expm-methods function

Matrix Exponential

Compute the exponential of a matrix.

methods

expm(x)

Arguments

• x: a matrix, typically inheriting from the dMatrix class.

Details

The exponential of a matrix is defined as the infinite Taylor series expm(A) = I + A + A^2/2! + A^3/3! + ... (although this is definitely not the way to compute it). The method for the dgeMatrix class uses Ward's diagonal Pade' approximation with three step preconditioning, a recommendation from Moler & Van Loan (1978) Nineteen dubious ways... .

Returns

The matrix exponential of x.

References

https://en.wikipedia.org/wiki/Matrix_exponential

Cleve Moler and Charles Van Loan (2003) Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Review 45 , 1, 3--49. tools:::Rd_expr_doi("10.1137/S00361445024180")

for historical reference mostly:

Moler, C. and Van Loan, C. (1978) Nineteen dubious ways to compute the exponential of a matrix. SIAM Review 20 , 4, 801--836. tools:::Rd_expr_doi("10.1137/1020098")

Eric W. Weisstein et al. (1999) Matrix Exponential. From MathWorld, https://mathworld.wolfram.com/MatrixExponential.html

Author(s)

This is a translation of the implementation of the corresponding Octave function contributed to the Octave project by A. Scottedward Hodel A.S.Hodel@Eng.Auburn.EDU . A bug in there has been fixed by Martin Maechler.

See Also

Package list("expm"), which provides newer (in some cases faster, more accurate) algorithms for computing the matrix exponential via its own (non-generic) function expm(). expm also implements logm(), sqrtm(), etc.

Generic function Schur.

Examples

(m1 <- Matrix(c(1,0,1,1), ncol = 2))
(e1 <- expm(m1)) ; e <- exp(1)
stopifnot(all.equal(e1@x, c(e,0,e,e), tolerance = 1e-15))
(m2 <- Matrix(c(-49, -64, 24, 31), ncol = 2))
(e2 <- expm(m2))
(m3 <- Matrix(cbind(0,rbind(6*diag(3),0))))# sparse!
(e3 <- expm(m3)) # upper triangular