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<H1 ALIGN="CENTER">LAPACK Users' Guide 
<BR>
Third Edition</H1>
<P ALIGN="CENTER"><STRONG>UPDATED: 22 Aug 1999</STRONG></P>
<P><STRONG>
<li><a href="http://www.cs.utk.edu/~eanderso/">E. Anderson</a>,
<li><a href="http://www.ms.uky.edu/~bai/">Z. Bai</a>,
<li><a href="http://www.mcs.anl.gov/people/bischof">C. Bischof</a>,
<li><a href="http://www.netlib.org/utk/people/SusanBlackford.html">S. Blackford</a>,
<li><a href="http://http.cs.berkeley.edu/~demmel/">J. Demmel</a>,
<li><a href="http://www.netlib.org/utk/people/JackDongarra.html">J. Dongarra</a>,
<li>J. Du Croz,
<li>A. Greenbaum,
<li><a href="http://www.nag.co.uk/Local/About/People/SvenHammarling/Shhome.asp">S. Hammarling</a>,
<li>A. McKenney,
<li>D. Sorensen

</STRONG><P>

<P>

<IMG ALIGN=BOTTOM SRC="_21926_tex2html_wrap2798.png"> 1999 by the
Society for Industrial and Applied Mathematics. Certain derivative work
portions have been copyrighted by the Numerical Algorithms Group Ltd.
<P>

The hardcopy version of <b>LAPACK Users' Guide, Third Edition</b> may be
ordered directly from <b>SIAM</b>. Mail ordering information and payment
to
<ul>
<li> SIAM
<br> Customer Service
<br> P.O. Box 7260
<br> Philadelphia, PA 19104
</ul>
<p> You can
also call 800-447-SIAM in USA and Canada or 215-382-9800 worldwide. You can
fax your order to 215-386-7999, send us e-mail
<a href="mailto:service@siam.org">service@siam.org</a>
or order over the web <a href="http://www.siam.org">www.siam.org</a>.
<p>
<table border cellspacing=0 cellpadding=5>
<tr>
 <th>Order Code: SE09</th><th>List Price: $39.00</th>
</tr>
<tr>
<th>ISBN:0-89871-447-8</th><th>SIAM Member Price: $31.20</th>
</tr>
</table>
</p>
<P>
The royalties from the sales of this book are being placed in a fund
to help students attend SIAM meetings and other SIAM related activities.
This fund is administered by SIAM and qualified individuals are
encouraged to
write directly to SIAM for guidelines.

<P>
<HR>
<BR>
<DIV ALIGN="CENTER">
<FONT SIZE="+1"><B>Dedication 
<BR></B></FONT>

</DIV>

<P>
This work is dedicated to Jim Wilkinson whose ideas and
spirit have given us inspiration and influenced the project
at every turn.

<HR>

<DIV ALIGN="CENTER">
<FONT SIZE="+1">Authors' Affiliations:

</FONT></DIV>
<BR>
<BR>
<DIV ALIGN="CENTER">
<FONT SIZE="+0">E. Anderson 
<BR>
<I>University of Tennessee, Knoxville</I> 
<BR>
<BR>
Z. Bai 
<BR>
<I>University of Kentucky and 
<BR>
University of California, Davis</I> 
<BR>
<BR>
C. Bischof 
<BR>
<I>Institute of Scientific Computing, Technical University Aachen, Germany</I> 
<BR> 
<BR>
L. S. Blackford (formerly L. S. Ostrouchov)
<BR>
<I>University of Tennessee, Knoxville</I> 
<BR> 
<BR>
J. Demmel 
<BR>
<I>University of California, Berkeley</I> 
<BR> 
<BR>
J. Dongarra 
<BR>
<I>University of Tennessee, Knoxville and 
<BR>
Oak Ridge National Laboratory</I> 
<BR>
<BR>
J. Du Croz 
<BR>
<I>Numerical Algorithms Group Ltd. (retired)</I> 
<BR> 
<BR>
A. Greenbaum 
<BR>
<I>University of Washington</I> 
<BR> 
<BR>
S. Hammarling 
<BR>
<I>Numerical Algorithms Group Ltd.</I> 
<BR> 
<BR>
A. McKenney 
<BR>
<BR>
D. Sorensen 
<BR>
<I>Rice University</I> 
<BR>

