:</STRONG>
|
||||
Setup of a hybrid three-level Schwarz preconditioner.</CAPTION>
|
||||
<TR><TD>
|
||||
<DIV ALIGN="CENTER">
|
||||
@@ -240,9 +240,9 @@ Setup of a hybrid three-level Schwarz preconditioner.</CAPTION>
|
||||
|
||||
<P>
|
||||
|
||||
<DIV ALIGN="CENTER"><A NAME="fig:ex_3la"></A><A NAME="979"></A>
|
||||
<DIV ALIGN="CENTER"><A NAME="fig:ex_3la"></A><A NAME="930"></A>
|
||||
<TABLE>
|
||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure:</STRONG>
|
||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 4:</STRONG>
|
||||
Setup of an additive three-level Schwarz preconditioner.</CAPTION>
|
||||
<TR><TD>
|
||||
<DIV ALIGN="CENTER">
|
||||
@@ -272,9 +272,9 @@ Setup of an additive three-level Schwarz preconditioner.</CAPTION>
|
||||
|
||||
<P>
|
||||
|
||||
<DIV ALIGN="CENTER"><A NAME="fig:ex_1l"></A><A NAME="981"></A>
|
||||
<DIV ALIGN="CENTER"><A NAME="fig:ex_1l"></A><A NAME="932"></A>
|
||||
<TABLE>
|
||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure:</STRONG>
|
||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 5:</STRONG>
|
||||
Setup of a one-level Schwarz preconditioner.</CAPTION>
|
||||
<TR><TD>
|
||||
<DIV ALIGN="CENTER">
|
||||
@@ -297,45 +297,37 @@ Setup of a one-level Schwarz preconditioner.</CAPTION>
|
||||
|
||||
<P>
|
||||
|
||||
<BR><B>Remark 3.</B> Any PSBLAS-based program using the basic preconditioners
|
||||
implemented in PSBLAS 2.0, i.e. the diagonal and block-Jacobi ones,
|
||||
can use the diagonal and block-Jacobi preconditioners
|
||||
implemented in MLD2P4 without any change in the code.
|
||||
The PSBLAS-based program must be only recompiled
|
||||
and linked to the MLD2P4 library.
|
||||
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2008-07-22
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@@ -25,34 +25,35 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<H1><A NAME="SECTION00080000000000000000"></A><A NAME="sec:userinterface"></A>
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@@ -61,7 +62,7 @@ User Interface
|
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</H1>
|
||||
|
||||
<P>
|
||||
The basic user interface of MLD2P4 consists of six routines. The four routines <code>mld_precinit</code>,
|
||||
The basic user interface of MLD2P4 consists of six routines. The four routines <code>mld_</code> <code>precinit</code>,
|
||||
<code>mld_precset</code>, <code>mld_precbld</code> and <code>mld_precaply</code> encapsulate all the functionalities
|
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for the setup and the application of any one-level and multi-level
|
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preconditioner implemented in the package.
|
||||
@@ -76,29 +77,29 @@ i.e.
|
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|
||||
<UL>
|
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<LI>the sparse matrix data structure, containing the matrix to be
|
||||
preconditioned, must be of type <code>mld_</code><I>x</I><code>spmat_type</code>
|
||||
with <I>x</I> = <code>s</code> for real single precision, <I>x</I> = <code>d</code>
|
||||
for real double precision, <I>x</I> = <code>c</code> for complex single precision,
|
||||
<I>x</I> = <code>z</code> for complex double precision;
|
||||
preconditioned, must be of type <code>mld_</code><SPAN CLASS="textit">x</SPAN><code>spmat_type</code>
|
||||
with <SPAN CLASS="textit">x</SPAN> = <code>s</code> for real single precision, <SPAN CLASS="textit">x</SPAN> = <code>d</code>
|
||||
for real double precision, <SPAN CLASS="textit">x</SPAN> = <code>c</code> for complex single precision,
|
||||
<SPAN CLASS="textit">x</SPAN> = <code>z</code> for complex double precision;
|
||||
</LI>
|
||||
<LI>the preconditioner data structure must be of type
|
||||
<code>mld_</code><I>x</I><code>prec_type</code>, with <I>x</I> =
|
||||
<code>mld_</code><SPAN CLASS="textit">x</SPAN><code>prec_type</code>, with <SPAN CLASS="textit">x</SPAN> =
|
||||
<code>s</code>, <code>d</code>, <code>c</code>, <code>z</code>, according to the sparse
|
||||
matrix data structure;
|
||||
</LI>
|
||||
<LI>the arrays containing the vectors <IMG
|
||||
<LI>the arrays containing the vectors <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img21.png"
|
||||
ALT="$v$"> and <IMG
|
||||
ALT="$v$"></SPAN> and <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img87.png"
|
||||
ALT="$w$"> involved in
|
||||
the preconditioner application <IMG
|
||||
ALT="$w$"></SPAN> involved in
|
||||
the preconditioner application <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="87" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img60.png"
|
||||
ALT="$w=M^{-1}v$"> must be of type
|
||||
<I>type</I><code>(</code><I>kind_parameter</I><code>)</code>, with <I>type</I> =
|
||||
<code>real</code>, <code>complex</code> and <I>kind_parameter</I> = <code>kind(1.e0)</code>,
|
||||
ALT="$w=M^{-1}v$"></SPAN> must be of type
|
||||
<SPAN CLASS="textit">type</SPAN><code>(</code><SPAN CLASS="textit">kind_parameter</SPAN><code>)</code>, with <SPAN CLASS="textit">type</SPAN> =
|
||||
<code>real</code>, <code>complex</code> and <SPAN CLASS="textit">kind_parameter</SPAN> = <code>kind(1.e0)</code>,
|
||||
<code>kind(1.d0)</code>, according to the sparse matrix and preconditioner
|
||||
data structure; note that the PSBLAS module <code>psb_base_mod</code>
|
||||
provides the constants <code>psb_spk_</code>
|
||||
@@ -106,91 +107,93 @@ i.e.
|
||||
</LI>
|
||||
<LI>real parameters defining the preconditioner must be declared
|
||||
according to the precision of the sparse matrix and preconditioner
|
||||
data structures (see Section <A HREF="node18.html#sec:precset"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A>).
|
||||
data structures (see Section <A HREF="node18.html#sec:precset">6.2</A>).
|
||||
</LI>
|
||||
</UL>
|
||||
A description of each routine is given in the remainder of this section.
|
||||
|
||||
<P>
|
||||
|
||||
<P>
|
||||
<BR><HR>
|
||||
<!--Table of Child-Links-->
|
||||
<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
|
||||
|
||||
<UL>
|
||||
<LI><A NAME="tex2html248"
|
||||
<UL CLASS="ChildLinks">
|
||||
<LI><A NAME="tex2html247"
|
||||
HREF="node16.html">Subroutine mld_precinit</A>
|
||||
<UL>
|
||||
<LI><A NAME="tex2html249"
|
||||
<LI><A NAME="tex2html248"
|
||||
HREF="node17.html">Arguments</A>
|
||||
</UL>
|
||||
<BR>
|
||||
<LI><A NAME="tex2html250"
|
||||
<LI><A NAME="tex2html249"
|
||||
HREF="node18.html">Subroutine mld_precset</A>
|
||||
<UL>
|
||||
<LI><A NAME="tex2html251"
|
||||
<LI><A NAME="tex2html250"
|
||||
HREF="node19.html">Arguments</A>
|
||||
</UL>
|
||||
<BR>
|
||||
<LI><A NAME="tex2html252"
|
||||
<LI><A NAME="tex2html251"
|
||||
HREF="node20.html">Subroutine mld_precbld</A>
|
||||
<UL>
|
||||
<LI><A NAME="tex2html253"
|
||||
<LI><A NAME="tex2html252"
|
||||
HREF="node21.html">Arguments</A>
|
||||
</UL>
|
||||
<BR>
|
||||
<LI><A NAME="tex2html254"
|
||||
<LI><A NAME="tex2html253"
|
||||
HREF="node22.html">Subroutine mld_precaply</A>
|
||||
<UL>
|
||||
<LI><A NAME="tex2html255"
|
||||
<LI><A NAME="tex2html254"
|
||||
HREF="node23.html">Arguments</A>
|
||||
</UL>
|
||||
<BR>
|
||||
<LI><A NAME="tex2html256"
|
||||
<LI><A NAME="tex2html255"
|
||||
HREF="node24.html">Subroutine mld_precfree</A>
|
||||
<UL>
|
||||
<LI><A NAME="tex2html257"
|
||||
<LI><A NAME="tex2html256"
|
||||
HREF="node25.html">Arguments</A>
|
||||
</UL>
|
||||
<BR>
|
||||
<LI><A NAME="tex2html258"
|
||||
<LI><A NAME="tex2html257"
|
||||
HREF="node26.html">Subroutine mld_precdescr</A>
|
||||
<UL>
|
||||
<LI><A NAME="tex2html259"
|
||||
<LI><A NAME="tex2html258"
|
||||
HREF="node27.html">Arguments</A>
|
||||
</UL></UL>
|
||||
<!--End of Table of Child-Links-->
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<HR>
|
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|
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<DIV CLASS="navigation"><HR>
|
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|
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<B> Next:</B> <A NAME="tex2html247"
|
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<B> Next:</B> <A NAME="tex2html246"
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|
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<B> Up:</B> <A NAME="tex2html243"
|
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<B> Up:</B> <A NAME="tex2html242"
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HREF="node14.html">Examples</A>
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<B> <A NAME="tex2html245"
|
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HREF="node1.html">Contents</A></B>
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<B> <A NAME="tex2html244"
|
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HREF="node2.html">Contents</A></B> </DIV>
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<!--End of Navigation Panel-->
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||||
<ADDRESS>
|
||||
Salvatore Filippone
|
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2008-07-22
|
||||
2008-07-23
|
||||
</ADDRESS>
|
||||
</BODY>
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</HTML>
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|
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+17
-16
@@ -1,4 +1,4 @@
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
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|
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original version by: Nikos Drakos, CBLU, University of Leeds
|
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@@ -25,34 +25,35 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<BODY >
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<B> Next:</B> <A NAME="tex2html271"
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<B> Up:</B> <A NAME="tex2html267"
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<B> Up:</B> <A NAME="tex2html266"
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HREF="node15.html">User Interface</A>
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<B> Previous:</B> <A NAME="tex2html261"
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<B> Previous:</B> <A NAME="tex2html260"
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HREF="node15.html">User Interface</A>
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<B> <A NAME="tex2html269"
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HREF="node1.html">Contents</A></B>
|
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<BR>
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<B> <A NAME="tex2html268"
|
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HREF="node2.html">Contents</A></B>
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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="SECTION00081000000000000000"></A><A NAME="sec:precinit"></A>
|
||||
@@ -76,15 +77,15 @@ according to the preconditioner type chosen by the user.
|
||||
<!--Table of Child-Links-->
|
||||
<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
|
||||
|
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<UL>
|
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<LI><A NAME="tex2html272"
|
||||
<UL CLASS="ChildLinks">
|
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<LI><A NAME="tex2html271"
|
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HREF="node17.html">Arguments</A>
|
||||
</UL>
|
||||
<!--End of Table of Child-Links-->
|
||||
<BR><HR>
|
||||
<ADDRESS>
|
||||
Salvatore Filippone
|
||||
2008-07-22
|
||||
2008-07-23
|
||||
</ADDRESS>
|
||||
</BODY>
|
||||
</HTML>
|
||||
|
||||
+27
-26
@@ -1,4 +1,4 @@
|
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
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|
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<!--Converted with LaTeX2HTML 2002-2-1 (1.71)
|
||||
original version by: Nikos Drakos, CBLU, University of Leeds
|
||||
@@ -24,34 +24,35 @@ original version by: Nikos Drakos, CBLU, University of Leeds
|
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</HEAD>
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<BODY >
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<A NAME="tex2html281"
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<DIV CLASS="navigation"><!--Navigation Panel-->
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HREF="node18.html">
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SRC="file:/usr/share/latex2html/icons/next.png"></A>
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SRC="file:/usr/share/latex2html/icons/up.png"></A>
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<B> Next:</B> <A NAME="tex2html282"
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<B> Next:</B> <A NAME="tex2html281"
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HREF="node18.html">Subroutine mld_precset</A>
|
||||
<B> Up:</B> <A NAME="tex2html278"
|
||||
<B> Up:</B> <A NAME="tex2html277"
|
||||
HREF="node16.html">Subroutine mld_precinit</A>
|
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<B> Previous:</B> <A NAME="tex2html274"
|
||||
<B> Previous:</B> <A NAME="tex2html273"
|
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HREF="node16.html">Subroutine mld_precinit</A>
|
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<B> <A NAME="tex2html280"
|
||||
HREF="node1.html">Contents</A></B>
|
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<BR>
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<B> <A NAME="tex2html279"
|
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HREF="node2.html">Contents</A></B>
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<BR>
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<BR></DIV>
|
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<!--End of Navigation Panel-->
|
||||
|
||||
<H3><A NAME="SECTION00081100000000000000">
|
||||
@@ -61,46 +62,46 @@ Arguments</A>
|
||||
<P>
|
||||
<TABLE CELLPADDING=3>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><code>p</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298><code>type(mld_</code><I>x</I><code>prec_type), intent(inout)</code>.</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><code>type(mld_</code><SPAN CLASS="textit">x</SPAN><code>prec_type), intent(inout)</code>.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34> </TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298>The preconditioner data structure. Note that <I>x</I>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340>The preconditioner data structure. Note that <SPAN CLASS="textit">x</SPAN>
|
||||
must be chosen according to the real/complex, single/double
|
||||
precision version of MLD2P4 under use.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><code>ptype</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298><code>character(len=*), intent(in)</code>.</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><code>character(len=*), intent(in)</code>.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34> </TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298>The type of preconditioner. Its values are specified
|
||||
in Table <A HREF="#tab:precinit"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A>.</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340>The type of preconditioner. Its values are specified
|
||||
in Table <A HREF="node13.html#tab:precinit">1</A>.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34> </TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298>Note that the strings are case insensitive.</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340>Note that the strings are case insensitive.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><code>info</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298><code>integer, intent(out)</code>.</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><code>integer, intent(out)</code>.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34> </TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298>Error code. If no error, 0 is returned. See Section <A HREF="node28.html#sec:errors"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A> for details.</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340>Error code. If no error, 0 is returned. See Section <A HREF="node28.html#sec:errors">7</A> for details.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><code>nlev</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298><code>integer, optional, intent(in)</code>.</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><code>integer, optional, intent(in)</code>.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34> </TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298>The number of levels of the multilevel preconditioner.
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340>The number of levels of the multilevel preconditioner.
|
||||
If <code>nlev</code> is not present and <code>ptype</code>=<code>'ML'</code>, <code>'ml'</code>,
|
||||
then <code>nlev</code>=2 is assumed. Otherwise, <code>nlev</code> is ignored.</TD>
|
||||
</TR>
|
||||
</TABLE>
|
||||
|
||||
<P>
|
||||
|
||||
<P>
|
||||
<BR><HR>
|
||||
<ADDRESS>
|
||||
Salvatore Filippone
|
||||
2008-07-22
|
||||
2008-07-23
|
||||
</ADDRESS>
|
||||
</BODY>
|
||||
</HTML>
|
||||
|
||||
+17
-16
@@ -1,4 +1,4 @@
|
||||
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
|
||||
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
|
||||
|
||||
<!--Converted with LaTeX2HTML 2002-2-1 (1.71)
|
||||
original version by: Nikos Drakos, CBLU, University of Leeds
|
||||
@@ -25,34 +25,35 @@ original version by: Nikos Drakos, CBLU, University of Leeds
|
||||
</HEAD>
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<BODY >
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<H2><A NAME="SECTION00082000000000000000"></A><A NAME="sec:precset"></A>
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@@ -76,15 +77,15 @@ contained in <code>val</code>.
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<!--Table of Child-Links-->
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<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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<LI><A NAME="tex2html295"
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<UL CLASS="ChildLinks">
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<!--End of Table of Child-Links-->
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<BR><HR>
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||||
<ADDRESS>
|
||||
Salvatore Filippone
|
||||
2008-07-22
|
||||
2008-07-23
|
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</ADDRESS>
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original version by: Nikos Drakos, CBLU, University of Leeds
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<BODY >
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<H3><A NAME="SECTION00082100000000000000">
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@@ -61,43 +62,38 @@ Arguments</A>
|
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<P>
|
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<TABLE CELLPADDING=3>
|
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<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><code>p</code></TD>
|
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298><code>type(mld_</code><I>x</I><code>prec_type), intent(inout)</code>.</TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><code>type(mld_</code><SPAN CLASS="textit">x</SPAN><code>prec_type), intent(inout)</code>.</TD>
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</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34> </TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298>The preconditioner data structure. Note that <I>x</I> must
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340>The preconditioner data structure. Note that <SPAN CLASS="textit">x</SPAN> must
|
||||
be chosen according to the real/complex, single/double precision
|
||||
version of MLD2P4 under use.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><code>what</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298><code>integer, intent(in)</code>.</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><code>integer, intent(in)</code>.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34> </TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298>The number identifying the parameter to be set.
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340>The number identifying the parameter to be set.
|
||||
A mnemonic constant has been associated to each of these
|
||||
numbers, as reported in Tables <A HREF="#tab:p_type"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A>-<A HREF="#tab:p_coarse"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A>.</TD>
|
||||
numbers, as reported in Tables <A HREF="#tab:p_type">2</A>-<A HREF="#tab:p_coarse">5</A>.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><code>val </code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298><code>integer</code> <I>or</I> <code>character(len=*)</code> <I>or</I>
|
||||
<code>real(psb_spk_)</code> <I>or</I> <code>real(psb_dpk_)</code>,
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><code>integer</code> <SPAN CLASS="textit">or</SPAN> <code>character(len=*)</code> <SPAN CLASS="textit">or</SPAN>
|
||||
<code>real(psb_spk_)</code> <SPAN CLASS="textit">or</SPAN> <code>real(psb_dpk_)</code>,
|
||||
<code>intent(in)</code>.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34> </TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298>The value of the parameter to be set. The list of allowed
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340>The value of the parameter to be set. The list of allowed
|
||||
values and the corresponding data types is given in
|
||||
Tables <A HREF="#tab:p_type"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A>-<A HREF="#tab:p_coarse"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A>.
|
||||
Tables <A HREF="#tab:p_type">2</A>-<A HREF="#tab:p_coarse">5</A>.
|
||||
When the value is of type <code>character(len=*)</code>,
|
||||
it is also treated as case insensitive.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><code>info</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298><code>integer, intent(out)</code>.</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><code>integer, intent(out)</code>.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34> </TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298>Error code. If no error, 0 is returned. See Section <A HREF="node28.html#sec:errors"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340>Error code. If no error, 0 is returned. See Section <A HREF="node28.html#sec:errors">7</A>
|
||||
for details.</TD>
|
||||
</TR>
|
||||
</TABLE>
|
||||
@@ -120,17 +116,20 @@ can be logically divided into four groups, i.e. parameters defining
|
||||
</LI>
|
||||
</OL>
|
||||
A list of the parameters that can be set, along with their allowed and
|
||||
default values, is given in Tables <A HREF="#tab:p_type"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A>-<A HREF="#tab:p_coarse"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A>.
|
||||
default values, is given in Tables <A HREF="#tab:p_type">2</A>-<A HREF="#tab:p_coarse">5</A>.
|
||||
For a detailed description of the meaning of the parameters, please
|
||||
refer to Section <A HREF="node10.html#sec:background"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A>.
|
||||
refer to Section <A HREF="node10.html#sec:background">4</A>.
|
||||
|
||||
<P>
|
||||
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER"><A NAME="1245"></A>
|
||||
<TABLE>
|
||||
<CAPTION><STRONG>Table 2:</STRONG>
|
||||
Parameters defining the type of multi-level preconditioner.
