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MATRIX_EXPONENTIAL\ Algorithms for the Matrix Exponential {#matrix_exponential-algorithms-for-the-matrix-exponential align=”center”} =====================================

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MATRIX_EXPONENTIAL is a C++ library which exhibits and compares some algorithms for approximating the matrix exponential function.

Formally, for a square matrix A and scalar t, the matrix exponential exp(A*t) can be defined as the sum:

exp(A*t) = sum ( 0 <= i < oo ) A\^i t\^i / i!

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.

MATRIX_EXPONENTIAL needs the R8LIB library. The test code for MATRIX_EXPONENTIAL requires the TEST_MATRIX_EXPONENTIAL library.

Licensing: {#licensing align=”center”}

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages: {#languages align=”center”}

MATRIX_EXPONENTIAL is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs: {#related-data-and-programs align=”center”}

EXPOKIT, a FORTRAN77 library which solves various forms of the matrix exponential problem, by Roger Sidje.

R8LIB, a C++ library which contains many utility routines using double precision real (R8) arithmetic.

TEST_MAT, a C++ library which defines test matrices.

TEST_MATRIX_EXPONENTIAL, a C++ library which defines a set of test cases for computing the matrix exponential.

Reference: {#reference align=”center”}

1. Alan Laub,\ Review of “Linear System Theory” by Joao Hespanha,\ SIAM Review,\ Volume 52, Number 4, December 2010, page 779-781.
2. Cleve Moler, Charles VanLoan,\ Nineteen Dubious Ways to Compute the Exponential of a Matrix, SIAM Review,\ Volume 20, Number 4, October 1978, pages 801-836.
3. Cleve Moler, Charles VanLoan,\ Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later,\ SIAM Review,\ Volume 45, Number 1, March 2003, pages 3-49.
4. Roger Sidje,\ EXPOKIT: Software Package for Computing Matrix Exponentials,\ ACM Transactions on Mathematical Software,\ Volume 24, Number 1, 1998, pages 130-156.
5. Robert Ward,\ Numerical computation of the matrix exponential with accuracy estimate,\ SIAM Journal on Numerical Analysis,\ Volume 14, Number 4, September 1977, pages 600-610.

Source Code: {#source-code align=”center”}

• matrix_exponential.cpp, the source code.
• matrix_exponential.hpp, the include file.

Examples and Tests: {#examples-and-tests align=”center”}

• matrix_exponential_prb.cpp, a sample calling program.
• matrix_exponential_prb_output.txt, the output file.

List of Routines: {#list-of-routines align=”center”}

• C8MAT_EXPM1 is essentially MATLAB’s built-in matrix exponential algorithm.
• R8MAT_EXPM1 is essentially MATLAB’s built-in matrix exponential algorithm.
• R8MAT_EXPM2 uses the Taylor series for the matrix exponential.
• R8MAT_EXPM3 approximates the matrix exponential using an eigenvalue approach.

You can go up one level to the C++ source codes.

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Last modified on 17 October 2012.