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improvement.html
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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1//EN"
"http://www.w3.org/TR/xhtml11/DTD/xhtml11.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head>
<meta http-equiv="Content-Type" content="application/xhtml+xml; charset=UTF-8" />
<meta name="generator" content="AsciiDoc 8.6.9" />
<title>Summary of improvements since LAPACK 3.0</title>
<style type="text/css">
/* Shared CSS for AsciiDoc xhtml11 and html5 backends */
/* Default font. */
body {
font-family: Georgia,serif;
}
/* Title font. */
h1, h2, h3, h4, h5, h6,
div.title, caption.title,
thead, p.table.header,
#toctitle,
#author, #revnumber, #revdate, #revremark,
#footer {
font-family: Arial,Helvetica,sans-serif;
}
body {
margin: 1em 5% 1em 5%;
}
a {
color: blue;
text-decoration: underline;
}
a:visited {
color: fuchsia;
}
em {
font-style: italic;
color: navy;
}
strong {
font-weight: bold;
color: #083194;
}
h1, h2, h3, h4, h5, h6 {
color: #527bbd;
margin-top: 1.2em;
margin-bottom: 0.5em;
line-height: 1.3;
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border-bottom: 2px solid silver;
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padding-top: 0.5em;
}
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float: left;
}
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clear: left;
}
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font-size: 1.0em;
}
div.sectionbody {
margin-left: 0;
}
hr {
border: 1px solid silver;
}
p {
margin-top: 0.5em;
margin-bottom: 0.5em;
}
ul, ol, li > p {
margin-top: 0;
}
ul > li { color: #aaa; }
ul > li > * { color: black; }
.monospaced, code, pre {
font-family: "Courier New", Courier, monospace;
font-size: inherit;
color: navy;
padding: 0;
margin: 0;
}
pre {
white-space: pre-wrap;
}
#author {
color: #527bbd;
font-weight: bold;
font-size: 1.1em;
}
#email {
}
#revnumber, #revdate, #revremark {
}
#footer {
font-size: small;
border-top: 2px solid silver;
padding-top: 0.5em;
margin-top: 4.0em;
}
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float: left;
padding-bottom: 0.5em;
}
#footer-badges {
float: right;
padding-bottom: 0.5em;
}
#preamble {
margin-top: 1.5em;
margin-bottom: 1.5em;
}
div.imageblock, div.exampleblock, div.verseblock,
div.quoteblock, div.literalblock, div.listingblock, div.sidebarblock,
div.admonitionblock {
margin-top: 1.0em;
margin-bottom: 1.5em;
}
div.admonitionblock {
margin-top: 2.0em;
margin-bottom: 2.0em;
margin-right: 10%;
color: #606060;
}
div.content { /* Block element content. */
padding: 0;
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/* Block element titles. */
div.title, caption.title {
color: #527bbd;
font-weight: bold;
text-align: left;
margin-top: 1.0em;
margin-bottom: 0.5em;
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div.title + * {
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td div.title:first-child {
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div.content div.title:first-child {
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div.content + div.title {
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div.sidebarblock > div.content {
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padding: 0.5em;
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div.listingblock > div.content {
border: 1px solid #dddddd;
border-left: 5px solid #f0f0f0;
background: #f8f8f8;
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margin-left: 1.0em;
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div.verseblock > div.attribution {
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text-align: left;
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div.verseblock + div.attribution {
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div.admonitionblock .icon {
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div.exampleblock > div.content {
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padding-left: 0.5em;
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div.imageblock div.content { padding-left: 0; }
span.image img { border-style: none; vertical-align: text-bottom; }
a.image:visited { color: white; }
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margin-bottom: 0.8em;
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dt {
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margin-bottom: 0;
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ul, ol {