|
||||
</CAPTION>
|
||||
<TR><TD>
|
||||
<DIV ALIGN="CENTER">
|
||||
<TABLE CELLPADDING=3 BORDER="1">
|
||||
<TABLE CELLPADDING=3 BORDER="1" ALIGN="CENTER">
|
||||
<TR><TD ALIGN="LEFT"><code>what</code></TD>
|
||||
<TD ALIGN="LEFT"><SMALL>DATA TYPE</SMALL></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=57><code>val</code></TD>
|
||||
@@ -149,7 +148,7 @@ refer to Section <A HREF="node10.html#sec:background"><IMG ALIGN="BOTTOM"
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=57><TT>'DIAG'</TT> <TT>'BJAC'</TT> <TT>'AS'</TT></TD>
|
||||
<TD ALIGN="LEFT"><TT>'AS'</TT></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Basic one-level preconditioner (i.e. smoother): diagonal,
|
||||
block Jacobi, AS</TD>
|
||||
block Jacobi, AS.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT"><code>mld_smoother_pos_</code></TD>
|
||||
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
|
||||
@@ -160,11 +159,21 @@ refer to Section <A HREF="node10.html#sec:background"><IMG ALIGN="BOTTOM"
|
||||
</TR>
|
||||
</TABLE>
|
||||
</DIV>
|
||||
</TD></TR>
|
||||
</TABLE>
|
||||
</DIV><P></P>
|
||||
<BR>
|
||||
|
||||
<P>
|
||||
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER"><A NAME="1247"></A>
|
||||
<TABLE>
|
||||
<CAPTION><STRONG>Table 3:</STRONG>
|
||||
Parameters defining the one-level preconditioner used as smoother.
|
||||
</CAPTION>
|
||||
<TR><TD>
|
||||
<DIV ALIGN="CENTER">
|
||||
<TABLE CELLPADDING=3 BORDER="1">
|
||||
<TABLE CELLPADDING=3 BORDER="1" ALIGN="CENTER">
|
||||
<TR><TD ALIGN="LEFT"><code>what</code></TD>
|
||||
<TD ALIGN="LEFT"><SMALL>DATA TYPE</SMALL></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=91><code>val</code></TD>
|
||||
@@ -173,10 +182,10 @@ refer to Section <A HREF="node10.html#sec:background"><IMG ALIGN="BOTTOM"
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT"><code>mld_sub_ovr_</code></TD>
|
||||
<TD ALIGN="LEFT"><code>integer</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=91>any integer number <IMG
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=91>any int. num. <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="32" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img88.png"
|
||||
ALT="$\ge 0$"></TD>
|
||||
ALT="$\ge 0$"></SPAN></TD>
|
||||
<TD ALIGN="LEFT">1</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Number of overlap layers.</TD>
|
||||
</TR>
|
||||
@@ -201,44 +210,44 @@ refer to Section <A HREF="node10.html#sec:background"><IMG ALIGN="BOTTOM"
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=91><TT>'ILU'</TT> <TT>'MILU'</TT> <TT>'ILUT'</TT>
|
||||
<TT>'UMF'</TT> <TT>'SLU'</TT></TD>
|
||||
<TD ALIGN="LEFT"><TT>'UMF'</TT></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Local solver: ILU(<IMG
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Local solver: ILU(<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img34.png"
|
||||
ALT="$p$">), MILU(<IMG
|
||||
ALT="$p$"></SPAN>), MILU(<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img34.png"
|
||||
ALT="$p$">), ILU(<IMG
|
||||
ALT="$p$"></SPAN>), ILU(<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="27" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img35.png"
|
||||
ALT="$p,t$">), LU from UMFPACK, LU from SuperLU
|
||||
ALT="$p,t$"></SPAN>), LU from UMFPACK, LU from SuperLU
|
||||
(plus triangular solve).</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT"><code>mld_sub_fillin_</code></TD>
|
||||
<TD ALIGN="LEFT"><code>integer</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=91>Any int. num. <IMG
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=91>Any int. num. <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="32" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img88.png"
|
||||
ALT="$\ge 0$"></TD>
|
||||
ALT="$\ge 0$"></SPAN></TD>
|
||||
<TD ALIGN="LEFT">0</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Fill-in level <IMG
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Fill-in level <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img34.png"
|
||||
ALT="$p$"> of the incomplete LU factorizations.</TD>
|
||||
ALT="$p$"></SPAN> of the incomplete LU factorizations.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT"><code>mld_sub_iluthrs_</code></TD>
|
||||
<TD ALIGN="LEFT"><code>real(</code><I>kind_parameter</I><code>)</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=91>Any real num. <IMG
|
||||
<TD ALIGN="LEFT"><code>real(</code><SPAN CLASS="textit">kind_parameter</SPAN><code>)</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=91>Any real num. <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="32" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img88.png"
|
||||
ALT="$\ge 0$"></TD>
|
||||
<TD ALIGN="LEFT"><TT>0.e0</TT> (or <TT>0.d0</TT>)</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Drop tolerance <IMG
|
||||
ALT="$\ge 0$"></SPAN></TD>
|
||||
<TD ALIGN="LEFT">0</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Drop tolerance <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img89.png"
|
||||
ALT="$t$"> in the ILU(<IMG
|
||||
ALT="$t$"></SPAN> in the ILU(<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="27" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img35.png"
|
||||
ALT="$p,t$">) factorization.</TD>
|
||||
ALT="$p,t$"></SPAN>) factorization.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT"><code>mld_sub_ren_</code></TD>
|
||||
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
|
||||
@@ -250,11 +259,21 @@ refer to Section <A HREF="node10.html#sec:background"><IMG ALIGN="BOTTOM"
|
||||
</TR>
|
||||
</TABLE>
|
||||
</DIV>
|
||||
</TD></TR>
|
||||
</TABLE>
|
||||
</DIV><P></P>
|
||||
<BR>
|
||||
|
||||
<P>
|
||||
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER"><A NAME="1249"></A>
|
||||
<TABLE>
|
||||
<CAPTION><STRONG>Table 4:</STRONG>
|
||||
Parameters defining the aggregation algorithm.
|
||||
</CAPTION>
|
||||
<TR><TD>
|
||||
<DIV ALIGN="CENTER">
|
||||
<TABLE CELLPADDING=3 BORDER="1">
|
||||
<TABLE CELLPADDING=3 BORDER="1" ALIGN="CENTER">
|
||||
<TR><TD ALIGN="LEFT"><code>what</code></TD>
|
||||
<TD ALIGN="LEFT"><SMALL>DATA TYPE</SMALL></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=65><code>val</code></TD>
|
||||
@@ -271,62 +290,86 @@ refer to Section <A HREF="node10.html#sec:background"><IMG ALIGN="BOTTOM"
|
||||
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=65><TT>'SMOOTH'</TT> <TT>'RAW'</TT></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=74><TT>'SMOOTH'</TT></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Type of aggregation: smoothed or raw, i.e. using the tentative prolongator.</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Type of aggregation: smoothed, raw (i.e. using the tentative prolongator).</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT"><code>mld_aggr_thresh_</code></TD>
|
||||
<TD ALIGN="LEFT"><code>real(</code><I>kind_parameter</I><code>)</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=65>Any real num. <IMG
|
||||
<TD ALIGN="LEFT"><code>real(</code><SPAN CLASS="textit">kind_parameter</SPAN><code>)</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=65>Any real num. <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="56" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img90.png"
|
||||
ALT="$\in [0, 1]$"></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=74><TT>0.e0</TT> (or <TT>0.d0</TT>)</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>The threshold <IMG
|
||||
ALT="$\in [0, 1]$"></SPAN></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=74>0</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>The threshold <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="13" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img91.png"
|
||||
ALT="$\theta$"> in the aggregation algorithm.</TD>
|
||||
ALT="$\theta$"></SPAN> in the aggregation algorithm.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT"><code>mld_aggr_eig_</code></TD>
|
||||
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=65><TT>'A_NORMI'</TT></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=74><TT>'A_NORMI'</TT></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Estimate of the maximum eigenvalue of <IMG
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Estimate of the eigenvalue <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="50" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img85.png"
|
||||
ALT="$D^{-1}A$">
|
||||
for the smoothed aggregation. Currently, only the infinity norm of
|
||||
ALT="$D^{-1}A$"></SPAN> with largest modulus,
|
||||
to build the damping parameter <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img84.png"
|
||||
ALT="$\omega$"></SPAN> in the smoothed aggregation.
|
||||
Currently, only the infinity norm of
|
||||
the matrix is available.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT"><code>mld_aggr_damp_</code></TD>
|
||||
<TD ALIGN="LEFT"><code>real(</code><I>kind_parameter</I><code>)</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=65>Any real num. <IMG
|
||||
WIDTH="32" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
<TD ALIGN="LEFT"><code>real(</code><SPAN CLASS="textit">kind_parameter</SPAN><code>)</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=65>Any real num.</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=74><!-- MATH
|
||||
$4/(3||D^{-1}A||_\infty)$
|
||||
-->
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="124" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img92.png"
|
||||
ALT="$>0$"></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=74><TT>4.e0/3.e0</TT> (or <TT>4.d0/3.d0</TT>)</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>The damping parameter <IMG
|
||||
ALT="$4/(3\vert\vert D^{-1}A\vert\vert _\infty)$"></SPAN></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>The damping parameter <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img84.png"
|
||||
ALT="$\omega$"> in the aggregation algorithm.</TD>
|
||||
ALT="$\omega$"></SPAN> in the smoothed aggregation algorithm.
|
||||
If the user specifies a negative value, then <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img84.png"
|
||||
ALT="$\omega$"></SPAN> is set to its default
|
||||
value; otherwise, <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img84.png"
|
||||
ALT="$\omega$"></SPAN> is set to the value provided by the
|
||||
user. In the latter case no estimate of the eigenvalue <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="50" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img85.png"
|
||||
ALT="$D^{-1}A$"></SPAN> with
|
||||
largest modulus is computed.</TD>
|
||||
</TR>
|
||||
</TABLE>
|
||||
</DIV>
|
||||
</TD></TR>
|
||||
</TABLE>
|
||||
</DIV><P></P>
|
||||
<BR>
|
||||
|
||||
<P>
|
||||
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER"><A NAME="1251"></A>
|
||||
<TABLE>
|
||||
<CAPTION><STRONG>Table 5:</STRONG>
|
||||
Parameters defining the coarse-space correction at the coarsest
|
||||
level.</CAPTION>
|
||||
<TR><TD>
|
||||
<DIV ALIGN="CENTER">
|
||||
<TABLE CELLPADDING=3 BORDER="1">
|
||||
<TABLE CELLPADDING=3 BORDER="1" ALIGN="CENTER">
|
||||
<TR><TD ALIGN="LEFT"><code>what</code></TD>
|
||||
<TD ALIGN="LEFT"><SMALL>DATA TYPE</SMALL></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=91><code>val</code></TD>
|
||||
<TD ALIGN="LEFT"><SMALL>DEFAULT</SMALL></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198><SMALL>COMMENTS</SMALL></TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT"><code>mld_coarse_mat_</code></TD>
|
||||
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=91><TT>'DISTR'</TT> <TT>'REPL'</TT></TD>
|
||||
<TD ALIGN="LEFT"><TT>'DISTR'</TT></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Coarsest matrix: distributed among the processors or replicated on each of them.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT"><code>mld_coarse_solve_</code></TD>
|
||||
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=91><TT>'BJAC'</TT> <TT>'UMF'</TT> <TT>'SLU'</TT>
|
||||
@@ -334,9 +377,15 @@ refer to Section <A HREF="node10.html#sec:background"><IMG ALIGN="BOTTOM"
|
||||
<TD ALIGN="LEFT"><TT>'BJAC'</TT></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Solver used at the coarsest level: block Jacobi, sequential LU from UMFPACK,
|
||||
sequential LU from SuperLU, distributed LU from SuperLU_Dist.
|
||||
If the coarsest matrix is distributed, only <TT>'BJAC'</TT> and <TT>'SLUDIST'</TT>
|
||||
can be chosen; if it is replicated, only <TT>'BJAC'</TT> or <TT>'SLUDIST'</TT> can
|
||||
be selected.</TD>
|
||||
With <TT>'SLUDIST'</TT> the coarsest matrix
|
||||
must be distributed; with <TT>'UMF'</TT> or
|
||||
<TT>'SLU'</TT> it must be replicated.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT"><code>mld_coarse_mat_</code></TD>
|
||||
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=91><TT>'DISTR'</TT> <TT>'REPL'</TT></TD>
|
||||
<TD ALIGN="LEFT"><TT>'DISTR'</TT></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Coarsest matrix: distributed among the processors or replicated on each of them.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT"><code>mld_coarse_subsolve_</code></TD>
|
||||
<TD ALIGN="LEFT"><code>character(len=*)</code></TD>
|
||||
@@ -344,90 +393,96 @@ refer to Section <A HREF="node10.html#sec:background"><IMG ALIGN="BOTTOM"
|
||||
<TT>'UMF'</TT> <TT>'SLU'</TT></TD>
|
||||
<TD ALIGN="LEFT"><TT>'UMF'</TT></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Solver for the diagonal blocks of the coarse matrix, in case the block Jacobi solver
|
||||
is chosen as coarsest-level solver: ILU(<IMG
|
||||
is chosen as coarsest-level solver: ILU(<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img34.png"
|
||||
ALT="$p$">), MILU(<IMG
|
||||
ALT="$p$"></SPAN>), MILU(<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img34.png"
|
||||
ALT="$p$">), ILU(<IMG
|
||||
ALT="$p$"></SPAN>), ILU(<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="27" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img35.png"
|
||||
ALT="$p,t$">), LU from UMFPACK,
|
||||
ALT="$p,t$"></SPAN>), LU from UMFPACK,
|
||||
LU from SuperLU, plus triangular solve.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT"><code>mld_coarse_sweeps_</code></TD>
|
||||
<TD ALIGN="LEFT"><code>integer</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=91>Any int. num. <IMG
|
||||
WIDTH="32" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img92.png"
|
||||
ALT="$>0$"></TD>
|
||||
<TD ALIGN="LEFT"><TT>4</TT></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=91>Any int. num. <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="31" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img93.png"
|
||||
ALT="$> 0$"></SPAN></TD>
|
||||
<TD ALIGN="LEFT">4</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Number of Block-Jacobi sweeps when 'BJAC' is used as coarsest-level solver.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT"><code>mld_coarse_fillin_</code></TD>
|
||||
<TD ALIGN="LEFT"><code>integer</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=91>Any int. num. <IMG
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=91>Any int. num. <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="32" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img88.png"
|
||||
ALT="$\ge 0$"></TD>
|
||||
<TD ALIGN="LEFT"><TT>0</TT></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Fill-in level <IMG
|
||||
ALT="$\ge 0$"></SPAN></TD>
|
||||
<TD ALIGN="LEFT">0</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Fill-in level <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img34.png"
|
||||
ALT="$p$"> of the incomplete LU factorizations.</TD>
|
||||
ALT="$p$"></SPAN> of the incomplete LU factorizations.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT"><code>mld_coarse_iluthrs_</code></TD>
|
||||
<TD ALIGN="LEFT"><code>real(</code><I>kind_parameter</I><code>)</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=91>Any real. num. <IMG
|
||||
<TD ALIGN="LEFT"><code>real(</code><SPAN CLASS="textit">kind_parameter</SPAN><code>)</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=91>Any real. num. <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="32" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img88.png"
|
||||
ALT="$\ge 0$"></TD>
|
||||
<TD ALIGN="LEFT"><TT>0.d0</TT> (or <TT>0.e0</TT>)</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Drop tolerance <IMG
|
||||
ALT="$\ge 0$"></SPAN></TD>
|
||||
<TD ALIGN="LEFT">0</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=198>Drop tolerance <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img89.png"
|
||||
ALT="$t$"> in the ILU(<IMG
|
||||
ALT="$t$"></SPAN> in the ILU(<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="27" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img35.png"
|
||||
ALT="$p,t$">) factorization.</TD>
|
||||
ALT="$p,t$"></SPAN>) factorization.</TD>
|
||||
</TR>
|
||||
</TABLE>
|
||||
</DIV>
|
||||
</TD></TR>
|
||||
</TABLE>
|
||||
</DIV><P></P>
|
||||
<BR>
|
||||
|
||||
<P>
|
||||
|
||||
<HR>
|
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|
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<B> <A NAME="tex2html302"
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HREF="node2.html">Contents</A></B> </DIV>
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<ADDRESS>
|
||||
Salvatore Filippone
|
||||
2008-07-22
|
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2008-07-23
|
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+80
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original version by: Nikos Drakos, CBLU, University of Leeds
|
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@@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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<HTML>
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<H1><A NAME="SECTION00020000000000000000"></A><A NAME="sec:overview"></A>
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<BR>
|
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General Overview
|
||||
</H1>
|
||||
|
||||
<P>
|
||||
The M<SMALL>ULTI-</SMALL>L<SMALL>EVEL </SMALL>D<SMALL>OMAIN </SMALL>D<SMALL>ECOMPOSITION </SMALL>P<SMALL>ARALLEL </SMALL>P<SMALL>RECONDITIONERS </SMALL>P<SMALL>ACKAGE BASED ON
|
||||
</SMALL>PSBLAS (MLD2P4) provides <I>multi-level Schwarz preconditioners</I> [<A
|
||||
HREF="node30.html#dd2_96">15</A>],
|
||||
to be used in the iterative solutions of sparse linear systems:
|
||||
<BR>
|
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<DIV ALIGN="RIGHT">
|
||||
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||||
<!-- MATH
|
||||
\begin{equation}
|
||||
Ax=b,
|
||||
\end{equation}
|
||||
-->
|
||||
<TABLE WIDTH="100%" ALIGN="CENTER">
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<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="system1"></A><IMG
|
||||
WIDTH="57" HEIGHT="30" BORDER="0"
|
||||
SRC="img1.png"
|
||||
ALT="\begin{displaymath}
|
||||
Ax=b,
|
||||
\end{displaymath}"></TD>
|
||||
<TD WIDTH=10 ALIGN="RIGHT">
|
||||
(1)</TD></TR>
|
||||
</TABLE>
|
||||
<BR CLEAR="ALL"></DIV><P></P>
|
||||
where <IMG
|
||||
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img2.png"
|
||||
ALT="$A$"> is a square, real or complex, sparse matrix with a symmetric sparsity pattern.
|
||||
These preconditioners have the following general features:
|
||||
<H2><A NAME="SECTION00020000000000000000">
|
||||
Contents</A>
|
||||
</H2>
|
||||
<!--Table of Contents-->
|
||||
|
||||
<UL CLASS="TofC">
|
||||
<LI><A NAME="tex2html68"
|
||||
HREF="node3.html">General Overview</A>
|
||||
<LI><A NAME="tex2html69"
|
||||
HREF="node4.html">Code Distribution</A>
|
||||
<LI><A NAME="tex2html70"
|
||||
HREF="node5.html">Configuring and Building MLD2P4</A>
|
||||
<UL>
|
||||
<LI>both <I>additive and hybrid multilevel</I> variants are implemented,
|
||||
i.e. variants that are additive among the levels and inside each level, and variants
|
||||
that are multiplicative among the levels and additive inside each level;
|
||||
the basic Additive Schwarz (AS) preconditioners are obtained by considering only one level;
|
||||
</LI>
|
||||
<LI>a <I>purely algebraic</I> approach is used to
|
||||
generate a sequence of coarse-level corrections to a basic AS preconditioner, without
|
||||
explicitly using any information on the geometry of the original problem (e.g. the
|
||||
discretization of a PDE). The <I>smoothed aggregation</I> technique is applied
|
||||
as algebraic coarsening strategy [<A
|
||||
HREF="node30.html#BREZINA_VANEK">1</A>,<A
|
||||
HREF="node30.html#VANEK_MANDEL_BREZINA">19</A>].
|
||||
</LI>
|
||||
<LI><A NAME="tex2html71"
|
||||
HREF="node6.html">Prerequisites</A>
|
||||
<LI><A NAME="tex2html72"
|
||||
HREF="node7.html">Optional third party libraries</A>
|
||||
<LI><A NAME="tex2html73"
|
||||
HREF="node8.html">Configuration options</A>
|
||||
<LI><A NAME="tex2html74"
|
||||
HREF="node9.html">Example and test programs</A>
|
||||
</UL>
|
||||
|
||||
<P>
|
||||
The package is written in <I>Fortran 95</I>, following an
|
||||
<I>object-oriented approach</I> through the exploitation of features
|
||||
such as abstract data type creation, functional
|
||||
overloading and dynamic memory management.
