Matrix Exponential
Computes the exponential of a matrix.
expm(A, np = 128) logm(A)
A |
numeric square matrix. |
np |
number of points to use on the unit circle. |
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’.
Matrix of the same size as A.
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).
Idea and Matlab code for a cubic root by Nick Trefethen in his “10 digits 1 page” project.
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.
N. J. Higham (2008). Matrix Functions: Theory and Computation. SIAM Society for Industrial and Applied Mathematics.
expm::expm
## 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 <- MtPlease choose more modern alternatives, such as Google Chrome or Mozilla Firefox.