Matrix Exponential

Description

Compute the exponential of a matrix.

Usage

expm(x)

Arguments

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.

Value

The matrix exponential of x.

Note

The expm package contains newer (partly faster and more accurate) algorithms for expm() and includes logm and sqrtm.

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.

References

http://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.

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

See Also

Schur; additionally, expm, logm, etc in package expm.

Examples

(m1 <- Matrix(c(1,0,1,1), nc = 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), nc = 2))
(e2 <- expm(m2))
(m3 <- Matrix(cbind(0,rbind(6*diag(3),0))))# sparse!
(e3 <- expm(m3)) # upper triangular