Matrix (version 1.5-4)

expm: Matrix Exponential

Description

Compute the exponential of a matrix.

Usage

expm(x)

Value

The matrix exponential of x.

Arguments

x

a matrix, typically inheriting from the dMatrix class.

Author

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

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.

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.

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

See Also

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

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