|
||||
The parallel implementation is based
|
||||
on a Single Program Multiple Data (SPMD) paradigm for distributed-memory architectures.
|
||||
Single and double precision implementations of MLD2P4 are available for both the
|
||||
real and the complex case, that can be used through a single interface.
|
||||
|
||||
<P>
|
||||
MLD2P4 has been designed to implement scalable and easy-to-use multilevel preconditioners
|
||||
in the context of the <I>PSBLAS (Parallel Sparse BLAS)
|
||||
computational framework</I> [<A
|
||||
HREF="node30.html#psblas_00">12</A>].
|
||||
PSBLAS is a library originally developed to address the parallel implementation of
|
||||
iterative solvers for sparse linear system, by providing basic linear algebra
|
||||
operators and data management facilities for distributed sparse matrices; it
|
||||
also includes parallel Krylov solvers, built on the top of the basic PSBLAS kernels.
|
||||
The preconditioners available in MLD2P4 can be used with these Krylov solvers.
|
||||
The choice of PSBLAS has been mainly motivated by the need of having
|
||||
a portable and efficient software infrastructure implementing ``de facto'' standard
|
||||
parallel sparse linear algebra kernels, to pursue goals such as performance,
|
||||
portability, modularity ed extensibility in the development of the preconditioner
|
||||
package. On the other hand, the implementation of MLD2P4 has led to some
|
||||
revisions and extentions of the PSBLAS kernels, leading to the
|
||||
recent PSBLAS 2.0 version [<A
|
||||
HREF="node30.html#PSBLASGUIDE">11</A>]. The inter-process comunication required
|
||||
by MLD2P4 is encapsulated into the PSBLAS routines, except few cases where
|
||||
MPI [<A
|
||||
HREF="node30.html#MPI1">16</A>] is explicitly called. Therefore, MLD2P4 can be run on any parallel
|
||||
machine where PSBLAS and MPI implementations are available.
|
||||
|
||||
<P>
|
||||
MLD2P4 has a layered and modular software architecture where three main layers can be identified.
|
||||
The lower layer consists of the PSBLAS kernels, the middle one implements
|
||||
the construction and application phases of the preconditioners, and the upper one
|
||||
provides a uniform and easy-to-use interface to all the preconditioners.
|
||||
This architecture allows for different levels of use of the package:
|
||||
few black-box routines at the upper layer allow non-expert users to easily
|
||||
build any preconditioner available in MLD2P4 and to apply it within a PSBLAS Krylov solver.
|
||||
On the other hand, the routines of the middle and lower layer can be used and extended
|
||||
by expert users to build new versions of multi-level Schwarz preconditioners.
|
||||
We provide here a description of the upper-layer routines, but not of the
|
||||
medium-layer ones.
|
||||
|
||||
<P>
|
||||
This guide is organized as follows. The notational conventions used in
|
||||
the guide and in the naming of the MLD2P4 routines are reported in
|
||||
Section <A HREF="node3.html#sec:conventions"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A>. Information on the distribution of the source code and the
|
||||
related license is given in Section <A HREF="node4.html#sec:distribution"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A>, while details on the configuration
|
||||
and installation of package are given in Section <A HREF="node5.html#sec:building"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A>. A description of
|
||||
multi-level Schwarz preconditioners based on smoothed aggregation is provided
|
||||
in Section <A HREF="node10.html#sec:background"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A>, to help the users in choosing among the different preconditioners
|
||||
implemented in MLD2P4. The basics for building and applying the preconditioners
|
||||
with the Krylov solvers implemented in PSBLAS are reported in Section <A HREF="node13.html#sec:started"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A>, where the
|
||||
Fortran 95 codes of a few sample programs are also shown. A reference guide for
|
||||
the upper-layer routines of MLD2P4, that are the user interface, is provided
|
||||
in Section <A HREF="node15.html#sec:userinterface"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A>. The error handling mechanism used by the package is briefly described
|
||||
in Section <A HREF="node28.html#sec:errors"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A>. The copyright terms concerning the distribution and modification
|
||||
of MLD2P4 are reported in Appendix <A HREF="node29.html#sec:license"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A>.
|
||||
|
||||
<P>
|
||||
<HR>
|
||||
<!--Navigation Panel-->
|
||||
<A NAME="tex2html89"
|
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HREF="node3.html">
|
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|
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SRC="file:/usr/share/latex2html/icons/next.png"></A>
|
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|
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|
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<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
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SRC="file:/usr/share/latex2html/icons/up.png"></A>
|
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|
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<BR>
|
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<B> Next:</B> <A NAME="tex2html90"
|
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HREF="node3.html">Notational Conventions</A>
|
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<B> Up:</B> <A NAME="tex2html86"
|
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HREF="userhtml.html">userhtml</A>
|
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<B> Previous:</B> <A NAME="tex2html80"
|
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HREF="node1.html">Contents</A>
|
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<B> <A NAME="tex2html88"
|
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HREF="node1.html">Contents</A></B>
|
||||
<!--End of Navigation Panel-->
|
||||
<LI><A NAME="tex2html75"
|
||||
HREF="node10.html">Multi-level Domain Decomposition Background</A>
|
||||
<UL>
|
||||
<LI><A NAME="tex2html76"
|
||||
HREF="node11.html">Multi-level Schwarz Preconditioners</A>
|
||||
<LI><A NAME="tex2html77"
|
||||
HREF="node12.html">Smoothed Aggregation</A>
|
||||
</UL>
|
||||
<BR>
|
||||
<LI><A NAME="tex2html78"
|
||||
HREF="node13.html">Getting Started</A>
|
||||
<UL>
|
||||
<LI><A NAME="tex2html79"
|
||||
HREF="node14.html">Examples</A>
|
||||
</UL>
|
||||
<BR>
|
||||
<LI><A NAME="tex2html80"
|
||||
HREF="node15.html">User Interface</A>
|
||||
<UL>
|
||||
<LI><A NAME="tex2html81"
|
||||
HREF="node16.html">Subroutine mld_precinit</A>
|
||||
<LI><A NAME="tex2html82"
|
||||
HREF="node18.html">Subroutine mld_precset</A>
|
||||
<LI><A NAME="tex2html83"
|
||||
HREF="node20.html">Subroutine mld_precbld</A>
|
||||
<LI><A NAME="tex2html84"
|
||||
HREF="node22.html">Subroutine mld_precaply</A>
|
||||
<LI><A NAME="tex2html85"
|
||||
HREF="node24.html">Subroutine mld_precfree</A>
|
||||
<LI><A NAME="tex2html86"
|
||||
HREF="node26.html">Subroutine mld_precdescr</A>
|
||||
</UL>
|
||||
<BR>
|
||||
<LI><A NAME="tex2html87"
|
||||
HREF="node28.html">Error Handling</A>
|
||||
<LI><A NAME="tex2html88"
|
||||
HREF="node29.html">License</A>
|
||||
<LI><A NAME="tex2html89"
|
||||
HREF="node30.html">Bibliography</A>
|
||||
</UL>
|
||||
<!--End of Table of Contents-->
|
||||
|
||||
<P>
|
||||
|
||||
<P>
|
||||
<BR><HR>
|
||||
<ADDRESS>
|
||||
Salvatore Filippone
|
||||
2008-07-22
|
||||
2008-07-23
|
||||
</ADDRESS>
|
||||
</BODY>
|
||||
</HTML>
|
||||
|
||||
+17
-16
@@ -1,4 +1,4 @@
|
||||
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
|
||||
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
|
||||
|
||||
<!--Converted with LaTeX2HTML 2002-2-1 (1.71)
|
||||
original version by: Nikos Drakos, CBLU, University of Leeds
|
||||
@@ -25,34 +25,35 @@ original version by: Nikos Drakos, CBLU, University of Leeds
|
||||
</HEAD>
|
||||
|
||||
<BODY >
|
||||
<!--Navigation Panel-->
|
||||
<A NAME="tex2html316"
|
||||
|
||||
<DIV CLASS="navigation"><!--Navigation Panel-->
|
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|
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HREF="node21.html">
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<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
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SRC="file:/usr/share/latex2html/icons/next.png"></A>
|
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<B> Up:</B> <A NAME="tex2html313"
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<B> Up:</B> <A NAME="tex2html312"
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<B> Previous:</B> <A NAME="tex2html306"
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<B> <A NAME="tex2html315"
|
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HREF="node1.html">Contents</A></B>
|
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<BR>
|
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<B> <A NAME="tex2html314"
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HREF="node2.html">Contents</A></B>
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<BR>
|
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<BR></DIV>
|
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<!--End of Navigation Panel-->
|
||||
|
||||
<H2><A NAME="SECTION00083000000000000000"></A><A NAME="sec:precbld"></A>
|
||||
@@ -75,15 +76,15 @@ the user through the routines <code>mld_precinit</code> and <code>mld_precset</c
|
||||
<!--Table of Child-Links-->
|
||||
<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
|
||||
|
||||
<UL>
|
||||
<LI><A NAME="tex2html318"
|
||||
<UL CLASS="ChildLinks">
|
||||
<LI><A NAME="tex2html317"
|
||||
HREF="node21.html">Arguments</A>
|
||||
</UL>
|
||||
<!--End of Table of Child-Links-->
|
||||
<BR><HR>
|
||||
<ADDRESS>
|
||||
Salvatore Filippone
|
||||
2008-07-22
|
||||
2008-07-23
|
||||
</ADDRESS>
|
||||
</BODY>
|
||||
</HTML>
|
||||
|
||||
+26
-26
@@ -1,4 +1,4 @@
|
||||
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
|
||||
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
|
||||
|
||||
<!--Converted with LaTeX2HTML 2002-2-1 (1.71)
|
||||
original version by: Nikos Drakos, CBLU, University of Leeds
|
||||
@@ -24,34 +24,35 @@ original version by: Nikos Drakos, CBLU, University of Leeds
|
||||
</HEAD>
|
||||
|
||||
<BODY >
|
||||
<!--Navigation Panel-->
|
||||
<A NAME="tex2html327"
|
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|
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<DIV CLASS="navigation"><!--Navigation Panel-->
|
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<B> Next:</B> <A NAME="tex2html328"
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<B> Next:</B> <A NAME="tex2html327"
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HREF="node22.html">Subroutine mld_precaply</A>
|
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<B> Up:</B> <A NAME="tex2html324"
|
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<B> Up:</B> <A NAME="tex2html323"
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HREF="node20.html">Subroutine mld_precbld</A>
|
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<B> Previous:</B> <A NAME="tex2html320"
|
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<B> Previous:</B> <A NAME="tex2html319"
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HREF="node20.html">Subroutine mld_precbld</A>
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<B> <A NAME="tex2html326"
|
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HREF="node1.html">Contents</A></B>
|
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<BR>
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<B> <A NAME="tex2html325"
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HREF="node2.html">Contents</A></B>
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|
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<BR></DIV>
|
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<!--End of Navigation Panel-->
|
||||
|
||||
<H3><A NAME="SECTION00083100000000000000">
|
||||
@@ -61,37 +62,36 @@ Arguments</A>
|
||||
<P>
|
||||
<TABLE CELLPADDING=3>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><code>a</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298><code>type(psb_</code><I>x</I><code>spmat_type), intent(in)</code>.</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><code>type(psb_</code><SPAN CLASS="textit">x</SPAN><code>spmat_type), intent(in)</code>.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34> </TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298>The sparse matrix structure containing the local part of the
|
||||
matrix to be preconditioned. Note that <I>x</I> must be chosen according
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340>The sparse matrix structure containing the local part of the
|
||||
matrix to be preconditioned. Note that <SPAN CLASS="textit">x</SPAN> must be chosen according
|
||||
to the real/complex,
|
||||
single/double precision version of MLD2P4 under use.
|
||||
See the PSBLAS User's Guide for details [<A
|
||||
HREF="node30.html#PSBLASGUIDE">11</A>].</TD>
|
||||
HREF="node30.html#PSBLASGUIDE">14</A>].</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><code>desc_a</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298><code>type(psb_desc_type), intent(in)</code>.</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><code>type(psb_desc_type), intent(in)</code>.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34> </TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298>The communication descriptor of <code>a</code>. See the PSBLAS User's Guide for
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340>The communication descriptor of <code>a</code>. See the PSBLAS User's Guide for
|
||||
details [<A
|
||||
HREF="node30.html#PSBLASGUIDE">11</A>].</TD>
|
||||
HREF="node30.html#PSBLASGUIDE">14</A>].</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><code>p</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298><code>type(mld_</code><I>x</I><code>prec_type), intent(inout)</code>.</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><code>type(mld_</code><SPAN CLASS="textit">x</SPAN><code>prec_type), intent(inout)</code>.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34> </TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298>The preconditioner data structure. Note that <I>x</I> must be chosen according
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340>The preconditioner data structure. Note that <SPAN CLASS="textit">x</SPAN> must be chosen according
|
||||
to the real/complex, single/double precision version of MLD2P4 under use.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><code>info</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298><code>integer, intent(out)</code>.</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><code>integer, intent(out)</code>.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34> </TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298>Error code. If no error, 0 is returned. See Section <A HREF="node28.html#sec:errors"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A> for details.</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340>Error code. If no error, 0 is returned. See Section <A HREF="node28.html#sec:errors">7</A> for details.</TD>
|
||||
</TR>
|
||||
</TABLE>
|
||||
|
||||
@@ -100,7 +100,7 @@ single/double precision version of MLD2P4 under use.
|
||||
<BR><HR>
|
||||
<ADDRESS>
|
||||
Salvatore Filippone
|
||||
2008-07-22
|
||||
2008-07-23
|
||||
</ADDRESS>
|
||||
</BODY>
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</HTML>
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+26
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<H2><A NAME="SECTION00084000000000000000"></A><A NAME="sec:precaply"></A>
|
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@@ -71,17 +72,17 @@ Subroutine mld_precaply
|
||||
This routine computes <!-- MATH
|
||||
$y = op(M^{-1})\, x$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="117" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
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SRC="img93.png"
|
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ALT="$y = op(M^{-1}) x$">, where <IMG
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<SPAN CLASS="MATH"><IMG
|
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WIDTH="118" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
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SRC="img94.png"
|
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ALT="$y = op(M^{-1}) x$"></SPAN>, where <SPAN CLASS="MATH"><IMG
|
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WIDTH="23" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img59.png"
|
||||
ALT="$M$"> is a previously built
|
||||
preconditioner, stored into <code>p</code>, and <IMG
|
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WIDTH="22" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
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SRC="img94.png"
|
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ALT="$op$">
|
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ALT="$M$"></SPAN> is a previously built
|
||||
preconditioner, stored into <code>p</code>, and <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="21" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img95.png"
|
||||
ALT="$op$"></SPAN>
|
||||
denotes the preconditioner itself or its transpose, according to
|
||||
the value of <code>trans</code>.
|
||||
Note that, when MLD2P4 is used with a Krylov solver from PSBLAS,
|
||||
@@ -93,15 +94,15 @@ and hence it is completely transparent to the user.
|
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<!--Table of Child-Links-->
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<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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<LI><A NAME="tex2html341"
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<UL CLASS="ChildLinks">
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HREF="node23.html">Arguments</A>
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</UL>
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<!--End of Table of Child-Links-->
|
||||
<BR><HR>
|
||||
<ADDRESS>
|
||||
Salvatore Filippone
|
||||
2008-07-22
|
||||
2008-07-23
|
||||
</ADDRESS>
|
||||
</BODY>
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</HTML>
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+70
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@@ -1,4 +1,4 @@
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@@ -24,34 +24,35 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<B> Up:</B> <A NAME="tex2html346"
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<H3><A NAME="SECTION00084100000000000000">
|
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@@ -61,128 +62,130 @@ Arguments</A>
|
||||
<P>
|
||||
<TABLE CELLPADDING=3>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><code>p</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298><code>type(mld_</code><I>x</I><code>prec_type), intent(inout)</code>.</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><code>type(mld_</code><SPAN CLASS="textit">x</SPAN><code>prec_type), intent(inout)</code>.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34> </TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298>The preconditioner data structure, containing the local part of <IMG
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340>The preconditioner data structure, containing the local part of <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="23" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img59.png"
|
||||
ALT="$M$">.
|
||||
Note that <I>x</I> must be chosen according
|
||||
ALT="$M$"></SPAN>.
|
||||
Note that <SPAN CLASS="textit">x</SPAN> must be chosen according
|
||||
to the real/complex, single/double precision version of MLD2P4 under use.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><code>x</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298><I>type</I><code>(</code><I>kind_parameter</I><code>), dimension(:), intent(in)</code>.</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><SPAN CLASS="textit">type</SPAN><code>(</code><SPAN CLASS="textit">kind_parameter</SPAN><code>), dimension(:), intent(in)</code>.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34> </TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298>The local part of the vector <IMG
|
||||
WIDTH="14" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img95.png"
|
||||
ALT="$x$">. Note that <I>type</I> and
|
||||
<I>kind_parameter</I> must be chosen according
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340>The local part of the vector <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img96.png"
|
||||
ALT="$x$"></SPAN>. Note that <SPAN CLASS="textit">type</SPAN> and
|
||||
<SPAN CLASS="textit">kind_parameter</SPAN> must be chosen according
|
||||
to the real/complex, single/double precision version of MLD2P4 under use.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><code>y</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298><I>type</I><code>(</code><I>kind_parameter</I><code>), dimension(:), intent(out)</code>.</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><SPAN CLASS="textit">type</SPAN><code>(</code><SPAN CLASS="textit">kind_parameter</SPAN><code>), dimension(:), intent(out)</code>.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34> </TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298>The local part of the vector <IMG
|
||||
WIDTH="14" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img96.png"
|
||||
ALT="$y$">. Note that <I>type</I> and
|
||||
<I>kind_parameter</I> must be chosen according
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340>The local part of the vector <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img97.png"
|
||||
ALT="$y$"></SPAN>. Note that <SPAN CLASS="textit">type</SPAN> and
|
||||
<SPAN CLASS="textit">kind_parameter</SPAN> must be chosen according
|
||||
to the real/complex, single/double precision version of MLD2P4 under use.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><code>desc_a</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298><code>type(psb_desc_type), intent(in)</code>.</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><code>type(psb_desc_type), intent(in)</code>.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34> </TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298>The communication descriptor associated to the matrix to be
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340>The communication descriptor associated to the matrix to be
|
||||
preconditioned.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><code>info</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298><code>integer, intent(out)</code>.</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><code>integer, intent(out)</code>.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34> </TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298>Error code. If no error, 0 is returned. See Section <A HREF="node28.html#sec:errors"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A> for details.</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340>Error code. If no error, 0 is returned. See Section <A HREF="node28.html#sec:errors">7</A> for details.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><code>trans</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298><code>character(len=1), optional, intent(in).</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><code>character(len=1), optional, intent(in).</code></TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34> </TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298>If <code>trans</code> = <code>'N','n'</code> then <!-- MATH
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340>If <code>trans</code> = <code>'N','n'</code> then <!-- MATH
|
||||
$op(M^{-1}) = M^{-1}$
|
||||
-->
|
||||
<IMG
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="132" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img97.png"
|
||||
ALT="$op(M^{-1}) = M^{-1}$">;
|
||||
SRC="img98.png"
|
||||
ALT="$op(M^{-1}) = M^{-1}$"></SPAN>;
|
||||
if <code>trans</code> = <code>'T','t'</code> then <!-- MATH
|
||||
$op(M^{-1}) = M^{-T}$
|
||||
-->
|
||||
<IMG
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="135" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img98.png"
|
||||
ALT="$op(M^{-1}) = M^{-T}$">
|
||||
(transpose of <IMG
|
||||
WIDTH="48" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img99.png"
|
||||
ALT="$M^{-1})$">; if <code>trans</code> = <code>'C','c'</code> then <!-- MATH
|
||||
ALT="$op(M^{-1}) = M^{-T}$"></SPAN>
|
||||
(transpose of <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="48" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img100.png"
|
||||
ALT="$M^{-1})$"></SPAN>; if <code>trans</code> = <code>'C','c'</code> then <!-- MATH
|
||||
$op(M^{-1}) = M^{-C}$
|
||||
-->
|
||||
<IMG
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="136" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img100.png"
|
||||
ALT="$op(M^{-1}) = M^{-C}$">
|
||||
(conjugate transpose of <IMG
|
||||
SRC="img101.png"
|
||||
ALT="$op(M^{-1}) = M^{-C}$"></SPAN>
|
||||
(conjugate transpose of <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="48" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img99.png"
|
||||
ALT="$M^{-1})$">.</TD>
|
||||
SRC="img100.png"
|
||||
ALT="$M^{-1})$"></SPAN>.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><code>work</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298><I>type</I><code>(</code><I>kind_parameter</I><code>), dimension(:), optional, target</code>.</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><SPAN CLASS="textit">type</SPAN><code>(</code><SPAN CLASS="textit">kind_parameter</SPAN><code>), dimension(:), optional, target</code>.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34> </TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298>Workspace. Its size should be at
|
||||
least <code>4 * psb_cd_get_local_cols(desc_a)</code> (see the PSBLAS User's Guide).
|
||||
Note that <I>type</I> and <I>kind_parameter</I> must be chosen according
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340>Workspace. Its size should be at
|
||||
least <code>4 * psb_cd_get_local_</code> <code>cols(desc_a)</code> (see the PSBLAS User's Guide).
|
||||
Note that <SPAN CLASS="textit">type</SPAN> and <SPAN CLASS="textit">kind_parameter</SPAN> must be chosen according
|
||||
to the real/complex, single/double precision version of MLD2P4 under use.</TD>
|
||||
</TR>
|
||||
</TABLE>
|
||||
|
||||
<P>
|
||||
<HR>
|
||||
|
||||
<P>
|
||||
|
||||
<DIV CLASS="navigation"><HR>
|
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<A NAME="tex2html350"
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SRC="file:/usr/share/latex2html/icons/next.png"></A>
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HREF="node22.html">
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<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
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SRC="file:/usr/share/latex2html/icons/up.png"></A>
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<B> Next:</B> <A NAME="tex2html350"
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HREF="node24.html">Subroutine mld_precfree</A>
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<B> Up:</B> <A NAME="tex2html347"
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<B> Up:</B> <A NAME="tex2html346"
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HREF="node22.html">Subroutine mld_precaply</A>
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HREF="node2.html">Contents</A></B> </DIV>
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<!--End of Navigation Panel-->
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<ADDRESS>
|
||||
Salvatore Filippone
|
||||
2008-07-22
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2008-07-23
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</ADDRESS>
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<H2><A NAME="SECTION00085000000000000000"></A><A NAME="sec:precfree"></A>
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@@ -74,15 +75,15 @@ This routine deallocates the preconditioner data structure.