</FONT></DIV>

<P>

<BR><HR>
<!--Table of Child-Links-->
<A NAME="CHILD_LINKS"></A>

<UL>
<LI><A NAME="tex2html3682"
 HREF="node1.html">Contents</A>
<LI><A NAME="tex2html3683"
 HREF="node2.html">List of Tables</A>
<UL>
<LI><A NAME="tex2html3684"
 HREF="node3.html">Preface to the Third Edition</A>
<LI><A NAME="tex2html3685"
 HREF="node4.html">Preface to the Second Edition</A>
</UL>
<LI><A NAME="tex2html3686"
 HREF="node5.html">Guide</A>
<UL>
<LI><A NAME="tex2html3687"
 HREF="node6.html">Essentials</A>
<UL>
<LI><A NAME="tex2html3688"
 HREF="node7.html">LAPACK</A>
<LI><A NAME="tex2html3689"
 HREF="node8.html">Problems that LAPACK can Solve</A>
<LI><A NAME="tex2html3690"
 HREF="node9.html">Computers for which LAPACK is Suitable</A>
<LI><A NAME="tex2html3691"
 HREF="node10.html">LAPACK Compared with LINPACK and EISPACK</A>
<LI><A NAME="tex2html3692"
 HREF="node11.html">LAPACK and the BLAS</A>
<LI><A NAME="tex2html3693"
 HREF="node12.html">Availability of LAPACK</A>
<LI><A NAME="tex2html3694"
 HREF="node13.html">Commercial Use of LAPACK</A>
<LI><A NAME="tex2html3695"
 HREF="node14.html">Installation of LAPACK</A>
<LI><A NAME="tex2html3696"
 HREF="node15.html">Documentation for LAPACK</A>
<LI><A NAME="tex2html3697"
 HREF="node16.html">Support for LAPACK</A>
<LI><A NAME="tex2html3698"
 HREF="node17.html">Errata in LAPACK</A>
<LI><A NAME="tex2html3699"
 HREF="node18.html">Other Related Software</A>
</UL>
<LI><A NAME="tex2html3700"
 HREF="node19.html">Contents of LAPACK</A>
<UL>
<LI><A NAME="tex2html3701"
 HREF="node20.html">What's new in version 3.0?</A>
<LI><A NAME="tex2html3702"
 HREF="node21.html">Structure of LAPACK</A>
<UL>
<LI><A NAME="tex2html3703"
 HREF="node22.html">Levels of Routines</A>
<LI><A NAME="tex2html3704"
 HREF="node23.html">Data Types and Precision</A>
<LI><A NAME="tex2html3705"
 HREF="node24.html">Naming Scheme</A>
</UL>
<LI><A NAME="tex2html3706"
 HREF="node25.html">Driver Routines</A>
<UL>
<LI><A NAME="tex2html3707"
 HREF="node26.html">Linear Equations</A>
<LI><A NAME="tex2html3708"
 HREF="node27.html">Linear Least Squares (LLS) Problems</A>
<LI><A NAME="tex2html3709"
 HREF="node28.html">Generalized Linear Least Squares (LSE and GLM) Problems</A>
<LI><A NAME="tex2html3710"
 HREF="node29.html">Standard Eigenvalue and Singular Value Problems</A>
<LI><A NAME="tex2html3711"
 HREF="node33.html">Generalized Eigenvalue and Singular Value
Problems</A>
</UL>
<LI><A NAME="tex2html3712"
 HREF="node37.html">Computational Routines</A>
<UL>
<LI><A NAME="tex2html3713"
 HREF="node38.html">Linear Equations</A>
<LI><A NAME="tex2html3714"
 HREF="node39.html">Orthogonal Factorizations and Linear Least Squares Problems</A>
<LI><A NAME="tex2html3715"
 HREF="node45.html">Generalized Orthogonal Factorizations and Linear Least Squares Problems</A>
<LI><A NAME="tex2html3716"
 HREF="node48.html">Symmetric Eigenproblems</A>
<LI><A NAME="tex2html3717"
 HREF="node49.html">Nonsymmetric Eigenproblems</A>
<LI><A NAME="tex2html3718"
 HREF="node53.html">Singular Value Decomposition</A>
<LI><A NAME="tex2html3719"
 HREF="node54.html">Generalized Symmetric Definite Eigenproblems</A>
<LI><A NAME="tex2html3720"
 HREF="node55.html">Generalized Nonsymmetric Eigenproblems</A>
<LI><A NAME="tex2html3721"
 HREF="node59.html">Generalized (or Quotient) Singular Value Decomposition</A>
</UL>
</UL>
<LI><A NAME="tex2html3722"
 HREF="node60.html">Performance of LAPACK</A>
<UL>
<LI><A NAME="tex2html3723"
 HREF="node61.html">Factors that Affect Performance</A>
<UL>
<LI><A NAME="tex2html3724"
 HREF="node62.html">Vectorization</A>
<LI><A NAME="tex2html3725"
 HREF="node63.html">Data Movement</A>
<LI><A NAME="tex2html3726"
 HREF="node64.html">Parallelism</A>
