expm: Matrix Exponential In pracma: Practical Numerical Math Functions

Description

Computes the exponential of a matrix.

Usage

 1 2 3 expm(A, np = 128) logm(A) 

Arguments

 A numeric square matrix. np number of points to use on the unit circle.

Details

For an analytic function f and a matrix A the expression f(A) can be computed by the Cauchy integral

f(A) = (2 π i)^{-1} \int_G (zI-A)^{-1} f(z) dz

where G is a closed contour around the eigenvalues of A.

Here this is achieved by taking G to be a circle and approximating the integral by the trapezoid rule.

logm is a fake at the moment as it computes the matrix logarithm through taking the logarithm of its eigenvalues; will be replaced by an approach using Pade interpolation.

Another more accurate and more reliable approach for computing these functions can be found in the R package ‘expm’.

Value

Matrix of the same size as A.

Note

This approach could be used for other analytic functions, but a point to consider is which branch to take (e.g., for the logm function).

Author(s)

Idea and Matlab code for a cubic root by Nick Trefethen in his “10 digits 1 page” project, for realization see file ‘cuberootA.m’ at http://people.maths.ox.ac.uk/trefethen/tda.html.

References

Moler, C., and Ch. Van Loan (2003). Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later. SIAM Review, Vol. 1, No. 1, pp. 1–46. [Available at CiteSeer, citeseer.ist.psu.edu]

N. J. Higham (2008). Matrix Functions: Theory and Computation. SIAM Society for Industrial and Applied Mathematics.

expm::expm
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ## The Ward test cases described in the help for expm::expm agree up to ## 10 digits with the values here and with results from Matlab's expm ! A <- matrix(c(-49, -64, 24, 31), 2, 2) expm(A) # -0.7357588 0.5518191 # -1.4715176 1.1036382 A1 <- matrix(c(10, 7, 8, 7, 7, 5, 6, 5, 8, 6, 10, 9, 7, 5, 9, 10), nrow = 4, ncol = 4, byrow = TRUE) expm(logm(A1)) logm(expm(A1)) ## System of linear differential equations: y' = M y (y = c(y1, y2, y3)) M <- matrix(c(2,-1,1, 0,3,-1, 2,1,3), 3, 3, byrow=TRUE) M C1 <- 0.5; C2 <- 1.0; C3 <- 1.5 t <- 2.0; Mt <- expm(t * M) yt <- Mt