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<BR><HR>
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<ADDRESS>
|
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Salvatore Filippone
|
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2008-07-22
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2008-07-23
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</ADDRESS>
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<B> Next:</B> <A NAME="tex2html373"
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HREF="node26.html">Subroutine mld_precdescr</A>
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<H3><A NAME="SECTION00085100000000000000">
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@@ -61,26 +62,27 @@ Arguments</A>
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<P>
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<TABLE CELLPADDING=3>
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<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><code>p</code></TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298><code>type(mld_</code><I>x</I><code>prec_type), intent(inout)</code>.</TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298><code>type(mld_</code><SPAN CLASS="textit">x</SPAN><code>prec_type), intent(inout)</code>.</TD>
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</TR>
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<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34> </TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298>The preconditioner data structure. Note that <I>x</I> must be chosen according
|
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298>The preconditioner data structure. Note that <SPAN CLASS="textit">x</SPAN> must be chosen according
|
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to the real/complex, single/double precision version of MLD2P4 under use.</TD>
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</TR>
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<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><code>info</code></TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298><code>integer, intent(out)</code>.</TD>
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</TR>
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<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34> </TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298>Error code. If no error, 0 is returned. See Section <A HREF="node28.html#sec:errors"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
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SRC="file:/usr/share/latex2html/icons/crossref.png"></A> for details.</TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=298>Error code. If no error, 0 is returned. See Section <A HREF="node28.html#sec:errors">7</A> for details.</TD>
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<P>
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<P>
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<BR><HR>
|
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<ADDRESS>
|
||||
Salvatore Filippone
|
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2008-07-22
|
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2008-07-23
|
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</ADDRESS>
|
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</BODY>
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original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -24,34 +24,35 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<H2><A NAME="SECTION00086000000000000000"></A><A NAME="sec:precdescr"></A>
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@@ -67,23 +68,23 @@ Subroutine mld_precdescr
|
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</DIV>
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||||
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<P>
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This routine prints a description of the preconditioner
|
||||
to the standard output or to a file.
|
||||
This routine prints a description of the preconditioner to the standard output or
|
||||
to a file. It must be called after <code>mld_precbld</code> has been called.
|
||||
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<P>
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<BR><HR>
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<!--Table of Child-Links-->
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HREF="node27.html">Arguments</A>
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</UL>
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<!--End of Table of Child-Links-->
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<BR><HR>
|
||||
<ADDRESS>
|
||||
Salvatore Filippone
|
||||
2008-07-22
|
||||
2008-07-23
|
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</ADDRESS>
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</BODY>
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</HTML>
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+21
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<!--Converted with LaTeX2HTML 2002-2-1 (1.71)
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original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -24,34 +24,35 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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</HEAD>
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<BODY >
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<B> Up:</B> <A NAME="tex2html390"
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HREF="node26.html">Subroutine mld_precdescr</A>
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<B> Previous:</B> <A NAME="tex2html387"
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<B> Previous:</B> <A NAME="tex2html386"
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HREF="node26.html">Subroutine mld_precdescr</A>
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HREF="node1.html">Contents</A></B>
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<H3><A NAME="SECTION00086100000000000000">
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@@ -61,24 +62,23 @@ Arguments</A>
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<P>
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||||
<TABLE CELLPADDING=3>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><code>p</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=300><code>type(mld_</code><I>x</I><code>prec_type), intent(in)</code>.</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><code>type(mld_</code><SPAN CLASS="textit">x</SPAN><code>prec_type), intent(in)</code>.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34> </TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=300>The preconditioner data structure. Note that <I>x</I> must be chosen according
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340>The preconditioner data structure. Note that <SPAN CLASS="textit">x</SPAN> must be chosen according
|
||||
to the real/complex, single/double precision version of MLD2P4 under use.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><code>info</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=300><code>integer, intent(out)</code>.</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><code>integer, intent(out)</code>.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34> </TD>
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<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=300>Error code. If no error, 0 is returned. See Section <A HREF="node28.html#sec:errors"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||||
SRC="file:/usr/share/latex2html/icons/crossref.png"></A> for details.</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340>Error code. If no error, 0 is returned. See Section <A HREF="node28.html#sec:errors">7</A> for details.</TD>
|
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</TR>
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||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34><code>iout</code></TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=300><code>integer, intent(in), optional</code>.</TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340><code>integer, intent(in), optional</code>.</TD>
|
||||
</TR>
|
||||
<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=34> </TD>
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=300>The id of the file where the preconditioner description
|
||||
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=340>The id of the file where the preconditioner description
|
||||
will be printed; the default is the standard output.</TD>
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</TR>
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</TABLE>
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@@ -87,7 +87,7 @@ Arguments</A>
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||||
<BR><HR>
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||||
<ADDRESS>
|
||||
Salvatore Filippone
|
||||
2008-07-22
|
||||
2008-07-23
|
||||
</ADDRESS>
|
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</BODY>
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</HTML>
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||||
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<H1><A NAME="SECTION00090000000000000000"></A><A NAME="sec:errors"></A>
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@@ -61,7 +62,7 @@ Error Handling
|
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</H1>
|
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|
||||
<P>
|
||||
The error handling in MLD2P4 is based on the PSBLAS version 2 error
|
||||
The error handling in MLD2P4 is based on the PSBLAS (version 2) error
|
||||
handling. Error conditions are signaled via an integer argument
|
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<code>info</code>; whenever an error condition is detected, an error trace
|
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stack is built by the library up to the top-level, user-callable
|
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@@ -69,16 +70,17 @@ routine. This routine will then decide, according to the user
|
||||
preferences, whether the error should be handled by terminating the
|
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program or by returning the error condition to the user code, which
|
||||
will then take action, and whether
|
||||
an error message should be printed. These options may be set by using
|
||||
an error message should be printed. These options may be set by using
|
||||
the PSBLAS error handling routines; for further details see the PSBLAS
|
||||
user's guide.
|
||||
User's Guide [<A
|
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HREF="node30.html#PSBLASGUIDE">14</A>].
|
||||
|
||||
<P>
|
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|
||||
<BR><HR>
|
||||
<ADDRESS>
|
||||
Salvatore Filippone
|
||||
2008-07-22
|
||||
2008-07-23
|
||||
</ADDRESS>
|
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</BODY>
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<H1><A NAME="SECTION000100000000000000000"></A><A NAME="sec:license"></A>
|
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@@ -100,12 +101,10 @@ ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
|
||||
POSSIBILITY OF SUCH DAMAGE.
|
||||
</PRE>
|
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|
||||
|
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<P>
|
||||
<BR><HR>
|
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<ADDRESS>
|
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Salvatore Filippone
|
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2008-07-22
|
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2008-07-23
|
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</ADDRESS>
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@@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds
|
||||
Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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<HTML>
|
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<HEAD>
|
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<TITLE>Notational Conventions</TITLE>
|
||||
<META NAME="description" CONTENT="Notational Conventions">
|
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<TITLE>General Overview</TITLE>
|
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<META NAME="description" CONTENT="General Overview">
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<H1><A NAME="SECTION00030000000000000000"></A><A NAME="sec:conventions"></A>
|
||||
<H1><A NAME="SECTION00030000000000000000"></A><A NAME="sec:overview"></A>
|
||||
<BR>
|
||||
Notational Conventions
|
||||
General Overview
|
||||
</H1>
|
||||
|
||||
<P>
|
||||
- caratteri tipografici usati nella guida (vedi guida ML recente e guida Aztec)
|
||||
<BR> - convenzioni sui nomi di routine (anche differenza nei nomi tra high-level e
|
||||
medium-level?), strutture dati, moduli, costanti, etc. (vedi guida psblas)
|
||||
<BR> - versione reale e complessa, singola e doppia precisione
|
||||
The M<SMALL>ULTI-</SMALL>L<SMALL>EVEL </SMALL>D<SMALL>OMAIN </SMALL>D<SMALL>ECOMPOSITION </SMALL>P<SMALL>ARALLEL </SMALL>P<SMALL>RECONDITIONERS </SMALL>P<SMALL>ACKAGE BASED ON
|
||||
</SMALL>PSBLAS (MLD2P4) provides <SPAN CLASS="textit">multi-level Schwarz preconditioners</SPAN> [<A
|
||||
HREF="node30.html#dd2_96">20</A>],
|
||||
to be used in the iterative solutions of sparse linear systems:
|
||||
<BR>
|
||||
<BR><HR>
|
||||
<DIV ALIGN="RIGHT" CLASS="mathdisplay">
|
||||
|
||||
<!-- MATH
|
||||
\begin{equation}
|
||||
Ax=b,
|
||||
\end{equation}
|
||||
-->
|
||||
<A NAME="system1"></A>
|
||||
<TABLE WIDTH="100%" ALIGN="CENTER">
|
||||
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="system1"></A><IMG
|
||||
WIDTH="57" HEIGHT="30" BORDER="0"
|
||||
SRC="img1.png"
|
||||
ALT="\begin{displaymath}
|
||||
Ax=b,
|
||||
\end{displaymath}"></TD>
|
||||
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
|
||||
(<SPAN CLASS="arabic">1</SPAN>)</TD></TR>
|
||||
</TABLE>
|
||||
<BR CLEAR="ALL"></DIV><P></P>
|
||||
where <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img2.png"
|
||||
ALT="$A$"></SPAN> is a square, real or complex, sparse matrix with a symmetric sparsity pattern.
|
||||
These preconditioners have the following general features:
|
||||
|
||||
<UL>
|
||||
<LI>both <SPAN CLASS="textit">additive and hybrid multilevel</SPAN> variants are implemented,
|
||||
i.e. variants that are additive among the levels and inside each level, and variants
|
||||
that are multiplicative among the levels and additive inside each level;
|
||||
the basic Additive Schwarz (AS) preconditioners are obtained by considering only one level;
|
||||
</LI>
|
||||
<LI>a <SPAN CLASS="textit">purely algebraic</SPAN> approach is used to
|
||||
generate a sequence of coarse-level corrections to a basic AS preconditioner, without
|
||||
explicitly using any information on the geometry of the original problem (e.g. the
|
||||
discretization of a PDE). The <SPAN CLASS="textit">smoothed aggregation</SPAN> technique is applied
|
||||
as algebraic coarsening strategy [<A
|
||||
HREF="node30.html#BREZINA_VANEK">1</A>,<A
|
||||
HREF="node30.html#VANEK_MANDEL_BREZINA">24</A>].
|
||||
</LI>
|
||||
</UL>
|
||||
|
||||
<P>
|
||||
The package is written in <SPAN CLASS="textit">Fortran 95</SPAN>, following an
|
||||
<SPAN CLASS="textit">object-oriented approach</SPAN> through the exploitation of features
|
||||
such as abstract data type creation, functional
|
||||
overloading and dynamic memory management.
|
||||
The parallel implementation is based
|
||||
on a Single Program Multiple Data (SPMD) paradigm for distributed-memory architectures.
|
||||
Single and double precision implementations of MLD2P4 are available for both the
|
||||
real and the complex case, that can be used through a single interface.
|
||||
|
||||
<P>
|
||||
MLD2P4 has been designed to implement scalable and easy-to-use multilevel preconditioners
|
||||
in the context of the <SPAN CLASS="textit">PSBLAS (Parallel Sparse BLAS)
|
||||
computational framework</SPAN> [<A
|
||||
HREF="node30.html#psblas_00">15</A>].
|
||||
PSBLAS is a library originally developed to address the parallel implementation of
|
||||
iterative solvers for sparse linear system, by providing basic linear algebra
|
||||
operators and data management facilities for distributed sparse matrices; it
|
||||
also includes parallel Krylov solvers, built on the top of the basic PSBLAS kernels.
|
||||
The preconditioners available in MLD2P4 can be used with these Krylov solvers.
|
||||
The choice of PSBLAS has been mainly motivated by the need of having
|
||||
a portable and efficient software infrastructure implementing ``de facto'' standard
|
||||
parallel sparse linear algebra kernels, to pursue goals such as performance,
|
||||
portability, modularity ed extensibility in the development of the preconditioner
|
||||
package. On the other hand, the implementation of MLD2P4 has led to some
|
||||
revisions and extentions of the PSBLAS kernels, leading to the
|
||||
recent PSBLAS 2.0 version [<A
|
||||
HREF="node30.html#PSBLASGUIDE">14</A>]. The inter-process comunication required
|
||||
by MLD2P4 is encapsulated into the PSBLAS routines, except few cases where
|
||||
MPI [<A
|
||||
HREF="node30.html#MPI1">21</A>] is explicitly called. Therefore, MLD2P4 can be run on any parallel
|
||||
machine where PSBLAS and MPI implementations are available.
|
||||
|
||||
<P>
|
||||
MLD2P4 has a layered and modular software architecture where three main layers can be identified.
|
||||
The lower layer consists of the PSBLAS kernels, the middle one implements
|
||||
the construction and application phases of the preconditioners, and the upper one
|
||||
provides a uniform and easy-to-use interface to all the preconditioners.
|
||||
This architecture allows for different levels of use of the package:
|
||||
few black-box routines at the upper layer allow non-expert users to easily
|
||||
build any preconditioner available in MLD2P4 and to apply it within a PSBLAS Krylov solver.
|
||||
On the other hand, the routines of the middle and lower layer can be used and extended
|
||||
by expert users to build new versions of multi-level Schwarz preconditioners.
|
||||
We provide here a description of the upper-layer routines, but not of the
|
||||
medium-layer ones.
|
||||
|
||||
<P>
|
||||
This guide is organized as follows. General information on the distribution of the source code
|
||||
is reported in Section <A HREF="node4.html#sec:distribution">2</A>, while details on the configuration
|
||||
and installation of the package are given in Section <A HREF="node5.html#sec:building">3</A>. A description of
|
||||
multi-level Schwarz preconditioners based on smoothed aggregation is provided
|
||||
in Section <A HREF="node10.html#sec:background">4</A>, to help the users in choosing among the different preconditioners
|
||||
implemented in MLD2P4. The basics for building and applying the preconditioners
|
||||
with the Krylov solvers implemented in PSBLAS are reported in Section <A HREF="node13.html#sec:started">5</A>, where the
|
||||
Fortran 95 codes of a few sample programs are also shown. A reference guide for
|
||||
the upper-layer routines of MLD2P4, that are the user interface, is provided
|
||||
in Section <A HREF="node15.html#sec:userinterface">6</A>. The error handling mechanism used by the package is briefly described
|
||||
in Section <A HREF="node28.html#sec:errors">7</A>. The copyright terms concerning the distribution and modification
|
||||
of MLD2P4 are reported in Appendix <A HREF="node29.html#sec:license">A</A>.
|
||||
|
||||
<P>
|
||||
|
||||
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|
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|
||||
<ADDRESS>
|
||||
Salvatore Filippone
|
||||
2008-07-22
|
||||
2008-07-23
|
||||
</ADDRESS>
|
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</BODY>
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</HTML>
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+56
-26
@@ -1,4 +1,4 @@
|
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
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|
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<!--Converted with LaTeX2HTML 2002-2-1 (1.71)
|
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original version by: Nikos Drakos, CBLU, University of Leeds
|
||||
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<H2><A NAME="SECTION000110000000000000000">
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Bibliography</A>
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</H2><DL COMPACT><DD><P></P><DT><A NAME="BREZINA_VANEK">1</A>
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||||
</H2><DL COMPACT><DD>
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<P>
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<P></P><DT><A NAME="BREZINA_VANEK">1</A>
|
||||
<DD>
|
||||
M. Brezina, P. Vanek,
|
||||
<EM>A Black-Box Iterative Solver Based on a Two-Level Schwarz Method</EM>,
|
||||
@@ -110,57 +114,83 @@ ACM Transactions on Mathematical Software, 30, 2004, 196-199.
|
||||
J.W. Demmel, S.C. Eisenstat, J.R. Gilbert, X.S. Li and J.W.H. Liu,
|
||||
A supernodal approach to sparse partial pivoting,
|
||||
SIAM Journal on Matrix Analysis and Applications, 20, 3, 1999, 720-755.
|
||||
<P></P><DT><A NAME="EFSTATHIOU">10</A>
|
||||
<P></P><DT><A NAME="blas3">10</A>
|
||||
<DD>
|
||||
J. J. Dongarra, J. Du Croz, I. S. Duff, S. Hammarling,
|
||||
<SPAN CLASS="textit">A set of Level 3 Basic Linear Algebra Subprograms</SPAN>,
|
||||
ACM Transactions on Mathematical Software, 16, 1990, 1-17.
|
||||
<P></P><DT><A NAME="blas2">11</A>
|
||||
<DD>
|
||||
J. J. Dongarra, J. Du Croz, S. Hammarling, R. J. Hanson,
|
||||
<SPAN CLASS="textit">An extended set of FORTRAN Basic Linear Algebra Subprograms</SPAN>,
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||||
ACM Transactions on Mathematical Software, 14, 1988, 1-17.