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ol.arabic {
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ol.loweralpha {
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ol.upperalpha {
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ol.lowerroman {
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ol.upperroman {
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tfoot {
font-weight: bold;
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td > div.verse {
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div.hdlist {
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margin-bottom: 0.8em;
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div.hdlist tr {
padding-bottom: 15px;
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dt.hdlist1.strong, td.hdlist1.strong {
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vertical-align: top;
font-style: normal;
padding-right: 0.8em;
color: navy;
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td.hdlist2 {
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div.hdlist.compact tr {
margin: 0;
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.comment {
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.footnote, .footnoteref {
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span.footnote, span.footnoteref {
vertical-align: super;
}
#footnotes {
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padding: 7px 0 0 0;
}
#footnotes div.footnote {
margin: 0 0 5px 0;
}
#footnotes hr {
border: none;
border-top: 1px solid silver;
height: 1px;
text-align: left;
margin-left: 0;
width: 20%;
min-width: 100px;
}
div.colist td {
padding-right: 0.5em;
padding-bottom: 0.3em;
vertical-align: top;
}
div.colist td img {
margin-top: 0.3em;
}
@media print {
#footer-badges { display: none; }
}
#toc {
margin-bottom: 2.5em;
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color: #527bbd;
font-size: 1.1em;
font-weight: bold;
margin-top: 1.0em;
margin-bottom: 0.1em;
}
div.toclevel0, div.toclevel1, div.toclevel2, div.toclevel3, div.toclevel4 {
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margin-bottom: 0;
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div.toclevel4 {
margin-left: 6em;
font-size: 0.9em;
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span.aqua { color: aqua; }
span.black { color: black; }
span.blue { color: blue; }
span.fuchsia { color: fuchsia; }
span.gray { color: gray; }
span.green { color: green; }
span.lime { color: lime; }
span.maroon { color: maroon; }
span.navy { color: navy; }
span.olive { color: olive; }
span.purple { color: purple; }
span.red { color: red; }
span.silver { color: silver; }
span.teal { color: teal; }
span.white { color: white; }
span.yellow { color: yellow; }
span.aqua-background { background: aqua; }
span.black-background { background: black; }
span.blue-background { background: blue; }
span.fuchsia-background { background: fuchsia; }
span.gray-background { background: gray; }
span.green-background { background: green; }
span.lime-background { background: lime; }
span.maroon-background { background: maroon; }
span.navy-background { background: navy; }
span.olive-background { background: olive; }
span.purple-background { background: purple; }
span.red-background { background: red; }
span.silver-background { background: silver; }
span.teal-background { background: teal; }
span.white-background { background: white; }
span.yellow-background { background: yellow; }
span.big { font-size: 2em; }
span.small { font-size: 0.6em; }
span.underline { text-decoration: underline; }
span.overline { text-decoration: overline; }
span.line-through { text-decoration: line-through; }
div.unbreakable { page-break-inside: avoid; }
/*
* xhtml11 specific
*
* */
div.tableblock {
margin-top: 1.0em;
margin-bottom: 1.5em;
}
div.tableblock > table {
border: 3px solid #527bbd;
}
thead, p.table.header {
font-weight: bold;
color: #527bbd;
}
p.table {
margin-top: 0;
}
/* Because the table frame attribute is overriden by CSS in most browsers. */
div.tableblock > table[frame="void"] {
border-style: none;
}
div.tableblock > table[frame="hsides"] {
border-left-style: none;
border-right-style: none;
}
div.tableblock > table[frame="vsides"] {
border-top-style: none;
border-bottom-style: none;
}
/*
* html5 specific
*
* */
table.tableblock {
margin-top: 1.0em;
margin-bottom: 1.5em;
}
thead, p.tableblock.header {
font-weight: bold;
color: #527bbd;
}
