expm

Matrix exponential

Syntax

Y = expm(X)

Description

Y = expm(X) computes the matrix exponential of X. Although it is not computed this way, if X has a full set of eigenvectors V with corresponding eigenvalues D, then [V,D] = eig(X) and

expm(X) = V*diag(exp(diag(D)))/V

Use exp for the element-by-element exponential.

Examples

Compare Exponentials

Compute and compare the exponential of A with the matrix exponential of A.

A = [1 1 0; 0 0 2; 0 0 -1];
exp(A)

ans = 3×3

    2.7183    2.7183    1.0000
    1.0000    1.0000    7.3891
    1.0000    1.0000    0.3679

expm(A)

ans = 3×3

    2.7183    1.7183    1.0862
         0    1.0000    1.2642
         0         0    0.3679

Notice that the diagonal elements of the two results are equal, which is true for any triangular matrix. The off-diagonal elements, including those below the diagonal, are different.

Input Arguments

X — Input matrix

square matrix

Input matrix, specified as a square matrix.

Data Types: single | double
Complex Number Support: Yes

Algorithms

The algorithm expm uses is described in [1] and [2].

Note

Matrix Exponentials illustrates the use of Padé approximation, Taylor series approximation, and eigenvalues and eigenvectors to compute the matrix exponential. References [3] and [4] describe and compare many algorithms for computing a matrix exponential.

References

[1] Higham, N. J., “The Scaling and Squaring Method for the Matrix Exponential Revisited,” SIAM J. Matrix Anal. Appl., 26(4) (2005), pp. 1179–1193.

[2] Al-Mohy, A. H. and N. J. Higham, “A new scaling and squaring algorithm for the matrix exponential,” SIAM J. Matrix Anal. Appl., 31(3) (2009), pp. 970–989.

[3] Golub, G. H. and C. F. Van Loan, Matrix Computation, p. 384, Johns Hopkins University Press, 1983.

[4] Moler, C. B. and C. F. Van Loan, “Nineteen Dubious Ways to Compute the Exponential of a Matrix,” SIAM Review 20, 1978, pp. 801–836. Reprinted and updated as “Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later,” SIAM Review 45, 2003, pp. 3–49.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• Code generation does not support sparse matrix inputs for this function.

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

The expm function fully supports GPU arrays. To run the function on a GPU, specify the input data as a gpuArray (Parallel Computing Toolbox). For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Version History

Introduced before R2006a

R2023b: Improved algorithm for single-precision matrices

The expm function has an improved algorithm for input matrices that are single precision. The new algorithm can use fewer terms in the Padé approximation of some matrix exponentials.