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||||
<P></P><DT><A NAME="BLACS">12</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
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||||
Tennessee, March 1995 (updated May 1997).
|
||||
<P></P><DT><A NAME="EFSTATHIOU">13</A>
|
||||
<DD>
|
||||
E. Efstathiou, J. G. Gander,
|
||||
<EM>Why Restricted Additive Schwarz Converges Faster than Additive Schwarz</EM>,
|
||||
BIT Numerical Mathematics, 43, 2003, 945-959.
|
||||
<P></P><DT><A NAME="PSBLASGUIDE">11</A>
|
||||
<P></P><DT><A NAME="PSBLASGUIDE">14</A>
|
||||
<DD>
|
||||
S. Filippone, A. Buttari,
|
||||
<EM>PSBLAS-2.3 User's Guide. A Reference Guide for the Parallel Sparse BLAS Library</EM>,
|
||||
available from <TT>http://www.ce.uniroma2.it/psblas/</TT>.
|
||||
<P></P><DT><A NAME="psblas_00">12</A>
|
||||
<P></P><DT><A NAME="psblas_00">15</A>
|
||||
<DD>
|
||||
S. Filippone, M. Colajanni,
|
||||
<EM>PSBLAS: A Library for Parallel Linear Algebra
|
||||
Computation on Sparse Matrices</EM>,
|
||||
ACM Transactions on Mathematical Software, 26, 4, 2000, 527-550.
|
||||
<P></P><DT><A NAME="SUPERLUDIST">13</A>
|
||||
<P></P><DT><A NAME="MPI2">16</A>
|
||||
<DD>
|
||||
W. Gropp, S. Huss-Lederman, A. Lumsdaine, E. Lusk, B. Nitzberg, W. Saphir, M. Snir,
|
||||
<EM>MPI: The Complete Reference. Volume 2 - The MPI-2 Extensions</EM>,
|
||||
MIT Press, 1998.
|
||||
<P></P><DT><A NAME="blas1">17</A>
|
||||
<DD>
|
||||
C. L. Lawson, R. J. Hanson, D. Kincaid, F. T. Krogh,
|
||||
<SPAN CLASS="textit">Basic Linear Algebra Subprograms for FORTRAN usage</SPAN>,
|
||||
ACM Transactions on Mathematical Software, 5, 1979, 308-323.
|
||||
<P></P><DT><A NAME="SUPERLUDIST">18</A>
|
||||
<DD>
|
||||
X. S. Li, J. W. Demmel, <EM>SuperLU_DIST: A Scalable Distributed-memory
|
||||
Sparse Direct Solver for Unsymmetric Linear Systems</EM>,
|
||||
ACM Transactions on Mathematical Software, 29, 2, 2003, 110-140.
|
||||
<P></P><DT><A NAME="Saad_book">14</A>
|
||||
<P></P><DT><A NAME="Saad_book">19</A>
|
||||
<DD>
|
||||
Y. Saad,
|
||||
<I>Iterative methods for sparse linear systems</I>, 2nd edition,
|
||||
<SPAN CLASS="textit">Iterative methods for sparse linear systems</SPAN>, 2nd edition,
|
||||
SIAM, 2003
|
||||
|
||||
<P>
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||||
<P></P><DT><A NAME="dd2_96">15</A>
|
||||
<P></P><DT><A NAME="dd2_96">20</A>
|
||||
<DD>
|
||||
B. Smith, P. Bjorstad, W. Gropp,
|
||||
<EM>Domain Decomposition: Parallel Multilevel Methods for Elliptic
|
||||
Partial Differential Equations</EM>,
|
||||
Cambridge University Press, 1996.
|
||||
<P></P><DT><A NAME="MPI1">16</A>
|
||||
<P></P><DT><A NAME="MPI1">21</A>
|
||||
<DD>
|
||||
M. Snir, S. Otto, S. Huss-Lederman, D. Walker, J. Dongarra,
|
||||
<EM>MPI: The Complete Reference. Volume 1 - The MPI Core</EM>, second edition,
|
||||
MIT Press, 1998.
|
||||
<P></P><DT><A NAME="StubenGMD69_99">17</A>
|
||||
<P></P><DT><A NAME="StubenGMD69_99">22</A>
|
||||
<DD>
|
||||
K. Stüben,
|
||||
<EM>Algebraic Multigrid (AMG): an Introduction with Applications</EM>,
|
||||
in A. Schüller, U. Trottenberg, C. Oosterlee, editors, Multigrid,
|
||||
Academic Press, 2000.
|
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<P></P><DT><A NAME="TUMINARO_TONG">18</A>
|
||||
<P></P><DT><A NAME="TUMINARO_TONG">23</A>
|
||||
<DD>
|
||||
R. S. Tuminaro, C. Tong,
|
||||
<EM>Parallel Smoothed Aggregation Multigrid: Aggregation Strategies on Massively Parallel Machines</EM>,
|
||||
in J. Donnelley, editor, Proceedings of SuperComputing 2000, Dallas, 2000.
|
||||
<P></P><DT><A NAME="VANEK_MANDEL_BREZINA">19</A>
|
||||
<P></P><DT><A NAME="VANEK_MANDEL_BREZINA">24</A>
|
||||
<DD>
|
||||
P. Vanek, J. Mandel and M. Brezina,
|
||||
<EM>Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems</EM>,
|
||||
@@ -173,7 +203,7 @@ Computing, 56, 1996, 179-196.
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<BR><HR>
|
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<ADDRESS>
|
||||
Salvatore Filippone
|
||||
2008-07-22
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2008-07-23
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</ADDRESS>
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</HTML>
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original version by: Nikos Drakos, CBLU, University of Leeds
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</HEAD>
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<H1><A NAME="SECTION000120000000000000000">
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@@ -65,13 +66,13 @@ Copyright © 1997, 1998, 1999,
|
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Mathematics Department, Macquarie University, Sydney.
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<P>
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The command line arguments were: <BR>
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<STRONG>latex2html</STRONG> <TT>-dir ../html userhtml.tex</TT>
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<STRONG>latex2html</STRONG> <TT>-html_version 4.0 -dir ../../html -mkdir -debug userhtml.tex</TT>
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<P>
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The translation was initiated by Salvatore Filippone on 2008-07-22
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The translation was initiated by Salvatore Filippone on 2008-07-23
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<BR><HR>
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<ADDRESS>
|
||||
Salvatore Filippone
|
||||
2008-07-22
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2008-07-23
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</ADDRESS>
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</BODY>
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</HTML>
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
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original version by: Nikos Drakos, CBLU, University of Leeds
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@@ -25,34 +25,35 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<H1><A NAME="SECTION00040000000000000000"></A><A NAME="sec:distribution"></A>
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@@ -61,25 +62,25 @@ Code Distribution
|
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</H1>
|
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|
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<P>
|
||||
MLD2P4 is available from our project web site
|
||||
MLD2P4 is available from the web site
|
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<BLOCKQUOTE>
|
||||
<TT>http://www.mld2p4.it
|
||||
<TT>http://www.mld2p4.it</TT>
|
||||
|
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</TT></BLOCKQUOTE>
|
||||
where you will also find contact points for further information and
|
||||
bug reports.
|
||||
</BLOCKQUOTE>
|
||||
where contact points for further information can be also found.
|
||||
To report bugs or ask general usage questions, please, send an email to
|
||||
<TT>bugreport@mld2p4.it</TT>.
|
||||
|
||||
<P>
|
||||
The software is available under a modified BSD license, as specified
|
||||
in appendix <A HREF="node29.html#sec:license"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
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SRC="file:/usr/share/latex2html/icons/crossref.png"></A>; please note that some of the optional
|
||||
in Appendix <A HREF="node29.html#sec:license">A</A>; please note that some of the optional
|
||||
third party libraries may be licensed under a different and more
|
||||
stringent license, most notably the GPL, and this should be taken into
|
||||
account when treating derived works.
|
||||
<BR><HR>
|
||||
<ADDRESS>
|
||||
Salvatore Filippone
|
||||
2008-07-22
|
||||
2008-07-23
|
||||
</ADDRESS>
|
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</BODY>
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</HTML>
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<H1><A NAME="SECTION00050000000000000000"></A><A NAME="sec:building"></A>
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@@ -63,18 +64,18 @@ Configuring and Building MLD2P4
|
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To build MLD2P4 it is necessary to set up a Makefile with appropriate
|
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values for your system; this is done by means of the <code>configure</code>
|
||||
script. The distribution also includes the autoconf and automake
|
||||
sources employed to generate the script, but this should not normally
|
||||
be needed to build the software.
|
||||
sources employed to generate the script, but usually this is not needed
|
||||
to build the software.
|
||||
|
||||
<P>
|
||||
MLD2P4 is implemented almost entirely in Fortran 95, with some
|
||||
interfaces to external libraries in C; we require the Fortran compiler
|
||||
to support the Fortran 95 standard plus the extension TR15581, which
|
||||
interfaces to external libraries in C; the Fortran compiler
|
||||
must support the Fortran 95 standard plus the extension TR15581, which
|
||||
enhances the usability of <code>ALLOCATABLE</code> variables. Most modern
|
||||
Fortran compilers support this language level. In particular, this is
|
||||
supported by the GNU Fortran compiler as of version 4.2.0; however we
|
||||
recommend to use the latest available release (4.3.1 at the time of
|
||||
this writing§).
|
||||
this writing).
|
||||
The software defines data types and interfaces for
|
||||
real and complex data, in both single and double precision.
|
||||
|
||||
@@ -83,21 +84,21 @@ real and complex data, in both single and double precision.
|
||||
<!--Table of Child-Links-->
|
||||
<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
|
||||
|
||||
<UL>
|
||||
<LI><A NAME="tex2html127"
|
||||
<UL CLASS="ChildLinks">
|
||||
<LI><A NAME="tex2html126"
|
||||
HREF="node6.html">Prerequisites</A>
|
||||
<LI><A NAME="tex2html128"
|
||||
<LI><A NAME="tex2html127"
|
||||
HREF="node7.html">Optional third party libraries</A>
|
||||
<LI><A NAME="tex2html129"
|
||||
<LI><A NAME="tex2html128"
|
||||
HREF="node8.html">Configuration options</A>
|
||||
<LI><A NAME="tex2html130"
|
||||
<LI><A NAME="tex2html129"
|
||||
HREF="node9.html">Example and test programs</A>
|
||||
</UL>
|
||||
<!--End of Table of Child-Links-->
|
||||
<BR><HR>
|
||||
<ADDRESS>
|
||||
Salvatore Filippone
|
||||
2008-07-22
|
||||
2008-07-23
|
||||
</ADDRESS>
|
||||
</BODY>
|
||||
</HTML>
|
||||
|
||||
+64
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||||
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|
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|
||||
@@ -25,113 +25,129 @@ original version by: Nikos Drakos, CBLU, University of Leeds
|
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|
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|
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|
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|
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|
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<H2><A NAME="SECTION00051000000000000000">
|
||||
Prerequisites</A>
|
||||
</H2>
|
||||
|
||||
<P>
|
||||
The following base libraries are needed:
|
||||
<DL>
|
||||
<DT><STRONG>BLAS</STRONG></DT>
|
||||
<DD>The Basic Linear Algebra subprograms. Many vendors provide
|
||||
optimized versions; if no vendor version is available for a given
|
||||
platform, the ATLAS software <code>http://www.netlib.org/atlas</code>
|
||||
<DD>The Basic Linear Algebra subprograms [<A
|
||||
HREF="node30.html#blas3">10</A>,<A
|
||||
HREF="node30.html#blas3">10</A>,<A
|
||||
HREF="node30.html#blas1">17</A>].
|
||||
Many vendors provide optimized versions; if no vendor version is
|
||||
available for a given platform, the ATLAS software
|
||||
(<code>http://math-atlas.sourceforge.net/</code>)
|
||||
may be employed. The reference BLAS from Netlib
|
||||
<code>http://www.netlib.org/blas</code> are meant to define the standard
|
||||
behaviour of the BLAS interface, so they not optimized for any
|
||||
particular plaftorm, and should only be used as a last
|
||||
resort. Note that BLAS computation form a relatively small part of
|
||||
(<code>http://www.netlib.org/blas</code>) are meant to define the standard
|
||||
behaviour of the BLAS interface, so they are not optimized for any
|
||||
particular plaftorm, and should only be used as a last
|
||||
resort. Note that BLAS computation form a relatively small part of
|
||||
the MLD2P4/PSBLAS computations; they are however critical when using
|
||||
preconditioners based on the UMFPACK or SuperLU third party
|
||||
libraries.
|
||||
</DD>
|
||||
<DT><STRONG>MPI</STRONG></DT>
|
||||
<DD>A version of MPI is available on most high performance
|
||||
computing system; we only require version 1.1.
|
||||
<DD>A version of MPI [<A
|
||||
HREF="node30.html#MPI2">16</A>,<A
|
||||
HREF="node30.html#MPI1">21</A>] is available on most high performance
|
||||
computing system; only version 1.1 is required.
|
||||
</DD>
|
||||
<DT><STRONG>BLACS</STRONG></DT>
|
||||
<DD>The Basic Linear Algebra Communication Subroutines are
|
||||
available in source form from <code>http://www.netlib.org/blacs</code>;
|
||||
<DD>The Basic Linear Algebra Communication Subroutines
|
||||
[<A
|
||||
HREF="node30.html#BLACS">12</A>] are available in source form from <code>http://www.netlib.org/blacs</code>;
|
||||
some vendors include them in their parallel computing
|
||||
support libraries.
|
||||
|
||||
</DD>
|
||||
<DT><STRONG>PSBLAS</STRONG></DT>
|
||||
<DD>Parallel Sparse BLAS [<A
|
||||
HREF="node30.html#PSBLASGUIDE">14</A>,<A
|
||||
HREF="node30.html#psblas_00">15</A>] is
|
||||
available from
|
||||
<BR><code>http://www.ce.uniroma2.it/psblas</code>; indeed, all the
|
||||
prerequisites listed so far are also prerequisites of PSBLAS.
|
||||
Version 2.3 (or later) is required. To build the MLD2P4 library
|
||||
it is necessary to get access to
|
||||
the source PSBLAS directory employed to build the version under use; after
|
||||
the build process completes, only the compiled form of the library is
|
||||
necessary to build user applications.
|
||||
</DD>
|
||||
</DL>
|
||||
|
||||
<P>
|
||||
The MLD2P4 software requires PSBLAS version 2.3 (at least), available
|
||||
from <code>http://www.ce.uniroma2.it/psblas</code>; indeed, all the
|
||||
prerequisites listed so far are also prerequisites of PSBLAS. Please
|
||||
note that to build the MLD2P4 library it is necessary to get access to
|
||||
the source PSBLAS directory used to build the version under use; after
|
||||
the build process completes, only the compiled form of the library is
|
||||
necessary to build user applications.
|
||||
|
||||
<P>
|
||||
Please note that all the libraries listed so fare (BLAS, MPI, BLACS,
|
||||
PSBLAS) must have Fortran interfaces compatible with the MLD2P4;
|
||||
Please note that the four previous libraries must have Fortran
|
||||
interfaces compatible with MLD2P4;
|
||||
usually this means that they should all be built with the same
|
||||
compiler.
|
||||
compiler as MLD2P4.
|
||||
|
||||
<P>
|
||||
<HR>
|
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<DIV CLASS="navigation"><HR>
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|
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<B> Previous:</B> <A NAME="tex2html132"
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<B> <A NAME="tex2html140"
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HREF="node1.html">Contents</A></B>
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<B> <A NAME="tex2html139"
|
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HREF="node2.html">Contents</A></B> </DIV>
|
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<!--End of Navigation Panel-->
|
||||
<ADDRESS>
|
||||
Salvatore Filippone
|
||||
2008-07-22
|
||||
2008-07-23
|
||||
</ADDRESS>
|
||||
</BODY>
|
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</HTML>
|
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|
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+30
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@@ -1,4 +1,4 @@
|
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
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|
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<!--Converted with LaTeX2HTML 2002-2-1 (1.71)
|
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original version by: Nikos Drakos, CBLU, University of Leeds
|
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@@ -25,34 +25,35 @@ original version by: Nikos Drakos, CBLU, University of Leeds
|
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|
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<BODY >
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<B> <A NAME="tex2html152"
|
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HREF="node1.html">Contents</A></B>
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<!--End of Navigation Panel-->
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|
||||
<H2><A NAME="SECTION00052000000000000000">
|
||||
@@ -67,23 +68,29 @@ for multilevel preconditioners may change to reflect their presence.
|
||||
<P>
|
||||
<DL>
|
||||
<DT><STRONG>UMFPACK</STRONG></DT>
|
||||
<DD>A sparse direct factorization package available from
|
||||
<DD>[<A
|
||||
HREF="node30.html#UMFPACK">8</A>]
|
||||
A sparse direct factorization package available from
|
||||
<BR> <code>http://www.cise.ufl.edu/research/sparse/umfpack/</code>;
|
||||
provides serial factorization and solution for double
|
||||
precision real and complex double precision data. We have tested
|
||||
provides serial factorization and triangular system solution for double
|
||||
precision real and complex data. We have tested
|
||||
versions 4.4 and 5.1;
|
||||
</DD>
|
||||
<DT><STRONG>SuperLU</STRONG></DT>
|
||||
<DD>A sparse direct factorization package available from
|
||||
<DD>[<A
|
||||
HREF="node30.html#SUPERLU">9</A>]
|
||||
A sparse direct factorization package available from
|
||||
<BR> <code>http://crd.lbl.gov/~xiaoye/SuperLU/</code>; provides serial
|
||||
factorization and solution for single and double precision real and
|
||||
complex data. We have tested versions 3.0 and 3.1.
|
||||
factorization and triangular system solution for single and double precision,
|
||||
real and complex data. We have tested versions 3.0 and 3.1.
|
||||
</DD>
|
||||
<DT><STRONG>SuperLU_Dist</STRONG></DT>
|
||||
<DD>A sparse direct factorization package available
|
||||
<DD>[<A
|
||||
HREF="node30.html#SUPERLUDIST">18</A>]
|
||||
A sparse direct factorization package available
|
||||
from the same site as SuperLU; provides parallel factorization and
|
||||
solution for real and complex double precision data. We have tested
|
||||
version 2.1.
|
||||
triangular system solution for double precision real and complex data.
|
||||
We have tested version 2.1.
|
||||
</DD>
|
||||
</DL>
|
||||
|
||||
@@ -91,7 +98,7 @@ for multilevel preconditioners may change to reflect their presence.
|
||||
<BR><HR>
|
||||
<ADDRESS>
|
||||
Salvatore Filippone
|
||||
2008-07-22
|
||||
2008-07-23
|
||||
</ADDRESS>
|
||||
</BODY>
|
||||
</HTML>
|
||||
|
||||
+20
-18
@@ -1,4 +1,4 @@
|
||||
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
|
||||
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
|
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|
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<!--Converted with LaTeX2HTML 2002-2-1 (1.71)
|
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original version by: Nikos Drakos, CBLU, University of Leeds
|
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@@ -25,34 +25,35 @@ original version by: Nikos Drakos, CBLU, University of Leeds
|
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<H2><A NAME="SECTION00053000000000000000">
|
||||
@@ -68,9 +69,10 @@ As a minimal example consider the following:
|
||||
<PRE>
|
||||
./configure --with-psblas=/home/user/PSBLAS/psblas-2.3
|
||||
</PRE>
|
||||
This assumes that the various MPI compilers and support libraries are
|
||||
available in the standard directories on the system. Note that the
|
||||
PSBLAS build directory must be specified with an <EM>absolute</EM> path.
|
||||
which assumes that the various MPI compilers and support libraries are
|
||||
available in the standard directories on the system, and specifies
|
||||
only the PSBLAS build directory (note that the latter directory must
|
||||
be specified with an <EM>absolute</EM> path).