p.tableblock {
margin-top: 0;
}
table.tableblock {
border-width: 3px;
border-spacing: 0px;
border-style: solid;
border-color: #527bbd;
border-collapse: collapse;
}
th.tableblock, td.tableblock {
border-width: 1px;
padding: 4px;
border-style: solid;
border-color: #527bbd;
}
table.tableblock.frame-topbot {
border-left-style: hidden;
border-right-style: hidden;
}
table.tableblock.frame-sides {
border-top-style: hidden;
border-bottom-style: hidden;
}
table.tableblock.frame-none {
border-style: hidden;
}
th.tableblock.halign-left, td.tableblock.halign-left {
text-align: left;
}
th.tableblock.halign-center, td.tableblock.halign-center {
text-align: center;
}
th.tableblock.halign-right, td.tableblock.halign-right {
text-align: right;
}
th.tableblock.valign-top, td.tableblock.valign-top {
vertical-align: top;
}
th.tableblock.valign-middle, td.tableblock.valign-middle {
vertical-align: middle;
}
th.tableblock.valign-bottom, td.tableblock.valign-bottom {
vertical-align: bottom;
}
/*
* manpage specific
*
* */
body.manpage h1 {
padding-top: 0.5em;
padding-bottom: 0.5em;
border-top: 2px solid silver;
border-bottom: 2px solid silver;
}
body.manpage h2 {
border-style: none;
}
body.manpage div.sectionbody {
margin-left: 3em;
}
@media print {
body.manpage div#toc { display: none; }
}
</style>
<script type="text/javascript">
/*<+'])');
// Function that scans the DOM tree for header elements (the DOM2
// nodeIterator API would be a better technique but not supported by all
// browsers).
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if (mo && (i.getAttribute("class") || i.getAttribute("className")) != "float") {
result[result.length] = new TocEntry(i, getText(i), mo[1]-1);
}
iterate(i);
}
}
}
iterate(el);
return result;
}
var toc = document.getElementById("toc");
if (!toc) {
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// Delete existing TOC entries in case we're reloading the TOC.
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if (entry.nodeName.toLowerCase() == 'div'
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// Rebuild TOC entries.
var entries = tocEntries(document.getElementById("content"), toclevels);
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if (entries.length == 0)
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},
/////////////////////////////////////////////////////////////////////
// Footnotes generator
/////////////////////////////////////////////////////////////////////
/* Based on footnote generation code from:
* http://www.brandspankingnew.net/archive/2005/07/format_footnote.html
*/
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<body class="article">
<div id="header">
<h1>Summary of improvements since LAPACK 3.0</h1>
</div>
<div id="content">
<div class="sect1">
<h2 id="_1_improvements_in_lapack_3_1_0">1. Improvements in LAPACK 3.1.0</h2>
<div class="sectionbody">
<div class="paragraph"><p>See <a href="lapack-3.1.0.changes">Release Notes for 3.1.0</a> for more details</p></div>
<div class="olist arabic"><ol class="arabic">
<li>
<p>
<strong>Hessenberg QR algorithm with the small bulge multi-shift QR algorithm
together with aggressive early deflation.</strong> This is an implementation of the
2003 SIAM SIAG LA Prize winning algorithm of Braman, Byers and Mathias,
that significantly speeds up the nonsymmetric eigenproblem.
</p>
</li>
<li>
<p>
<strong>Improvements of the Hessenberg reduction subroutines.</strong> These accelerate the
first phase of the nonsymmetric eigenvalue problem.
</p>
</li>
<li>
<p>
<strong>New MRRR eigenvalue algorithms that also support subset computations.</strong>
These implementations of the 2006 SIAM SIAG LA Prize winning algorithm of
Dhillon and Parlett are also significantly more accurate than the version
in LAPACK 3.0.
</p>
</li>
<li>
<p>
<strong>Mixed precision iterative refinement subroutines for exploiting fast single
precision hardware.</strong> On platforms like the Cell processor that do single
precision much faster than double, linear systems can be solved many times
faster. Even on commodity processors there is a factor of 2 in speed
between single and double precision. These are prototype routines in the
sense that their interfaces might changed based on user feedback.
</p>
</li>
<li>
<p>
<strong>New partial column norm updating strategy for QR factorization with column
pivoting</strong>. This fixes a subtle numerical bug dating back to LINPACK that can
give completely wrong results.
