MatrixExp {msm}R Documentation

Matrix exponential

Description

Calculates the exponential of a square matrix.

Usage

MatrixExp(mat, t = 1, n = 20, k = 3, method="pade")

Arguments

mat

A square matrix

t

An optional scaling factor, or a vector or scaling factors, for the eigenvalues of mat

n

Number of terms in the series approximation to the exponential

k

Underflow correction factor, for the series approximation

method

"pade" for the Pade approximation, or "series" for the power series approximation. Ignored unless mat has repeated eigenvalues.

Details

The exponential E of a square matrix M is calculated as

E = U exp(D) U^{-1}

where D is a diagonal matrix with the eigenvalues of M on the diagonal, exp(D) is a diagonal matrix with the exponentiated eigenvalues of M on the diagonal, and U is a matrix whose columns are the eigenvectors of M.

This method of calculation is used if M has distinct eigenvalues. I If M has repeated eigenvalues, then its eigenvector matrix may be non-invertible. In this case, the matrix exponential is calculated using the Pade approximation defined by Moler and van Loan (2003), or the less robust power series approximation,

exp(M) = I + M + M^2/2 + M^3 / 3! + M^4 / 4! + ...

For a continuous-time homogeneous Markov process with transition intensity matrix Q, the probability of occupying state s at time u + t conditional on occupying state r at time u is given by the (r,s) entry of the matrix exp(tQ).

The implementation of the Pade approximation was taken from JAGS by Martyn Plummer (http://www-fis.iarc.fr/~martyn/software/jags).

The series approximation method was adapted from the corresponding function in Jim Lindsey's R package rmutil (http://popgen.unimaas.nl/~jlindsey/rcode.html).

Value

The exponentiated matrix exp(mat). Or, if t is a vector of length 2 or more, an array of exponentiated matrices.

References

Cox, D. R. and Miller, H. D. The theory of stochastic processes, Chapman and Hall, London (1965)

Moler, C and van Loan, C (2003). Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Review 45, 3–49.
At http://epubs.siam.org/sam-bin/dbq/article/41801


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