expm-methods: Matrix Exponential

In Matrix: Sparse and Dense Matrix Classes and Methods

Matrix Exponential

Description

Compute the exponential of a matrix.

Usage

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...”.

Value

The matrix exponential of x.

Author(s)

This is a translation of the implementation of the corresponding Octave function contributed to the Octave project by A. Scottedward Hodel [e-mail]. A bug in there has been fixed by Martin Maechler.

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. \Sexpr[results=rd]{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for historical reference mostly: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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1137/1020098")}

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

See Also

Package 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

Matrix documentation built on Aug. 13, 2024, 3:01 p.m.