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condition.html
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<html>
<head>
<title>
CONDITION - Matrix Condition Number Estimation
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
CONDITION <br> Matrix Condition Number Estimation
</h1>
<hr>
<p>
<b>CONDITION</b>
is a C++ library which
implements methods for computing or estimating the condition number of a matrix.
</p>
<p>
Let ||*|| be a matrix norm, let A be an invertible matrix, and inv(A) the inverse of A.
The condition number of A with respect to the norm ||*|| is defined to be
<pre>
kappa(A) = ||A|| * ||inv(A)||
</pre>
</p>
<p>
If A is not invertible, the condition number is taken to be infinity.
</p>
<p>
Facts about the condition number include:
<ul>
<li>
1 <= kappa(A) for all matrices A.
</li>
<li>
1 = kappa(I), where I is the identity matrix.
</li>
<li>
for the L2 matrix norm, the condition number of any orthogonal matrix is 1.
</li>
<li>
for the L2 matrix norm, the condition number is the ratio of the maximum
to minimum singular values;
</li>
</ul>
</p>
<p>
The <b>CONDITION</b> library needs access to a copy of the R8LIB library.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this
web page are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>CONDITION</b> is available in
<a href = "../../c_src/condition/condition.html">a C version</a> and
<a href = "../../cpp_src/condition/condition.html">a C++ version</a> and
<a href = "../../f77_src/condition/condition.html">a FORTRAN77 version</a> and
<a href = "../../f_src/condition/condition.html">a FORTRAN90 version</a> and
<a href = "../../m_src/condition/condition.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/linpack_d/linpack_d.html">
LINPACK_D</a>,
a C++ library which
solves linear systems using double precision real arithmetic;
</p>
<p>
<a href = "../../f77_src/napack/napack.html">
NAPACK</a>,
a FORTRAN77 library which
includes many routines for applied numerical linear algebra tasks,
including the matrix condition number,
by William Hager.
</p>
<p>
<a href = "../../cpp_src/r8lib/r8lib.html">
R8LIB</a>,
a C++ library which
contains many utility routines using double precision real (R8) arithmetic.
</p>
<p>
<a href = "../../cpp_src/test_mat/test_mat.html">
TEST_MAT</a>,
a C++ library which
defines test matrices for which some of the determinant, eigenvalues, inverse,
null vectors, P*L*U factorization or linear system solution are already known.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Alan Cline, Cleve Moler, Pete Stewart, James Wilkinson,<br>
An estimate for the Condition Number of a Matrix,<br>
Technical Report TM-310,<br>
Argonne National Laboratory, 1977.
</li>
<li>
Alan Cline, Russell Rew,<br>
A set of counterexamples to three condition number estimators,<br>
SIAM Journal on Scientific and Statistical Computing,<br>
Volume 4, Number 4, December 1983, pages 602-611.
</li>
<li>
William Hager,<br>
Condition Estimates,<br>
SIAM Journal on Scientific and Statistical Computing,<br>
Volume 5, Number 2, June 1984, pages 311-316.
</li>
<li>
Nicholas Higham,<br>
A survey of condition number estimation for triangular matrices,<br>
SIAM Review,<br>
Volume 9, Number 4, December 1987, pages 575-596.
</li>
<li>
Diane OLeary,<br>
Estimating matrix condition numbers,<br>
SIAM Journal on Scientific and Statistical Computing,<br>
Volume 1, Number 2, June 1980, pages 205-209.
</li>
<li>
Pete Stewart,<br>
Efficient Generation of Random Orthogonal Matrices With an Application
to Condition Estimators,<br>
SIAM Journal on Numerical Analysis,<br>
Volume 17, Number 3, June 1980, pages 403-409.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "condition.cpp">condition.cpp</a>, the source code.
</li>
<li>
<a href = "condition.hpp">condition.hpp</a>, the include file.
</li>
<li>
<a href = "condition.sh">condition.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "condition_prb.cpp">condition_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "condition_prb.sh">condition_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "condition_prb_output.txt">condition_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>COMBIN</b> returns the COMBIN matrix.
</li>
<li>
<b>COMBIN_INVERSE</b> returns the inverse of the COMBIN matrix.
</li>
<li>
<b>CONDITION_HAGER</b> estimates the L1 condition number of a matrix.
</li>
<li>
<b>CONDITION_LINPACK</b> estimates the L1 condition number.
</li>
<li>
<b>CONDITION_SAMPLE1</b> estimates the L1 condition number of a matrix.
</li>
<li>
<b>CONEX1</b> returns the CONEX1 matrix.
</li>
<li>
<b>CONEX1_INVERSE</b> returns the inverse of the CONEX1 matrix.
</li>
<li>
<b>CONEX2</b> returns the CONEX2 matrix.
</li>
<li>
<b>CONEX2_INVERSE</b> returns the inverse of the CONEX2 matrix.
</li>
<li>
<b>CONEX3</b> returns the CONEX3 matrix.
</li>
<li>
<b>CONEX3_INVERSE</b> returns the inverse of the CONEX3 matrix.
</li>
<li>
<b>CONEX4</b> returns the CONEX4 matrix.
</li>
<li>
<b>CONEX4_INVERSE</b> returns the inverse of the CONEX4 matrix.
</li>
<li>
<b>KAHAN</b> returns the KAHAN matrix.
</li>
<li>
<b>KAHAN_INVERSE</b> returns the inverse of the KAHAN matrix.
</li>
<li>
<b>R8GE_FA</b> performs a LINPACK-style PLU factorization of a R8GE matrix.
</li>
<li>
<b>R8GE_INVERSE</b> computes the inverse of a R8GE matrix factored by R8GE_FA.
</li>
<li>
<b>R8GE_SL</b> solves a R8GE system factored by R8GE_FA.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 04 October 2012.
</i>
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