</UL>
<LI><A NAME="tex2html3727"
 HREF="node65.html">The BLAS as the Key to Portability</A>
<LI><A NAME="tex2html3728"
 HREF="node66.html">Block Algorithms and their
Derivation</A>
<LI><A NAME="tex2html3729"
 HREF="node67.html">Examples of Block Algorithms in LAPACK</A>
<UL>
<LI><A NAME="tex2html3730"
 HREF="node68.html">Factorizations for Solving Linear Equations</A>
<LI><A NAME="tex2html3731"
 HREF="node69.html"><B><I>QR</I></B> Factorization</A>
<LI><A NAME="tex2html3732"
 HREF="node70.html">Eigenvalue Problems</A>
</UL>
<LI><A NAME="tex2html3733"
 HREF="node71.html">LAPACK Benchmark</A>
</UL>
<LI><A NAME="tex2html3734"
 HREF="node72.html">Accuracy and Stability</A>
<UL>
<LI><A NAME="tex2html3735"
 HREF="node73.html">Sources of Error in Numerical Calculations</A>
<UL>
<LI><A NAME="tex2html3736"
 HREF="node74.html">Further Details:  Floating Point Arithmetic</A>
</UL>
<LI><A NAME="tex2html3737"
 HREF="node75.html">How to Measure Errors</A>
<UL>
<LI><A NAME="tex2html3738"
 HREF="node76.html">Further Details:  How to Measure Errors</A>
</UL>
<LI><A NAME="tex2html3739"
 HREF="node77.html">Further Details:  How Error Bounds Are Derived</A>
<UL>
<LI><A NAME="tex2html3740"
 HREF="node78.html">Standard Error Analysis</A>
<LI><A NAME="tex2html3741"
 HREF="node79.html">Improved Error Bounds</A>
</UL>
<LI><A NAME="tex2html3742"
 HREF="node80.html">Error Bounds for Linear Equation Solving</A>
<UL>
<LI><A NAME="tex2html3743"
 HREF="node81.html">Further Details:  Error Bounds for Linear Equation Solving</A>
</UL>
<LI><A NAME="tex2html3744"
 HREF="node82.html">Error Bounds for Linear Least Squares Problems</A>
<UL>
<LI><A NAME="tex2html3745"
 HREF="node83.html">Further Details:  Error Bounds for Linear Least Squares
Problems</A>
</UL>
<LI><A NAME="tex2html3746"
 HREF="node84.html">Error Bounds for Generalized Least Squares Problems</A>
<UL>
<LI><A NAME="tex2html3747"
 HREF="node85.html">Linear Equality Constrained Least Squares Problem</A>
<LI><A NAME="tex2html3748"
 HREF="node87.html">General Linear Model Problem</A>
</UL>
<LI><A NAME="tex2html3749"
 HREF="node89.html">Error Bounds for the Symmetric Eigenproblem </A>
<UL>
<LI><A NAME="tex2html3750"
 HREF="node90.html">Further Details:  Error Bounds for the Symmetric Eigenproblem </A>
</UL>
<LI><A NAME="tex2html3751"
 HREF="node91.html">Error Bounds for the Nonsymmetric Eigenproblem </A>
<UL>
<LI><A NAME="tex2html3752"
 HREF="node92.html">Further Details:  Error Bounds for the Nonsymmetric Eigenproblem</A>
</UL>
<LI><A NAME="tex2html3753"
 HREF="node96.html">Error Bounds for the Singular Value Decomposition</A>
<UL>
<LI><A NAME="tex2html3754"
 HREF="node97.html">Further Details:  Error Bounds for the Singular Value Decomposition</A>
</UL>
<LI><A NAME="tex2html3755"
 HREF="node98.html">Error Bounds for the Generalized Symmetric Definite
Eigenproblem</A>
<UL>
<LI><A NAME="tex2html3756"
 HREF="node99.html">Further Details:  Error Bounds for the Generalized Symmetric Definite
Eigenproblem</A>
</UL>
<LI><A NAME="tex2html3757"
 HREF="node100.html">Error Bounds for the Generalized Nonsymmetric Eigenproblem</A>
<UL>
<LI><A NAME="tex2html3758"
 HREF="node101.html">Further Details: Error Bounds for the Generalized
Nonsymmetric Eigenproblem</A>
</UL>
<LI><A NAME="tex2html3759"
 HREF="node106.html">Error Bounds for the Generalized Singular Value Decomposition</A>
<UL>
<LI><A NAME="tex2html3760"
 HREF="node107.html">Further Details:  Error Bounds for the Generalized Singular Value Decomposition</A>
</UL>
<LI><A NAME="tex2html3761"
 HREF="node108.html">Error Bounds for Fast Level 3 BLAS</A>
</UL>
<LI><A NAME="tex2html3762"
 HREF="node109.html">Documentation and Software Conventions</A>