</p>
</li>
<li>
<p>
<strong>Thread safety:</strong> Removed all the SAVE and DATA statements (or provided
alternate routines without those statements), increasing reliability
on SMPs.
</p>
</li>
<li>
<p>
<strong>Additional support for matrices with NaN/subnormal elements, optimization
of the balancing subroutines, improving reliability.</strong>
</p>
</li>
</ol></div>
</div>
</div>
<div class="sect1">
<h2 id="_2_improvements_in_lapack_3_1_1">2. Improvements in LAPACK 3.1.1</h2>
<div class="sectionbody">
<div class="paragraph"><p>See <a href="lapack-3.1.1.changes">Release Notes for 3.1.1</a> for more details</p></div>
<div class="olist arabic"><ol class="arabic">
<li>
<p>
<strong>Add BLAS routines so that the BLAS provided is complete</strong>
</p>
</li>
<li>
<p>
<strong>Provide 5 flavours of SECOND and DSECND and ddd the TIMER variable in the make.inc to chose which routine to use</strong>
</p>
</li>
</ol></div>
</div>
</div>
<div class="sect1">
<h2 id="_2_improvements_in_lapack_3_2_0">2. Improvements in LAPACK 3.2.0</h2>
<div class="sectionbody">
<div class="paragraph"><p>See <a href="lapack-3.2.html">Release Notes for 3.2</a> for more details</p></div>
<div class="olist arabic"><ol class="arabic">
<li>
<p>
<strong>Extra Precise Iterative Refinement:</strong> New linear solvers that
“guarantee” fully accurate answers (or give a warning that the
answer cannot be trusted). The matrix types supported in this
release are: GE (general), SY (symmetric), PO (positive definite),
HE (Hermitian), and GB (general band) in all the relevant
precisions.
</p>
</li>
<li>
<p>
<strong>XBLAS, or portable “extra precise BLAS”:</strong> Our new linear
solvers in (1) depend on these to perform iterative refinement.
The XBLAS will be released in a separate
package.
</p>
</li>
<li>
<p>
<strong>Non-Negative Diagonals from Householder QR:</strong> The QR
factorization routines now guarantee that the diagonal is both real
and non-negative. Factoring a uniformly random matrix now correctly
generates an orthogonal Q from the Haar distribution.
</p>
</li>
<li>
<p>
<strong>High Performance QR and Householder Reflections on Low-Profile
Matrices:</strong> The auxiliary routines to apply Householder reflections
(e.g. DLARFB) automatically reduce the cost of QR from O(n<sup>3</sup>) to
O(n<sup>2</sup>+nb<sup>2</sup>) for matrices stored in a dense format for band
matrices with bandwidth b with no user interface changes. Other
users of these routines can see similar benefits in particular on
“narrow profile” matrices.
</p>
</li>
<li>
<p>
<strong>New fast and accurate Jacobi SVD:</strong> High accuracy SVD routine for
dense matrices, which can compute tiny singular values to many more
correct digits than xGESVD when the matrix has columns differing
widely in norm, and usually runs faster than xGESVD too.
</p>
</li>
<li>
<p>
<strong>Routines for Rectangular Full Packed format:</strong> The RFP format
(SF, HF, PF, TF) enables efficient routines with optimal storage for
symmetric, Hermitian or triangular matrices. Since these routines
utilize the Level 3 BLAS, they are generally much more efficient
than the existing packed storage routines (SP, HP, PP, TP).
</p>
</li>
<li>
<p>
<strong>Pivoted Cholesky:</strong> The Cholesky factorization with diagonal
pivoting for symmetric positive semi-definite matrices. Pivoting is
required for reliable rank detection.
</p>
</li>
<li>
<p>
<strong>Mixed precision iterative refinement</strong> routines for exploiting
fast single precision hardware: On platforms like the Cell processor
that do single precision much faster than double, linear systems can
be solved many times faster. Even on commodity processors there is a
factor of 2 in speed between single and double precision. The
matrix types supported in this release are: GE (general), PO
(positive definite).
