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<html>
<head>
<title>
MATRIX_EXPONENTIAL - Algorithms for the Matrix Exponential
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
MATRIX_EXPONENTIAL <br> Algorithms for the Matrix Exponential
</h1>
<hr>
<p>
<b>MATRIX_EXPONENTIAL</b>
is a MATLAB library which
exhibits and compares some algorithms for approximating the matrix
exponential function.
</p>
<p>
Formally, for a square matrix A and scalar t, the matrix exponential
exp(A*t) can be defined as the sum:
<blockquote>
exp(A*t) = sum ( 0 <= i < oo ) A^i t^i / i!
</blockquote>
</p>
<p>
The simplest form of the matrix exponential problem asks for the
value when t = 1. Even for this simple case, and for a matrix
of small order, it can be quite difficult to compute the matrix
exponential accurately.
</p>
<p>
<b>MATRIX_EXPONENTIAL</b> needs the R8LIB library.
The test code for <b>MATRIX_EXPONENTIAL</b> requires the TEST_MATRIX_EXPONENTIAL
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>MATRIX_EXPONENTIAL</b> is available in
<a href = "../../c_src/matrix_exponential/matrix_exponential.html">a C version</a> and
<a href = "../../cpp_src/matrix_exponential/matrix_exponential.html">a C++ version</a> and
<a href = "../../f77_src/matrix_exponential/matrix_exponential.html">a FORTRAN77 version</a> and
<a href = "../../f_src/matrix_exponential/matrix_exponential.html">a FORTRAN90 version</a> and
<a href = "../../m_src/matrix_exponential/matrix_exponential.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f77_src/expokit/expokit.html">
EXPOKIT</a>,
a FORTRAN77 library which
solves various forms of the matrix exponential problem,
by Roger Sidje.
</p>
<p>
<a href = "../../m_src/r8lib/r8lib.html">
R8LIB</a>,
a MATLAB library which
contains many utility routines using double precision real (R8) arithmetic.
</p>
<p>
<a href = "../../m_src/test_mat/test_mat.html">
TEST_MAT</a>,
a MATLAB library which
defines test matrices.
</p>
<p>
<a href = "../../m_src/test_matrix_exponential/test_matrix_exponential.html">
TEST_MATRIX_EXPONENTIAL</a>,
a MATLAB library which
defines a set of test cases for computing the matrix exponential.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Alan Laub,<br>
Review of "Linear System Theory" by Joao Hespanha,<br>
SIAM Review,<br>
Volume 52, Number 4, December 2010, page 779-781.
</li>
<li>
Cleve Moler, Charles VanLoan,<br>
Nineteen Dubious Ways to Compute the Exponential of a Matrix,
SIAM Review,<br>
Volume 20, Number 4, October 1978, pages 801-836.
</li>
<li>
Cleve Moler, Charles VanLoan,<br>
Nineteen Dubious Ways to Compute the Exponential of a Matrix,
Twenty-Five Years Later,<br>
SIAM Review,<br>
Volume 45, Number 1, March 2003, pages 3-49.
</li>
<li>
Roger Sidje,<br>
EXPOKIT: Software Package for Computing Matrix Exponentials,<br>
ACM Transactions on Mathematical Software,<br>
Volume 24, Number 1, 1998, pages 130-156.
</li>
<li>
Robert Ward,<br>
Numerical computation of the matrix exponential with accuracy estimate,<br>
SIAM Journal on Numerical Analysis,<br>
Volume 14, Number 4, September 1977, pages 600-610.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "c8mat_expm1.m">c8mat_expm1.m</a>,
is essentially MATLAB's built-in matrix exponential algorithm.
</li>
<li>
<a href = "c8mat_expm2.m">c8mat_expm2.m</a>,
estimates the matrix exponential using a Taylor series.
</li>
<li>
<a href = "c8mat_expm3.m">c8mat_expm3.m</a>,
estimates the matrix exponential using information from an
eigenvalue decomposition.
</li>
<li>
<a href = "r8mat_expm1.m">r8mat_expm1.m</a>,
is essentially MATLAB's built-in matrix exponential algorithm.
</li>
<li>
<a href = "r8mat_expm2.m">r8mat_expm2.m</a>,
estimates the matrix exponential using a Taylor series.
</li>
<li>
<a href = "r8mat_expm3.m">r8mat_expm3.m</a>,
estimates the matrix exponential using information from an
eigenvalue decomposition.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "matrix_exponential_test.m">matrix_exponential_test.m</a>,
calls all the tests;
</li>
<li>
<a href = "matrix_exponential_test01.m">matrix_exponential_test01.m</a>,
compares several algorithms on the real test matrices.
</li>
<li>
<a href = "matrix_exponential_test02.m">matrix_exponential_test02.m</a>,
compares several algorithms on the complex test matrices.
</li>
<li>
<a href = "matrix_exponential_test_output.txt">
matrix_exponential_test_output.txt</a>,
output from the tests;
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last modified on 05 March 2013.
</i>
<!-- John Burkardt -->
</body>
</html>