MATRIX_EXPONENTIAL
Algorithms for the Matrix Exponential

MATRIX_EXPONENTIAL is a FORTRAN90 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.

Licensing:

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

Languages:

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:

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

TEST_MAT, a FORTRAN90 library which defines test matrices.

TEST_MATRIX_EXPONENTIAL, a FORTRAN90 library which defines a set of test cases for computing the matrix exponential.

Reference:

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:

• matrix_exponential.f90, the source code.
• matrix_exponential.sh, BASH commands to compile the source code.

Examples and Tests:

• matrix_exponential_prb.f90, a sample calling program.
• matrix_exponential_prb.sh, BASH commands to compile and run the sample program.
• matrix_exponential_prb_output.txt, the output file.

List of Routines:

• EXPM1 is essentially MATLAB's built-in matrix exponential algorithm.
• EXPM2 uses the Taylor series for the matrix exponential.
• EXPM3 approximates the matrix exponential using an eigenvalue approach.
• R8_LOG_2 returns the logarithm base 2 of an R8.
• R8MAT_FSS factors and solves multiple R8GE systems.
• R8MAT_IDENTITY stores the identity matrix in an R8MAT.
• R8MAT_INSIGNIFICANT determines if an R8MAT is insignificant.
• R8MAT_MINVM computes inverse(A) * B for R8MAT's.
• R8MAT_NORM_L1 returns the matrix L1 norm of an R8MAT.
• R8MAT_NORM_LI returns the matrix L-oo norm of an R8MAT.
• R8MAT_PRINT prints an R8MAT.
• R8MAT_PRINT_SOME prints some of an R8MAT.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN90 source codes.