</p>
</li>
<li>
<p>
<strong>Some new variants added for the one sided factorization:</strong> LU
gets right-looking, left-looking, Crout and Recursive), QR gets
right-looking and left-looking, Cholesky gets left-looking,
right-looking and top-looking. Depending on the computer
architecture (or speed of the underlying BLAS), one of these
variants may be faster than the original LAPACK implementation.
</p>
</li>
<li>
<p>
<strong>More robust DQDS algorithm:</strong> Fixed some rare convergence
failures for the bidiagonal DQDS SVD routine.
</p>
</li>
<li>
<p>
<strong>Better documentation for the multishift Hessenberg QR algorithm
with early aggressive deflation, and various improvements of the
code.</strong>
</p>
</li>
</ol></div>
</div>
</div>
<div class="sect1">
<h2 id="_3_improvements_in_lapack_3_2_1">3. Improvements in LAPACK 3.2.1</h2>
<div class="sectionbody">
<div class="paragraph"><p>See <a href="lapack-3.2.1.html">Release Notes for 3.2.1</a> for more details</p></div>
<div class="olist arabic"><ol class="arabic">
<li>
<p>
<strong>change xLARRD (MRRR routines) to deal with some matrices from Godunov</strong>
</p>
</li>
</ol></div>
</div>
</div>
<div class="sect1">
<h2 id="_4_improvements_in_lapack_3_3_0">4. Improvements in LAPACK 3.3.0</h2>
<div class="sectionbody">
<div class="paragraph"><p>See <a href="lapack-3.3.0.html">Release Notes for 3.3.0</a> for more details</p></div>
<div class="olist arabic"><ol class="arabic">
<li>
<p>
<strong>Standard C language APIs for LAPACK:</strong> The standard interfaces include
support for Fortran and C data formats (column-major and row-major) to ease
interoperability with, and migration of, Fortran code. Full LAPACK
functionality is now accessible in a C-friendly manner. These new interfaces
are a key feature of the new LAPACK 3.3 release and the just released Intel
MKL 10.3 product. PLASMA 2.3 relies on it.
</p>
</li>
<li>
<p>
<strong>Computing the complete CS decomposition:</strong> Compute the complete CS
decomposition of a partitioned unitary matrix. The algorithm is designed
for numerical stability. See: Brian D. Sutton. Computing the complete CS
decomposition. Numer. Algorithms. 50 (2009), no. 1, 33–65.
</p>
</li>
<li>
<p>
<strong>Level-3 BLAS symmetric indefinite solve and symmetric indefinite inversion</strong>:
This enables better performance for xSYTRF and xSYTRI.
</p>
</li>
<li>
<p>
<strong>Thread safe xLAMCH</strong>: SLAMCH and DLAMCH were the only two routines not thread-safe
in LAPACK-3.2. This is fixed: all routines in LAPACK-3.3 are now thread-safe.
</p>
</li>
</ol></div>
</div>
</div>
<div class="sect1">
<h2 id="_4_improvements_in_lapack_3_3_1">4. Improvements in LAPACK 3.3.1</h2>
<div class="sectionbody">
<div class="paragraph"><p>See <a href="lapack-3.3.1.html">Release Notes for 3.3.1</a> for more details</p></div>
<div class="paragraph"><p><strong>Improve the CMAKE build system.</strong></p></div>
<div class="ulist"><ul>
<li>
<p>
Added a FindBLAS module to easily search for and link against various optimized BLAS implementations
</p>
</li>
<li>
<p>
Added proper default compiler flags for various commercial Fortran compilers
</p>
</li>
<li>
<p>
Check for and disable SIGFPE handlers (LAPACK code handles these internally) on known compilers
</p>
</li>
<li>
<p>
Remove testing drivers and BLAS source from nightly code coverage to get a more accurate representation of the LAPACK code getting exercised
</p>
</li>
<li>
<p>
Suppress harmless warning messages for various platforms and compilers
</p>