|
||||
The full set of options may be looked at by issuing the command
|
||||
<code>./configure --help</code>, which produces:
|
||||
<PRE>
|
||||
@@ -133,7 +135,7 @@ Optional Packages:
|
||||
--with-PACKAGE[=ARG] use PACKAGE [ARG=yes]
|
||||
--without-PACKAGE do not use PACKAGE (same as --with-PACKAGE=no)
|
||||
--with-psblas The source directory for PSBLAS, for example,
|
||||
--with-psblas=/opt/packages/psblas-2.2
|
||||
--with-psblas=/opt/packages/psblas-2.3
|
||||
--with-libs List additional link flags here. For example,
|
||||
--with-libs=-lspecial_system_lib or
|
||||
--with-libs=-L/path/to/libs
|
||||
@@ -206,7 +208,7 @@ make install
|
||||
<BR><HR>
|
||||
<ADDRESS>
|
||||
Salvatore Filippone
|
||||
2008-07-22
|
||||
2008-07-23
|
||||
</ADDRESS>
|
||||
</BODY>
|
||||
</HTML>
|
||||
|
||||
+21
-19
@@ -1,4 +1,4 @@
|
||||
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
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|
||||
original version by: Nikos Drakos, CBLU, University of Leeds
|
||||
@@ -24,38 +24,40 @@ original version by: Nikos Drakos, CBLU, University of Leeds
|
||||
</HEAD>
|
||||
|
||||
<BODY >
|
||||
<!--Navigation Panel-->
|
||||
<A NAME="tex2html175"
|
||||
|
||||
<DIV CLASS="navigation"><!--Navigation Panel-->
|
||||
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HREF="node10.html">
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SRC="file:/usr/share/latex2html/icons/next.png"></A>
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HREF="node10.html">Multi-level Domain Decomposition Background</A>
|
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<B> Up:</B> <A NAME="tex2html172"
|
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<B> Up:</B> <A NAME="tex2html171"
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HREF="node5.html">Configuring and Building MLD2P4</A>
|
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<B> Previous:</B> <A NAME="tex2html168"
|
||||
<B> Previous:</B> <A NAME="tex2html167"
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HREF="node8.html">Configuration options</A>
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<B> <A NAME="tex2html174"
|
||||
HREF="node1.html">Contents</A></B>
|
||||
<BR>
|
||||
<B> <A NAME="tex2html173"
|
||||
HREF="node2.html">Contents</A></B>
|
||||
<BR>
|
||||
<BR></DIV>
|
||||
<!--End of Navigation Panel-->
|
||||
|
||||
<H2><A NAME="SECTION00054000000000000000">
|
||||
Example and test programs</A>
|
||||
<H2><A NAME="SECTION00054000000000000000"></A><A NAME="sec:ex_and_test"></A>
|
||||
<BR>
|
||||
Example and test programs
|
||||
</H2>
|
||||
The package contains the <code>examples</code> and <code>tests</code> directories;
|
||||
both of them are further divided into <code>fileread</code> and
|
||||
@@ -75,13 +77,13 @@ both of them are further divided into <code>fileread</code> and
|
||||
</DD>
|
||||
</DL>
|
||||
The <code>fileread</code> directories contain sample programs that read
|
||||
sparse matrices from files, according to the Matrix Market storage
|
||||
format; the <code>pargen</code> instead generate matrices in full parallel
|
||||
mode from the discretization of a sample PDE.
|
||||
sparse matrices from files, according to the Matrix Market or the
|
||||
Harwell-Boeing storage format; the <code>pargen</code> instead generate
|
||||
matrices in full parallel mode from the discretization of a sample PDE.
|
||||
<BR><HR>
|
||||
<ADDRESS>
|
||||
Salvatore Filippone
|
||||
2008-07-22
|
||||
2008-07-23
|
||||
</ADDRESS>
|
||||
</BODY>
|
||||
</HTML>
|
||||
|
||||
+27
-115
@@ -1,4 +1,4 @@
|
||||
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
|
||||
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
|
||||
|
||||
<!--Converted with LaTeX2HTML 2002-2-1 (1.71)
|
||||
original version by: Nikos Drakos, CBLU, University of Leeds
|
||||
@@ -22,7 +22,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds
|
||||
</HEAD>
|
||||
|
||||
<BODY >
|
||||
<!--Navigation Panel-->
|
||||
|
||||
<DIV CLASS="navigation"><!--Navigation Panel-->
|
||||
<A NAME="tex2html13"
|
||||
HREF="node1.html">
|
||||
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
||||
@@ -32,130 +33,60 @@ original version by: Nikos Drakos, CBLU, University of Leeds
|
||||
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
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SRC="file:/usr/share/latex2html/icons/prev_g.png">
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HREF="node1.html">
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HREF="node2.html">
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<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
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SRC="file:/usr/share/latex2html/icons/contents.png"></A>
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||||
<BR>
|
||||
<B> Next:</B> <A NAME="tex2html14"
|
||||
HREF="node1.html">Contents</A>
|
||||
HREF="node1.html">Abstract</A>
|
||||
<B> <A NAME="tex2html12"
|
||||
HREF="node1.html">Contents</A></B>
|
||||
<BR>
|
||||
HREF="node2.html">Contents</A></B>
|
||||
<BR>
|
||||
<BR></DIV>
|
||||
<!--End of Navigation Panel-->
|
||||
|
||||
<P>
|
||||
|
||||
MLD2P4 User's and Reference Guidetitle
|
||||
|
||||
<BR>
|
||||
<TABLE WIDTH="100%">
|
||||
<TR><TD>
|
||||
<DIV ALIGN="RIGHT">
|
||||
<FONT SIZE="+4"><B>MLD2P4
|
||||
<BIG CLASS="XLARGE"><B>MLD2P4
|
||||
<BR>
|
||||
<BR>
|
||||
User's and Reference Guide
|
||||
</B></FONT>
|
||||
<BR>
|
||||
<BR>
|
||||
<BR>
|
||||
<BR><I><FONT SIZE="+2">A guide for the Multi-Level Domain Decomposition <BR>
|
||||
User's and Reference Guide</B></BIG>
|
||||
<BR><SPAN CLASS="textit"><BIG CLASS="LARGE">A guide for the Multi-Level Domain Decomposition <BR>
|
||||
<BR>
|
||||
Parallel Preconditioners Package
|
||||
based on PSBLAS</FONT></I>
|
||||
|
||||
</DIV></TD></TR>
|
||||
</TABLE>
|
||||
|
||||
<P>
|
||||
based on PSBLAS</BIG></SPAN>
|
||||
<B>Pasqua D'Ambra</B>
|
||||
<BR>
|
||||
<TABLE WIDTH="100%">
|
||||
<TR><TD>
|
||||
<DIV ALIGN="RIGHT">
|
||||
<FONT SIZE="+1"><B>Pasqua D'Ambra</B></FONT>
|
||||
<BR><FONT SIZE="+1">ICAR-CNR, Naples, Italy
|
||||
ICAR-CNR, Naples, Italy
|
||||
<BR>
|
||||
<BR>
|
||||
<BR>
|
||||
<BR><FONT SIZE="+1"><B>Daniela di Serafino</B></FONT>
|
||||
<BR><FONT SIZE="+1">Second University of Naples, Italy
|
||||
<BR><B>Daniela di Serafino</B>
|
||||
<BR>
|
||||
Second University of Naples, Italy
|
||||
<BR>
|
||||
<BR>
|
||||
<BR>
|
||||
<BR><FONT SIZE="+1"><B>Salvatore Filippone</B></FONT>
|
||||
<BR><FONT SIZE="+1">University of Rome ``Tor Vergata'', Italy
|
||||
|
||||
</FONT></FONT></FONT></DIV></TD></TR>
|
||||
</TABLE>
|
||||
|
||||
<P>
|
||||
<BR><B>Salvatore Filippone</B>
|
||||
<BR>
|
||||
<TABLE WIDTH="100%">
|
||||
<TR><TD>
|
||||
<DIV ALIGN="RIGHT">
|
||||
<FONT SIZE="+1">Software version: 1.0
|
||||
<BR>
|
||||
July 22, 2008
|
||||
|
||||
</FONT></DIV></TD></TR>
|
||||
</TABLE>
|
||||
|
||||
<P>
|
||||
University of Rome ``Tor Vergata'', Italy
|
||||
Software version: 1.0
|
||||
<BR>
|
||||
July 24, 2008
|
||||
|
||||
<BR>
|
||||
|
||||
<P>
|
||||
|
||||
<P>
|
||||
|
||||
<H3>Abstract:</H3>
|
||||
<DIV>
|
||||
MLD2P4 (M<SMALL>ULTI-</SMALL>L<SMALL>EVEL </SMALL>D<SMALL>OMAIN </SMALL>D<SMALL>ECOMPOSITION </SMALL>P<SMALL>ARALLEL </SMALL>P<SMALL>RECONDITIONERS </SMALL>P<SMALL>ACKAGE BASED ON
|
||||
</SMALL>PSBLAS) is a package of parallel algebraic multi-level preconditioners.
|
||||
It implements various versions of one-level additive and of multi-level additive
|
||||
and hybrid Schwarz algorithms. In the multi-level case, a purely algebraic approach
|
||||
is applied to generate coarse-level corrections, so that no geometric background is needed
|
||||
concerning the matrix to be preconditioned. The matrix is required to be square, real
|
||||
or complex, with a symmetric sparsity pattern.
|
||||
|
||||
<P>
|
||||
MLD2P4 has been designed to provide scalable and easy-to-use preconditioners in the
|
||||
context of the PSBLAS (Parallel Sparse Basic Linear Algebra Subprograms)
|
||||
computational framework and can be used in conjuction with the Krylov solvers
|
||||
available in this framework. MLD2P4 enables the user to easily specify different aspects
|
||||
of a generic algebraic multilevel Schwarz preconditioner, thus allowing to search
|
||||
for the ``best'' preconditioner for the problem at hand.
|
||||
|
||||
<P>
|
||||
The package has been designed employing object-oriented techniques,
|
||||
using Fortran 95, with interfaces to additional third party libraries
|
||||
such as UMFPACK, SuperLU and SuperLU_Dist, that
|
||||
can be exploited in building multi-level preconditioners. The parallel
|
||||
implementation is based on a Single Program Multiple Data (SPMD)
|
||||
paradigm for distributed-memory architectures; the inter-process data
|
||||
communication is based on MPI and is managed mainly through PSBLAS.
|
||||
|
||||
<P>
|
||||
This guide provides a brief description of the functionalities and
|
||||
the user interface of MLD2P4.
|
||||
</DIV>
|
||||
<P>
|
||||
|
||||
<P>
|
||||
|
||||
<BR><HR>
|
||||
<BR><HR>
|
||||
<!--Table of Child-Links-->
|
||||
<A NAME="CHILD_LINKS"></A>
|
||||
|
||||
<UL>
|
||||
<UL CLASS="ChildLinks">
|
||||
<LI><A NAME="tex2html15"
|
||||
HREF="node1.html">Contents</A>
|
||||
HREF="node1.html">Abstract</A>
|
||||
<LI><A NAME="tex2html16"
|
||||
HREF="node2.html">General Overview</A>
|
||||
HREF="node2.html">Contents</A>
|
||||
<LI><A NAME="tex2html17"
|
||||
HREF="node3.html">Notational Conventions</A>
|
||||
HREF="node3.html">General Overview</A>
|
||||
<LI><A NAME="tex2html18"
|
||||
HREF="node4.html">Code Distribution</A>
|
||||
<LI><A NAME="tex2html19"
|
||||
@@ -238,29 +169,10 @@ the user interface of MLD2P4.
|
||||
HREF="node31.html">About this document ...</A>
|
||||
</UL>
|
||||
<!--End of Table of Child-Links-->
|
||||
<HR>
|
||||
<!--Navigation Panel-->
|
||||
<A NAME="tex2html13"
|
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HREF="node1.html">
|
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<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
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SRC="file:/usr/share/latex2html/icons/next.png"></A>
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|
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SRC="file:/usr/share/latex2html/icons/up_g.png">
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<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
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SRC="file:/usr/share/latex2html/icons/prev_g.png">
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HREF="node1.html">
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<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
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||||
SRC="file:/usr/share/latex2html/icons/contents.png"></A>
|
||||
<BR>
|
||||
<B> Next:</B> <A NAME="tex2html14"
|
||||
HREF="node1.html">Contents</A>
|
||||
<B> <A NAME="tex2html12"
|
||||
HREF="node1.html">Contents</A></B>
|
||||
<!--End of Navigation Panel-->
|
||||
<BR><HR>
|
||||
<ADDRESS>
|
||||
Salvatore Filippone
|
||||
2008-07-22
|
||||
2008-07-23
|
||||
</ADDRESS>
|
||||
</BODY>
|
||||
</HTML>
|
||||
|
||||
+2
-2
@@ -84,9 +84,9 @@
|
||||
#
|
||||
|
||||
TOPFILE = userguide.tex
|
||||
SECFILE = title.tex abstract.tex overview.tex conventions.tex distribution.tex \
|
||||
SECFILE = title.tex abstract.tex overview.tex distribution.tex \
|
||||
building.tex background.tex gettingstarted.tex userinterface.tex \
|
||||
errors.tex bibliography.tex license.tex
|
||||
errors.tex bibliography.tex license.tex userhtml.tex
|
||||
FIGDIR = figures
|
||||
|
||||
XPDFFLAGS =
|
||||
|
||||
@@ -1,4 +1,5 @@
|
||||
\begin{abstract}
|
||||
\section*{Abstract}
|
||||
\addcontentsline{toc}{section}{Abstract}
|
||||
\textsc{MLD2P4 (Multi-Level Domain Decomposition Parallel Preconditioners Package based on
|
||||
PSBLAS}) is a package of parallel algebraic multi-level preconditioners.
|
||||
It implements various versions of one-level additive and of multi-level additive
|
||||
@@ -24,4 +25,3 @@ communication is based on MPI and is managed mainly through PSBLAS.
|
||||
|
||||
This guide provides a brief description of the functionalities and
|
||||
the user interface of MLD2P4.
|
||||
\end{abstract}
|
||||
|
||||
@@ -29,7 +29,7 @@ in a coarse space, which globally couples the information related to the single
|
||||
submatrices.
|
||||
|
||||
\emph{Two-level Schwarz} preconditioners are obtained
|
||||
by combining basic (one-level) Schwarz preconditioners with a coarse-level
|
||||
by combining basic (one-level) Sch\-warz preconditioners with a coarse-level
|
||||
correction. In this context, the one-level preconditioner is often
|
||||
called `smoother'. Different two-level preconditioners are obtained by varying the
|
||||
choice of the smoother and of the coarse-level correction, and the
|
||||
|
||||
@@ -1,10 +1,11 @@
|
||||
%\section{Bibliography\label{sec:bib}}
|
||||
\begin{thebibliography}{99}
|
||||
\addcontentsline{toc}{section}{\refname}
|
||||
\markboth{\textsc{MLD2P4 User's and Reference Guide}}
|
||||
{\textsc{References}}
|
||||
|
||||
%\let\refname\relax
|
||||
|
||||
\begin{thebibliography}{99}
|
||||
%
|
||||
%\bibitem{PARA04FOREST}
|
||||
%G.~Bella, S.~Filippone, A.~De Maio, A., Testa, M.:
|
||||
@@ -81,11 +82,21 @@ J.W.~Demmel, S.C.~Eisenstat, J.R.~Gilbert, X.S.~Li and J.W.H.~Liu,
|
||||
A supernodal approach to sparse partial pivoting,
|
||||
SIAM Journal on Matrix Analysis and Applications, 20, 3, 1999, 720--755.
|
||||
%
|
||||
%\bibitem{BLACS}
|
||||
%J.~J.~Dongarra and R.~C.~Whaley,
|
||||
%{\em A User's Guide to the BLACS v.~1.1},
|
||||
%Lapack Working Note 94, Tech.\ Rep.\ UT-CS-95-281, University of
|
||||
%Tennessee, March 1995 (updated May 1997).
|
||||
\bibitem{blas3}
|
||||
J.~J.~Dongarra, J.~Du Croz, I.~S.~Duff, S.~Hammarling,
|
||||
\emph{A set of Level 3 Basic Linear Algebra Subprograms},
|
||||
ACM Transactions on Mathematical Software, 16, 1990, 1--17.
|
||||
%
|
||||
\bibitem{blas2}
|
||||
J.~J.~Dongarra, J.~Du Croz, S.~Hammarling, R.~J.~Hanson,
|
||||
\emph{An extended set of FORTRAN Basic Linear Algebra Subprograms},
|
||||
ACM Transactions on Mathematical Software, 14, 1988, 1--17.
|
||||
%
|
||||
\bibitem{BLACS}
|
||||
J.~J.~Dongarra and R.~C.~Whaley,
|
||||
{\em A User's Guide to the BLACS v.~1.1},
|
||||
Lapack Working Note 94, Tech.\ Rep.\ UT-CS-95-281, University of
|
||||
Tennessee, March 1995 (updated May 1997).
|
||||
%
|
||||
%\bibitem{sblas_97}
|
||||
%I.~Duff, M.~Marrone, G.~Radicati and C.~Vittoli,
|
||||
@@ -114,6 +125,17 @@ S.~Filippone, M.~Colajanni,
|
||||
{\em PSBLAS: A Library for Parallel Linear Algebra
|
||||
Computation on Sparse Matrices},
|
||||
ACM Transactions on Mathematical Software, 26, 4, 2000, 527--550.
|
||||
%
|
||||
\bibitem{MPI2}
|
||||
W.~Gropp, S.~Huss-Lederman, A.~Lumsdaine, E.~Lusk, B.~Nitzberg, W.~Saphir, M.~Snir,
|
||||
{\em MPI: The Complete Reference. Volume 2 - The MPI-2 Extensions},
|
||||
MIT Press, 1998.
|
||||
%
|
||||
\bibitem{blas1}
|
||||
C.~L.~Lawson, R.~J.~Hanson, D.~Kincaid, F.~T.~Krogh,
|
||||
\emph{Basic Linear Algebra Subprograms for FORTRAN usage},
|
||||
ACM Transactions on Mathematical Software, 5, 1979, 308--323.
|
||||
%
|
||||
\bibitem{SUPERLUDIST}
|
||||
X.~S.~Li, J.~W.~Demmel, {\em SuperLU\_DIST: A Scalable Distributed-memory
|
||||
Sparse Direct Solver for Unsymmetric Linear Systems},
|
||||
@@ -164,7 +186,7 @@ Cambridge University Press, 1996.
|
||||
M.~Snir, S.~Otto, S.~Huss-Lederman, D.~Walker, J.~Dongarra,
|
||||
{\em MPI: The Complete Reference. Volume 1 - The MPI Core}, second edition,
|
||||
MIT Press, 1998.
|
||||
%
|
||||
%%
|
||||
\bibitem{StubenGMD69_99}
|
||||
K.~St\"{u}ben,
|
||||
{\em Algebraic Multigrid (AMG): an Introduction with Applications},
|
||||
|
||||
+50
-45
@@ -4,55 +4,56 @@
|
||||
To build MLD2P4 it is necessary to set up a Makefile with appropriate
|
||||
values for your system; this is done by means of the \verb|configure|
|
||||
script. The distribution also includes the autoconf and automake
|
||||
sources employed to generate the script, but this should not normally
|
||||
be needed to build the software.
|
||||
sources employed to generate the script, but usually this is not needed
|
||||
to build the software.
|
||||
|
||||
MLD2P4 is implemented almost entirely in Fortran~95, with some
|
||||
interfaces to external libraries in C; we require the Fortran compiler
|
||||
to support the Fortran~95 standard plus the extension TR15581, which
|
||||
interfaces to external libraries in C; the Fortran compiler
|
||||
must support the Fortran~95 standard plus the extension TR15581, which
|
||||
enhances the usability of \verb|ALLOCATABLE| variables. Most modern
|
||||
Fortran compilers support this language level. In particular, this is
|
||||
supported by the GNU Fortran compiler as of version 4.2.0; however we
|
||||
recommend to use the latest available release (4.3.1 at the time of
|
||||
this writing§).
|
||||
this writing).
|
||||
The software defines data types and interfaces for
|
||||
real and complex data, in both single and double precision.
|
||||
|
||||
\subsection{Prerequisites}
|
||||
|
||||
The following base libraries are needed:
|
||||
\begin{description}
|
||||
\item[BLAS] The Basic Linear Algebra subprograms. Many vendors provide
|
||||
optimized versions; if no vendor version is available for a given
|
||||
platform, the ATLAS software \verb!http://www.netlib.org/atlas!
|
||||
\item[BLAS] The Basic Linear Algebra subprograms \cite{blas3,blas3,blas1}.
|
||||
Many vendors provide optimized versions; if no vendor version is
|
||||
available for a given platform, the ATLAS software
|
||||
(\verb!http://math-atlas.sourceforge.net/!)
|
||||
may be employed. The reference BLAS from Netlib
|
||||
\verb|http://www.netlib.org/blas| are meant to define the standard
|
||||
behaviour of the BLAS interface, so they not optimized for any
|
||||
particular plaftorm, and should only be used as a last
|
||||
resort. Note that BLAS computation form a relatively small part of
|
||||
(\verb|http://www.netlib.org/blas|) are meant to define the standard
|
||||
behaviour of the BLAS interface, so they are not optimized for any
|
||||
particular plaftorm, and should only be used as a last
|
||||
resort. Note that BLAS computation form a relatively small part of
|
||||
the MLD2P4/PSBLAS computations; they are however critical when using
|
||||
preconditioners based on the UMFPACK or SuperLU third party
|
||||
libraries.