<UL>
<LI><A NAME="tex2html3763"
 HREF="node110.html">Design and Documentation of Argument Lists</A>
<UL>
<LI><A NAME="tex2html3764"
 HREF="node111.html">Structure of the Documentation</A>
<LI><A NAME="tex2html3765"
 HREF="node112.html">Order of Arguments</A>
<LI><A NAME="tex2html3766"
 HREF="node113.html">Argument Descriptions</A>
<LI><A NAME="tex2html3767"
 HREF="node114.html">Option Arguments</A>
<LI><A NAME="tex2html3768"
 HREF="node115.html">Problem Dimensions</A>
<LI><A NAME="tex2html3769"
 HREF="node116.html">Array Arguments</A>
<LI><A NAME="tex2html3770"
 HREF="node117.html">Work Arrays</A>
<LI><A NAME="tex2html3771"
 HREF="node118.html">LWORK Query</A>
<LI><A NAME="tex2html3772"
 HREF="node119.html">Error Handling and the Diagnostic Argument INFO</A>
</UL>
<LI><A NAME="tex2html3773"
 HREF="node120.html">Determining the Block Size for Block Algorithms</A>
<LI><A NAME="tex2html3774"
 HREF="node121.html">Matrix Storage Schemes</A>
<UL>
<LI><A NAME="tex2html3775"
 HREF="node122.html">Conventional Storage</A>
<LI><A NAME="tex2html3776"
 HREF="node123.html">Packed Storage</A>
<LI><A NAME="tex2html3777"
 HREF="node124.html">Band Storage</A>
<LI><A NAME="tex2html3778"
 HREF="node125.html">Tridiagonal and Bidiagonal Matrices</A>
<LI><A NAME="tex2html3779"
 HREF="node126.html">Unit Triangular Matrices</A>
<LI><A NAME="tex2html3780"
 HREF="node127.html">Real Diagonal Elements of Complex Matrices</A>
</UL>
<LI><A NAME="tex2html3781"
 HREF="node128.html">Representation of Orthogonal or Unitary Matrices</A>
</UL>
<LI><A NAME="tex2html3782"
 HREF="node129.html">Installing LAPACK Routines</A>
<UL>
<LI><A NAME="tex2html3783"
 HREF="node130.html">Points to Note</A>
<LI><A NAME="tex2html3784"
 HREF="node131.html">Installing ILAENV</A>
</UL>
<LI><A NAME="tex2html3785"
 HREF="node132.html">Troubleshooting</A>
<UL>
<LI><A NAME="tex2html3786"
 HREF="node133.html">Bug Report Checklist</A>
<LI><A NAME="tex2html3787"
 HREF="node134.html">Installation Debugging Hints</A>
<LI><A NAME="tex2html3788"
 HREF="node135.html">Common Errors in Calling LAPACK Routines</A>
<LI><A NAME="tex2html3789"
 HREF="node136.html">Failures Detected by LAPACK Routines</A>
<UL>
<LI><A NAME="tex2html3790"
 HREF="node137.html">Invalid Arguments and XERBLA</A>
<LI><A NAME="tex2html3791"
 HREF="node138.html">Computational Failures and INFO <B>&gt;</B> 0</A>
</UL>
<LI><A NAME="tex2html3792"
 HREF="node139.html">Wrong Results</A>
<LI><A NAME="tex2html3793"
 HREF="node140.html">Poor Performance</A>
</UL>
<LI><A NAME="tex2html3794"
 HREF="node141.html">Index of Driver and Computational Routines</A>
<UL>
<LI><A NAME="tex2html3795"
 HREF="node142.html">Notes</A>
</UL>
<LI><A NAME="tex2html3796"
 HREF="node143.html">Index of Auxiliary Routines</A>
<UL>
<LI><A NAME="tex2html3797"
 HREF="node144.html">Notes</A>
</UL>
<LI><A NAME="tex2html3798"
 HREF="node145.html">Quick Reference 
Guide to the BLAS</A>
<LI><A NAME="tex2html3799"
 HREF="node146.html">Converting from LINPACK or EISPACK</A>
<UL>
<LI><A NAME="tex2html3800"
 HREF="node147.html">Notes</A>
</UL>
<LI><A NAME="tex2html3801"
 HREF="node148.html">LAPACK Working Notes</A>
</UL>
<LI><A NAME="tex2html3802"
 HREF="node149.html">Specifications of Routines</A>
<UL>
<LI><A NAME="tex2html3803"
 HREF="node150.html">Notes</A>
</UL>
<LI><A NAME="tex2html3804"
 HREF="node151.html">Bibliography</A>
<LI><A NAME="tex2html3805"
 HREF="node152.html">Index by Keyword</A>
<LI><A NAME="tex2html3806"
 HREF="node153.html">About this document ...</A>
</UL>
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<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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