|
||||
\item[MPI] A version of MPI is available on most high performance
|
||||
computing system; we only require version 1.1.
|
||||
\item[BLACS] The Basic Linear Algebra Communication Subroutines are
|
||||
available in source form from \verb|http://www.netlib.org/blacs|;
|
||||
\item[MPI] A version of MPI \cite{MPI2,MPI1} is available on most high performance
|
||||
computing system; only version 1.1 is required.
|
||||
\item[BLACS] The Basic Linear Algebra Communication Subroutines
|
||||
\cite{BLACS} are available in source form from \verb|http://www.netlib.org/blacs|;
|
||||
some vendors include them in their parallel computing
|
||||
support libraries.
|
||||
\item[PSBLAS] Parallel Sparse BLAS \cite{PSBLASGUIDE,psblas_00} is
|
||||
available from \\ \verb|http://www.ce.uniroma2.it/psblas|; indeed, all the
|
||||
prerequisites listed so far are also prerequisites of PSBLAS.
|
||||
Version 2.3 (or later) is required. To build the MLD2P4 library
|
||||
it is necessary to get access to
|
||||
the source PSBLAS directory employed to build the version under use; after
|
||||
the build process completes, only the compiled form of the library is
|
||||
necessary to build user applications.
|
||||
\end{description}
|
||||
|
||||
The MLD2P4 software requires PSBLAS version 2.3 (at least), available
|
||||
from \verb|http://www.ce.uniroma2.it/psblas|; indeed, all the
|
||||
prerequisites listed so far are also prerequisites of PSBLAS. Please
|
||||
note that to build the MLD2P4 library it is necessary to get access to
|
||||
the source PSBLAS directory used to build the version under use; after
|
||||
the build process completes, only the compiled form of the library is
|
||||
necessary to build user applications.
|
||||
|
||||
Please note that all the libraries listed so fare (BLAS, MPI, BLACS,
|
||||
PSBLAS) must have Fortran interfaces compatible with the MLD2P4;
|
||||
Please note that the four previous libraries must have Fortran
|
||||
interfaces compatible with MLD2P4;
|
||||
usually this means that they should all be built with the same
|
||||
compiler.
|
||||
|
||||
compiler as MLD2P4.
|
||||
|
||||
\subsection{Optional third party libraries}
|
||||
|
||||
@@ -61,19 +62,22 @@ note that these are optional, but if you enable them some defaults
|
||||
for multilevel preconditioners may change to reflect their presence.
|
||||
|
||||
\begin{description}
|
||||
\item[UMFPACK] A sparse direct factorization package available from\\
|
||||
\item[UMFPACK] \cite{UMFPACK}
|
||||
A sparse direct factorization package available from \\
|
||||
\verb|http://www.cise.ufl.edu/research/sparse/umfpack/|;
|
||||
provides serial factorization and solution for double
|
||||
precision real and complex double precision data. We have tested
|
||||
provides serial factorization and triangular system solution for double
|
||||
precision real and complex data. We have tested
|
||||
versions 4.4 and 5.1;
|
||||
\item[SuperLU] A sparse direct factorization package available from \\
|
||||
\item[SuperLU] \cite{SUPERLU}
|
||||
A sparse direct factorization package available from \\
|
||||
\verb|http://crd.lbl.gov/~xiaoye/SuperLU/|; provides serial
|
||||
factorization and solution for single and double precision real and
|
||||
complex data. We have tested versions 3.0 and 3.1.
|
||||
\item[SuperLU\_Dist] A sparse direct factorization package available
|
||||
factorization and triangular system solution for single and double precision,
|
||||
real and complex data. We have tested versions 3.0 and 3.1.
|
||||
\item[SuperLU\_Dist] \cite{SUPERLUDIST}
|
||||
A sparse direct factorization package available
|
||||
from the same site as SuperLU; provides parallel factorization and
|
||||
solution for real and complex double precision data. We have tested
|
||||
version 2.1.
|
||||
triangular system solution for double precision real and complex data.
|
||||
We have tested version 2.1.
|
||||
\end{description}
|
||||
|
||||
\subsection{Configuration options}
|
||||
@@ -85,9 +89,10 @@ As a minimal example consider the following:
|
||||
\begin{verbatim}
|
||||
./configure --with-psblas=/home/user/PSBLAS/psblas-2.3
|
||||
\end{verbatim}
|
||||
This assumes that the various MPI compilers and support libraries are
|
||||
available in the standard directories on the system. Note that the
|
||||
PSBLAS build directory must be specified with an {\em absolute} path.
|
||||
which assumes that the various MPI compilers and support libraries are
|
||||
available in the standard directories on the system, and specifies
|
||||
only the PSBLAS build directory (note that the latter directory must
|
||||
be specified with an {\em absolute} path).
|
||||
The full set of options may be looked at by issuing the command
|
||||
\verb|./configure --help|, which produces:
|
||||
\begin{verbatim}
|
||||
@@ -150,7 +155,7 @@ Optional Packages:
|
||||
--with-PACKAGE[=ARG] use PACKAGE [ARG=yes]
|
||||
--without-PACKAGE do not use PACKAGE (same as --with-PACKAGE=no)
|
||||
--with-psblas The source directory for PSBLAS, for example,
|
||||
--with-psblas=/opt/packages/psblas-2.2
|
||||
--with-psblas=/opt/packages/psblas-2.3
|
||||
--with-libs List additional link flags here. For example,
|
||||
--with-libs=-lspecial_system_lib or
|
||||
--with-libs=-L/path/to/libs
|
||||
@@ -218,7 +223,7 @@ followed (optionally) by
|
||||
make install
|
||||
\end{verbatim}
|
||||
|
||||
\subsection{Example and test programs}
|
||||
\subsection{Example and test programs\label{sec:ex_and_test}}
|
||||
The package contains the \verb|examples| and \verb|tests| directories;
|
||||
both of them are further divided into \verb|fileread| and
|
||||
\verb|pargen| subdirectories. Their purpose is as follows:
|
||||
@@ -233,6 +238,6 @@ both of them are further divided into \verb|fileread| and
|
||||
implemented in the library.
|
||||
\end{description}
|
||||
The \verb|fileread| directories contain sample programs that read
|
||||
sparse matrices from files, according to the Matrix Market storage
|
||||
format; the \verb|pargen| instead generate matrices in full parallel
|
||||
mode from the discretization of a sample PDE.
|
||||
sparse matrices from files, according to the Matrix Market or the
|
||||
Harwell-Boeing storage format; the \verb|pargen| instead generate
|
||||
matrices in full parallel mode from the discretization of a sample PDE.
|
||||
|
||||
@@ -1,8 +0,0 @@
|
||||
\section{Notational Conventions\label{sec:conventions}}
|
||||
\markboth{\textsc{MLD2P4 User's and Reference Guide}}
|
||||
{\textsc{\ref{sec:conventions} Notational Conventions}}
|
||||
|
||||
- caratteri tipografici usati nella guida (vedi guida ML recente e guida Aztec) \\
|
||||
- convenzioni sui nomi di routine (anche differenza nei nomi tra high-level e
|
||||
medium-level?), strutture dati, moduli, costanti, etc. (vedi guida psblas) \\
|
||||
- versione reale e complessa, singola e doppia precisione\\
|
||||
@@ -3,15 +3,17 @@
|
||||
{\textsc{\ref{sec:distribution} Code Distribution}}
|
||||
|
||||
\noindent
|
||||
MLD2P4 is available from our project web site
|
||||
MLD2P4 is available from the web site
|
||||
\begin{quotation}
|
||||
\tt http://www.mld2p4.it
|
||||
\texttt{http://www.mld2p4.it}
|
||||
\end{quotation}
|
||||
where you will also find contact points for further information and
|
||||
bug reports.
|
||||
where contact points for further information can be also found.
|
||||
To report bugs or ask general usage questions, please, send an email to
|
||||
\texttt{bugreport@mld2p4.it}.
|
||||
|
||||
|
||||
The software is available under a modified BSD license, as specified
|
||||
in appendix~\ref{sec:license}; please note that some of the optional
|
||||
in Appendix~\ref{sec:license}; please note that some of the optional
|
||||
third party libraries may be licensed under a different and more
|
||||
stringent license, most notably the GPL, and this should be taken into
|
||||
account when treating derived works.
|
||||
|
||||
+4
-4
@@ -1,8 +1,8 @@
|
||||
\section{Error Handling}\label{sec:errors}
|
||||
\section{Error Handling\label{sec:errors}}
|
||||
\markboth{\textsc{MLD2P4 User's and Reference Guide}}
|
||||
{\textsc{\ref{sec:errors} Error handling}}
|
||||
|
||||
The error handling in MLD2P4 is based on the PSBLAS version 2 error
|
||||
The error handling in MLD2P4 is based on the PSBLAS (version 2) error
|
||||
handling. Error conditions are signaled via an integer argument
|
||||
\verb|info|; whenever an error condition is detected, an error trace
|
||||
stack is built by the library up to the top-level, user-callable
|
||||
@@ -10,9 +10,9 @@ routine. This routine will then decide, according to the user
|
||||
preferences, whether the error should be handled by terminating the
|
||||
program or by returning the error condition to the user code, which
|
||||
will then take action, and whether
|
||||
an error message should be printed. These options may be set by using
|
||||
an error message should be printed. These options may be set by using
|
||||
the PSBLAS error handling routines; for further details see the PSBLAS
|
||||
user's guide.
|
||||
User's Guide \cite{PSBLASGUIDE}.
|
||||
|
||||
%%% Local Variables:
|
||||
%%% mode: latex
|
||||
|
||||
+16
-13
@@ -7,7 +7,7 @@ Schwarz preconditioners with the Krylov solvers included in PSBLAS \cite{PSBLASG
|
||||
The following steps are required:
|
||||
\begin{enumerate}
|
||||
\item \emph{Declare the preconditioner data structure}. It is a derived data type,
|
||||
\verb|mld_|\-\emph{x}\verb|prec_type|, where \emph{x} may be \verb|s|, \verb|d|, \verb|c|
|
||||
\verb|mld_|\-\emph{x}\verb|prec_| \verb|type|, where \emph{x} may be \verb|s|, \verb|d|, \verb|c|
|
||||
or \verb|z|, according to the basic data type of the sparse matrix
|
||||
(\verb|s| = real single precision; \verb|d| = real double precision;
|
||||
\verb|c| = complex single precision; \verb|z| = complex double precision).
|
||||
@@ -36,7 +36,7 @@ The following steps are required:
|
||||
this step is completely transparent to the user, since \verb|mld_precaply| is called
|
||||
by the PSBLAS routine implementing the Krylov solver (\verb|psb_krylov|).
|
||||
\item \emph{Free the preconditioner data structure}. This is performed by
|
||||
the routine \verb|mld_precfree|. This step is complementary to step 1 and should
|
||||
the routine \verb|mld_| \verb|precfree|. This step is complementary to step 1 and should
|
||||
be performed when the preconditioner is no more used.
|
||||
\end{enumerate}
|
||||
A detailed description of the above routines is given in Section~\ref{sec:userinterface}.
|
||||
@@ -94,9 +94,10 @@ Multilevel &\verb|'ML'| & Multi-level hybrid preconditioner (additive
|
||||
Coarsest matrix: distributed among the processors.
|
||||
Coarsest-level solver:
|
||||
4 sweeps of the block-Jacobi solver,
|
||||
with LU factorization of the blocks
|
||||
with LU (or ILU) factorization of the blocks
|
||||
(UMFPACK for the double precision versions and
|
||||
SuperLU for the single precision ones) \\
|
||||
SuperLU for the single precision ones, if they have been
|
||||
installed; ILU(0), otherwise). \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
%}
|
||||
@@ -123,7 +124,7 @@ here for brevity; the statements concerning the deallocation of the PSBLAS
|
||||
data structure are neglected too.
|
||||
The complete code can be found in the example program file \verb|mld_dexample_ml.f90|,
|
||||
in the directory \verb|examples/fileread| of the MLD2P4 tree (see
|
||||
Section~\ref{sec:building}).
|
||||
Section~\ref{sec:ex_and_test}).
|
||||
For details on the use of the PSBLAS routines, see the PSBLAS User's
|
||||
Guide \cite{PSBLASGUIDE}.
|
||||
|
||||
@@ -221,12 +222,21 @@ included in the example program file \verb|mld_dexample_ml.f90| too.
|
||||
|
||||
Finally, Figure~\ref{fig:ex_1l} shows the setup of a one-level
|
||||
additive Schwarz preconditioner, i.e.\ RAS with overlap 2. The corresponding
|
||||
example program is available in \verb|mld_dexample_1lev.f90|.
|
||||
example program is available in \verb|mld_dexample_| \verb|1lev.f90|.
|
||||
|
||||
For all the previous preconditioners, example programs where the sparse matrix and
|
||||
the right-hand side are generated by discretizing a PDE with Dirichlet
|
||||
boundary conditions are also available in the directory \verb|examples/pdegen|.
|
||||
|
||||
\ \\
|
||||
\textbf{Remark 3.} Any PSBLAS-based program using the basic preconditioners
|
||||
implemented in PSBLAS 2.0, i.e.\ the diagonal and block-Jacobi ones,
|
||||
can use the diagonal and block-Jacobi preconditioners
|
||||
implemented in MLD2P4 without any change in the code.
|
||||
The PSBLAS-based program must be only recompiled
|
||||
and linked to the MLD2P4 library.
|
||||
\\
|
||||
|
||||
|
||||
\begin{figure}[tbh]
|
||||
\begin{center}
|
||||
@@ -291,13 +301,6 @@ boundary conditions are also available in the directory \verb|examples/pdegen|.
|
||||
\end{center}
|
||||
\end{figure}
|
||||
|
||||
\ \\
|
||||
\textbf{Remark 3.} Any PSBLAS-based program using the basic preconditioners
|
||||
implemented in PSBLAS 2.0, i.e.\ the diagonal and block-Jacobi ones,
|
||||
can use the diagonal and block-Jacobi preconditioners
|
||||
implemented in MLD2P4 without any change in the code.
|
||||
The PSBLAS-based program must be only recompiled
|
||||
and linked to the MLD2P4 library.
|
||||
|
||||
%%% Local Variables:
|
||||
%%% mode: latex
|
||||
|
||||
@@ -71,11 +71,9 @@ by expert users to build new versions of multi-level Schwarz preconditioners.
|
||||
We provide here a description of the upper-layer routines, but not of the
|
||||
medium-layer ones.
|
||||
|
||||
This guide is organized as follows. The notational conventions used in
|
||||
the guide and in the naming of the MLD2P4 routines are reported in
|
||||
Section~\ref{sec:conventions}. Information on the distribution of the source code and the
|
||||
related license is given in Section~\ref{sec:distribution}, while details on the configuration
|
||||
and installation of package are given in Section~\ref{sec:building}. A description of
|
||||
This guide is organized as follows. General information on the distribution of the source code
|
||||
is reported in Section~\ref{sec:distribution}, while details on the configuration
|
||||
and installation of the package are given in Section~\ref{sec:building}. A description of
|
||||
multi-level Schwarz preconditioners based on smoothed aggregation is provided
|
||||
in Section~\ref{sec:background}, to help the users in choosing among the different preconditioners
|
||||
implemented in MLD2P4. The basics for building and applying the preconditioners
|
||||
|
||||
+20
-7
@@ -3,10 +3,24 @@
|
||||
% $Id: title.tex 1999 2007-10-29 15:25:27Z sfilippo $
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
\ifx\pdfoutput\undefined % We're not running pdftex
|
||||
\else
|
||||
\ifcase\pdfoutput % We're not running pdftex
|
||||
{\Large\bfseries MLD2P4\\[.8ex] User's and Reference Guide}\\
|
||||
\emph{\large A guide for the Multi-Level Domain Decomposition \\[.6ex]
|
||||
Parallel Preconditioners Package
|
||||
based on PSBLAS}
|
||||
{\bfseries Pasqua D'Ambra}\\
|
||||
ICAR-CNR, Naples, Italy\\[3ex]
|
||||
{\bfseries Daniela di Serafino}\\
|
||||
Second University of Naples, Italy\\[3ex]
|
||||
{\bfseries Salvatore Filippone} \\
|
||||
University of Rome ``Tor Vergata'', Italy
|
||||
%\\[10ex]
|
||||
%\today
|
||||
Software version: 1.0\\
|
||||
%\today
|
||||
July 24, 2008
|
||||
\or
|
||||
\pdfbookmark{MLD2P4 User's and Reference Guide}{title}
|
||||
\fi
|
||||
\newlength{\centeroffset}
|
||||
%\setlength{\centeroffset}{-0.5\oddsidemargin}
|
||||
%\addtolength{\centeroffset}{0.5\evensidemargin}
|
||||
@@ -23,7 +37,6 @@ Parallel Preconditioners Package
|
||||
based on PSBLAS}
|
||||
\end{minipage}}
|
||||
|
||||
|
||||
\vspace{\stretch{1}}
|
||||
\noindent\hspace*{\centeroffset}\makebox[0pt][l]{\begin{minipage}{\textwidth}
|
||||
\flushright
|
||||
@@ -41,12 +54,12 @@ based on PSBLAS}
|
||||
\noindent\hspace*{\centeroffset}\makebox[0pt][l]{\begin{minipage}{\textwidth}
|
||||
\flushright
|
||||
\large Software version: 1.0\\
|
||||
\today
|
||||
%\today
|
||||
\large July 24, 2008
|
||||
\end{minipage}}
|
||||
|
||||
|
||||
%\addtolength{\textwidth}{\centeroffset}
|
||||
\vspace{\stretch{2}}
|
||||
\fi
|
||||
|
||||
\endinput
|
||||
|
||||
|
||||
+44
-7
@@ -37,7 +37,8 @@
|
||||
}
|
||||
|
||||
\setlength\textwidth{1.15\textwidth}
|
||||
% \setlength\evensidemargin{.7in}
|
||||
\setlength\oddsidemargin{0.3in}
|
||||
\setlength\evensidemargin{0.2in}
|
||||
% \newlength{\centeroffset}
|
||||
% \setlength{\centeroffset}{0.5\oddsidemargin}
|
||||
% \addtolength{\centeroffset}{0.5\evensidemargin}
|
||||
@@ -80,6 +81,8 @@
|
||||
\makeatother
|
||||
\newcommand{\example}{\stepcounter{example}%
|
||||
\section*{\examplename~\theexample}}
|
||||
\def\bsideways{\sidewaystable}
|
||||
\def\esideways{\endsidewaystable}
|
||||
|
||||
\newcommand{\precdata}{\hyperlink{precdata}{{\tt mld\_prec\_type}}}
|
||||
\newcommand{\descdata}{\hyperlink{descdata}{{\tt psb\_desc\_type}}}
|
||||
@@ -87,8 +90,45 @@
|
||||
\newcommand{\Ref}[1]{\mbox{(\ref{#1})}}
|
||||
|
||||
\begin{document}
|
||||
\include{title}
|
||||
%\cleardoublepage
|
||||
\pdfbookmark{MLD2P4 User's and Reference Guide}{title}
|
||||
\newlength{\centeroffset}
|
||||
%\setlength{\centeroffset}{-0.5\oddsidemargin}
|
||||
%\addtolength{\centeroffset}{0.5\evensidemargin}
|
||||
%\addtolength{\textwidth}{-\centeroffset}
|
||||
\thispagestyle{empty}
|
||||
\vspace*{\stretch{1}}
|
||||
\noindent\hspace*{\centeroffset}\makebox[0pt][l]{\begin{minipage}{\textwidth}
|
||||
\flushright
|
||||
{\Huge\bfseries MLD2P4\\[.8ex] User's and Reference Guide
|
||||
}
|
||||
\noindent\rule[-1ex]{\textwidth}{5pt}\\[2.5ex]
|
||||
\hfill\emph{\Large A guide for the Multi-Level Domain Decomposition \\[.6ex]
|
||||
Parallel Preconditioners Package
|
||||
based on PSBLAS}
|
||||
\end{minipage}}
|
||||
|
||||
\vspace{\stretch{1}}
|
||||
\noindent\hspace*{\centeroffset}\makebox[0pt][l]{\begin{minipage}{\textwidth}
|
||||
\flushright
|
||||
{\large\bfseries Pasqua D'Ambra}\\
|
||||
\large ICAR-CNR, Naples, Italy\\[3ex]
|
||||
{\large\bfseries Daniela di Serafino}\\
|
||||
\large Second University of Naples, Italy\\[3ex]
|
||||
{\large\bfseries Salvatore Filippone} \\
|
||||
\large University of Rome ``Tor Vergata'', Italy
|
||||
%\\[10ex]
|
||||
%\today
|
||||
\end{minipage}}
|
||||
|
||||
\vspace{\stretch{1}}
|
||||
\noindent\hspace*{\centeroffset}\makebox[0pt][l]{\begin{minipage}{\textwidth}
|
||||
\flushright
|
||||
\large Software version: 1.0\\
|
||||
%\today
|
||||
\large July 24, 2008
|
||||
\end{minipage}}
|
||||
%\addtolength{\textwidth}{\centeroffset}
|
||||
\vspace{\stretch{2}}
|
||||
\clearpage
|
||||
\ \\
|
||||
\thispagestyle{empty}
|
||||
@@ -112,16 +152,13 @@
|
||||
\setcounter{page}{1} % Chapters start on page 1
|
||||
|
||||
\include{overview}
|
||||
\include{conventions}
|
||||
\include{distribution}
|
||||
\include{building}
|
||||
\include{background}
|
||||
\include{gettingstarted}
|
||||
\include{userinterface}
|
||||
%\include{advanced}
|
||||
\include{errors}
|
||||
%\include{listofroutines}
|
||||
\cleardoublepage
|
||||
\clearpage
|
||||
\appendix
|
||||
\include{license}
|
||||
\cleardoublepage
|
||||
|
||||
+17
-3
@@ -80,6 +80,8 @@
|
||||
\makeatother
|
||||
\newcommand{\example}{\stepcounter{example}%
|
||||
\section*{\examplename~\theexample}}
|
||||
\def\bsideways{\begin{table}}
|
||||
\def\esideways{\end{table}}
|
||||
|
||||
\newcommand{\precdata}{\hyperlink{precdata}{{\tt mld\_prec\_type}}}
|
||||
\newcommand{\descdata}{\hyperlink{descdata}{{\tt psb\_desc\_type}}}
|
||||
@@ -87,8 +89,21 @@
|
||||
\newcommand{\Ref}[1]{\mbox{(\ref{#1})}}
|
||||
|
||||
\begin{document}
|
||||
\include{title}
|
||||
%\cleardoublepage
|
||||
{\Large\bfseries MLD2P4\\[.8ex] User's and Reference Guide}\\
|
||||
\emph{\large A guide for the Multi-Level Domain Decomposition \\[.6ex]
|
||||
Parallel Preconditioners Package
|
||||
based on PSBLAS}
|
||||
{\bfseries Pasqua D'Ambra}\\
|
||||
ICAR-CNR, Naples, Italy\\[3ex]
|
||||
{\bfseries Daniela di Serafino}\\
|
||||
Second University of Naples, Italy\\[3ex]
|
||||
{\bfseries Salvatore Filippone} \\
|
||||
University of Rome ``Tor Vergata'', Italy
|
||||
%\\[10ex]
|
||||
%\today
|
||||
Software version: 1.0\\
|
||||
%\today
|
||||
July 24, 2008
|
||||
\clearpage
|
||||
\ \\
|
||||
\thispagestyle{empty}
|
||||
@@ -112,7 +127,6 @@
|
||||
\setcounter{page}{1} % Chapters start on page 1
|
||||
|
||||
\include{overview}
|
||||
\include{conventions}
|
||||
\include{distribution}
|
||||
\include{building}
|
||||
\include{background}
|
||||
|
||||
+48
-40
@@ -1,11 +1,9 @@
|
||||
\section{User Interface\label{sec:userinterface}}
|
||||
\markboth{\textsc{MLD2P4 User's and Reference Guide}}
|
||||
{\textsc{\ref{sec:userinterface} User Interface}}
|
||||
\def\bsideways{\sidewaystable}
|
||||
\def\esideways{\endsidewaystable}
|
||||
|
||||
|
||||
The basic user interface of MLD2P4 consists of six routines. The four routines \verb|mld_precinit|,
|
||||
The basic user interface of MLD2P4 consists of six routines. The four routines \verb|mld_| \verb|precinit|,
|
||||
\verb|mld_precset|, \verb|mld_precbld| and \verb|mld_precaply| encapsulate all the functionalities
|
||||
for the setup and the application of any one-level and multi-level
|
||||
preconditioner implemented in the package.
|
||||
@@ -40,6 +38,7 @@ i.e.
|
||||
\end{itemize}
|
||||
A description of each routine is given in the remainder of this section.
|
||||
|
||||
\clearpage
|
||||
|
||||
\subsection{Subroutine mld\_precinit\label{sec:precinit}}
|
||||
|
||||
@@ -54,7 +53,7 @@ according to the preconditioner type chosen by the user.
|
||||
|
||||
\subsubsection*{Arguments}
|
||||
|
||||
\begin{tabular}{p{1.2cm}p{10.5cm}}
|
||||
\begin{tabular}{p{1.2cm}p{12cm}}
|
||||
\verb|p| & \verb|type(mld_|\emph{x}\verb|prec_type), intent(inout)|.\\
|
||||
& The preconditioner data structure. Note that \emph{x}
|
||||
must be chosen according to the real/complex, single/double
|
||||
@@ -71,6 +70,7 @@ according to the preconditioner type chosen by the user.
|
||||
then \verb|nlev|=2 is assumed. Otherwise, \verb|nlev| is ignored.\\
|
||||
\end{tabular}
|
||||
|
||||
\clearpage
|
||||
|
||||
\subsection{Subroutine mld\_precset\label{sec:precset}}
|
||||
|
||||
@@ -85,7 +85,7 @@ contained in \verb|val|.
|
||||
|
||||
\subsubsection*{Arguments}
|
||||
|
||||
\begin{tabular}{p{1.2cm}p{10.5cm}}
|
||||
\begin{tabular}{p{1.2cm}p{12cm}}
|
||||
\verb|p| & \verb|type(mld_|\emph{x}\verb|prec_type), intent(inout)|.\\
|
||||
& The preconditioner data structure. Note that \emph{x} must
|
||||
be chosen according to the real/complex, single/double precision
|
||||
@@ -136,7 +136,7 @@ refer to Section~\ref{sec:background}.
|
||||
%multi-level preconditioners. Non-expert users are recommended
|
||||
%to call \verb| mld_precset| without specifying \verb|ilev|.
|
||||
|
||||
\ifcase\pdfoutput\else\bsideways\fi
|
||||
\bsideways
|
||||
\begin{center}
|
||||
\begin{tabular}{|l|l|p{2cm}|l|p{7cm}|}
|
||||
\hline
|
||||
@@ -152,7 +152,7 @@ refer to Section~\ref{sec:background}.
|
||||
& \texttt{'DIAG'} \ \ \ \texttt{'BJAC'} \ \ \ \texttt{'AS'}
|
||||
& \texttt{'AS'}
|
||||
& Basic one-level preconditioner (i.e.\ smoother): diagonal,
|
||||
block Jacobi, AS \\ \hline
|
||||
block Jacobi, AS. \\ \hline
|
||||
\verb|mld_smoother_pos_| & \verb|character(len=*)|
|
||||
& \texttt{'PRE'} \ \ \ \texttt{'POST'} \ \ \ \texttt{'TWOSIDE'}
|
||||
& \texttt{'POST'}
|
||||
@@ -163,9 +163,9 @@ refer to Section~\ref{sec:background}.
|
||||
\end{center}
|
||||
\caption{Parameters defining the type of multi-level preconditioner.
|
||||
\label{tab:p_type}}
|
||||
\ifcase\pdfoutput\else\esideways\fi
|
||||
\esideways
|
||||
|
||||
\ifcase\pdfoutput\else\bsideways\fi
|
||||
\bsideways
|
||||
\begin{center}
|
||||
\begin{tabular}{|l|l|p{3.2cm}|l|p{7cm}|}
|
||||
\hline
|
||||
@@ -173,7 +173,7 @@ refer to Section~\ref{sec:background}.
|
||||
\textsc{comments} \\ \hline
|
||||
%\multicolumn{5}{|c|}{\emph{basic one-level preconditioner (smoother)}} \\ \hline
|
||||
\verb|mld_sub_ovr_| & \verb|integer|
|
||||
& any integer number $\ge 0$
|
||||
& any~int.~num.~$\ge 0$
|
||||
& 1
|
||||
& Number of overlap layers. \\ \hline
|
||||
\verb|mld_sub_restr_| & \verb|character(len=*)|
|
||||
@@ -200,7 +200,7 @@ refer to Section~\ref{sec:background}.
|
||||
& Fill-in level $p$ of the incomplete LU factorizations. \\ \hline
|
||||
\verb|mld_sub_iluthrs_| & \verb|real(|\emph{kind\_parameter}\verb|)|
|
||||
& Any~real~num.~$\ge 0$
|
||||
& \texttt{0.e0} (or \texttt{0.d0})
|
||||
& 0
|
||||
& Drop tolerance $t$ in the ILU($p,t$) factorization. \\ \hline
|
||||
\verb|mld_sub_ren_| & \verb|character(len=*)|
|
||||
& \texttt{'RENUM\_NONE'} \texttt{'RENUM\_GLOBAL'} %, \texttt{'RENUM_GPS'}
|
||||
@@ -213,9 +213,9 @@ refer to Section~\ref{sec:background}.
|
||||
\end{center}
|
||||
\caption{Parameters defining the one-level preconditioner used as smoother.
|
||||
\label{tab:p_smoother}}
|
||||
\ifcase\pdfoutput\else\esideways\fi
|
||||
\esideways
|
||||
|
||||
\ifcase\pdfoutput\else\bsideways\fi
|
||||
\bsideways
|
||||
\begin{center}
|
||||
\begin{tabular}{|l|l|p{2.3cm}|p{2.6cm}|p{7cm}|}
|
||||
\hline
|
||||
@@ -229,48 +229,53 @@ refer to Section~\ref{sec:background}.
|
||||
\verb|mld_aggr_kind_| & \verb|character(len=*)|
|
||||
& \texttt{'SMOOTH'} \hspace{2.5cm} \texttt{'RAW'}
|
||||
& \texttt{'SMOOTH'}
|
||||
& Type of aggregation: smoothed or raw, i.e.\ using the tentative prolongator. \\ \hline
|
||||
& Type of aggregation: smoothed, raw (i.e.\ using the tentative prolongator). \\ \hline
|
||||
\verb|mld_aggr_thresh_| & \verb|real(|\emph{kind\_parameter}\verb|)|
|
||||
& Any~real~num. $\in [0, 1]$
|
||||
& \texttt{0.e0} (or \texttt{0.d0})
|
||||
& 0
|
||||
& The threshold $\theta$ in the aggregation algorithm. \\ \hline
|
||||
\verb|mld_aggr_eig_| & \verb|character(len=*)|
|
||||
& \texttt{'A\_NORMI'}
|
||||
& \texttt{'A\_NORMI'}
|
||||
& Estimate of the maximum eigenvalue of $D^{-1}A$
|
||||
for the smoothed aggregation. Currently, only the infinity norm of
|
||||
& Estimate of the eigenvalue $D^{-1}A$ with largest modulus,
|
||||
to build the damping parameter $\omega$ in the smoothed aggregation.
|
||||
Currently, only the infinity norm of
|
||||
the matrix is available. \\ \hline
|
||||
\verb|mld_aggr_damp_| & \verb|real(|\emph{kind\_parameter}\verb|)|
|
||||
& Any~real~num. $>0$
|
||||
& \texttt{4.e0/3.e0} \hspace{1cm} (or \texttt{4.d0/3.d0})
|
||||
& The damping parameter $\omega$ in the aggregation algorithm. \\
|
||||
& Any~real~num.
|
||||
& $4/(3||D^{-1}A||_\infty)$
|
||||
& The damping parameter $\omega$ in the smoothed aggregation algorithm.
|
||||
If the user specifies a negative value, then $\omega$ is set to its default
|
||||
value; otherwise, $\omega$ is set to the value provided by the
|
||||
user. In the latter case no estimate of the eigenvalue $D^{-1}A$ with
|
||||
largest modulus is computed.\\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
\caption{Parameters defining the aggregation algorithm.
|
||||
\label{tab:p_aggregation}}
|
||||
\ifcase\pdfoutput\else\esideways\fi
|
||||
\esideways
|
||||
|
||||
\ifcase\pdfoutput\else\bsideways\fi
|
||||
\bsideways
|
||||
\begin{center}
|
||||
\begin{tabular}{|l|l|p{3.2cm}|l|p{7cm}|}
|
||||
\hline
|
||||
\verb|what| & \textsc{data type} & \verb|val| & \textsc{default} &
|
||||
\textsc{comments} \\ \hline
|
||||
%\multicolumn{5}{|c|}{\emph{coarse-space correction at the coarsest level}}\\ \hline
|
||||
\verb|mld_coarse_mat_| & \verb|character(len=*)|
|
||||
& \texttt{'DISTR'} \hspace{2.5cm} \texttt{'REPL'}
|
||||
& \texttt{'DISTR'}
|
||||
& Coarsest matrix: distributed among the processors or replicated on each of them. \\ \hline
|
||||
\verb|mld_coarse_solve_| & \verb|character(len=*)|
|
||||
& \texttt{'BJAC'} \hspace{2.5cm} \texttt{'UMF'} \hspace{2.5cm} \texttt{'SLU'}
|
||||
\hspace{2.5cm} \texttt{'SLUDIST'}
|
||||
& \texttt{'BJAC'}
|
||||
& Solver used at the coarsest level: block Jacobi, sequential LU from UMFPACK,
|
||||
sequential LU from SuperLU, distributed LU from SuperLU\_Dist.
|
||||
If the coarsest matrix is distributed, only \texttt{'BJAC'} and \texttt{'SLUDIST'}
|
||||
can be chosen; if it is replicated, only \texttt{'BJAC'} or \texttt{'SLUDIST'} can
|
||||
be selected. \\ \hline
|
||||
With \texttt{'SLUDIST'} the coarsest matrix
|
||||
must be distributed; with \texttt{'UMF'} or
|
||||
\texttt{'SLU'} it must be replicated. \\ \hline
|
||||
\verb|mld_coarse_mat_| & \verb|character(len=*)|
|
||||
& \texttt{'DISTR'} \hspace{2.5cm} \texttt{'REPL'}
|
||||
& \texttt{'DISTR'}
|
||||
& Coarsest matrix: distributed among the processors or replicated on each of them. \\ \hline
|
||||
\verb|mld_coarse_subsolve_| & \verb|character(len=*)|
|
||||
& \texttt{'ILU'} \hspace{2.5cm} \texttt{'MILU'} \hspace{2.5cm} \texttt{'ILUT'}
|
||||
\hspace{2.5cm} \texttt{'UMF'} \hspace{2.5cm} \texttt{'SLU'}
|
||||
@@ -280,25 +285,26 @@ refer to Section~\ref{sec:background}.
|
||||
LU from SuperLU, plus triangular solve. \\ \hline
|
||||
\verb|mld_coarse_sweeps_|& \verb|integer|
|
||||
& Any~int.~num.~$> 0$
|
||||
& \texttt{4}
|
||||
& Number of Block-Jacobi sweeps when 'BJAC' is used as coarsest-level solver. \\
|
||||
& 4
|
||||
& Number of Block-Jacobi sweeps when 'BJAC' is used as coarsest-level solver. \\ \hline
|
||||
\verb|mld_coarse_fillin_| & \verb|integer|
|
||||
& Any~int.~num.~$\ge 0$
|
||||
& \texttt{0}
|
||||
& 0
|
||||
& Fill-in level $p$ of the incomplete LU factorizations. \\ \hline
|
||||
\verb|mld_coarse_iluthrs_| & \verb|real(|\emph{kind\_parameter}\verb|)|
|
||||
& Any~real.~num.~$\ge 0$
|
||||
& \texttt{0.d0} (or \texttt{0.e0})
|
||||
& 0
|
||||
& Drop tolerance $t$ in the ILU($p,t$) factorization. \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
\caption{Parameters defining the coarse-space correction at the coarsest
|
||||
level.\label{tab:p_coarse}}
|
||||
\ifcase\pdfoutput\else\esideways\fi
|
||||
\esideways
|
||||
|
||||
|
||||
\clearpage
|
||||
|
||||
\subsection{Subroutine mld\_precbld\label{sec:precbld}}
|
||||
|
||||
\begin{center}
|
||||
@@ -311,7 +317,7 @@ the user through the routines \verb|mld_precinit| and \verb|mld_precset|.
|
||||
|
||||
\subsubsection*{Arguments}
|
||||
|
||||
\begin{tabular}{p{1.2cm}p{10.5cm}}
|
||||
\begin{tabular}{p{1.2cm}p{12cm}}
|
||||
\verb|a| & \verb|type(psb_|\emph{x}\verb|spmat_type), intent(in)|. \\
|
||||
& The sparse matrix structure containing the local part of the
|
||||
matrix to be preconditioned. Note that \emph{x} must be chosen according
|
||||
@@ -347,7 +353,7 @@ and hence it is completely transparent to the user.
|
||||
|
||||
\subsubsection*{Arguments}
|
||||
|
||||
\begin{tabular}{p{1.2cm}p{10.5cm}}
|
||||
\begin{tabular}{p{1.2cm}p{12cm}}
|
||||
\verb|p| & \verb|type(mld_|\emph{x}\verb|prec_type), intent(inout)|.\\
|
||||
& The preconditioner data structure, containing the local part of $M$.
|
||||
Note that \emph{x} must be chosen according
|
||||
@@ -372,11 +378,12 @@ and hence it is completely transparent to the user.
|
||||
(conjugate transpose of $M^{-1})$.\\
|
||||
\verb|work| & \emph{type}\verb|(|\emph{kind\_parameter}\verb|), dimension(:), optional, target|.\\
|
||||
& Workspace. Its size should be at
|
||||
least \verb|4 * psb_cd_get_local_cols(desc_a)| (see the PSBLAS User's Guide).
|
||||
least \verb|4 * psb_cd_get_local_| \verb|cols(desc_a)| (see the PSBLAS User's Guide).
|
||||
Note that \emph{type} and \emph{kind\_parameter} must be chosen according
|
||||
to the real/complex, single/double precision version of MLD2P4 under use.\\
|
||||
\end{tabular}
|
||||
|
||||
\clearpage
|
||||
|
||||
\subsection{Subroutine mld\_precfree\label{sec:precfree}}
|
||||
|
||||
@@ -397,6 +404,7 @@ This routine deallocates the preconditioner data structure.
|
||||
& Error code. If no error, 0 is returned. See Section~\ref{sec:errors} for details.\\
|
||||
\end{tabular}
|
||||
|
||||
\clearpage
|
||||
|
||||
\subsection{Subroutine mld\_precdescr\label{sec:precdescr}}
|
||||
|
||||
@@ -406,12 +414,12 @@ This routine deallocates the preconditioner data structure.
|
||||
\end{center}
|
||||
|
||||
\noindent
|
||||
This routine prints a description of the preconditioner
|
||||
to the standard output or to a file.
|
||||
This routine prints a description of the preconditioner to the standard output or
|
||||
to a file. It must be called after \verb|mld_precbld| has been called.
|
||||
|
||||
\subsubsection*{Arguments}
|
||||
|
||||
\begin{tabular}{p{1.2cm}p{10.6cm}}
|
||||
\begin{tabular}{p{1.2cm}p{12cm}}
|
||||
\verb|p| & \verb|type(mld_|\emph{x}\verb|prec_type), intent(in)|.\\
|
||||
& The preconditioner data structure. Note that \emph{x} must be chosen according
|
||||
to the real/complex, single/double precision version of MLD2P4 under use.\\
|
||||
|
||||
Reference in New Issue
Block a user