Matrix exponential
Description
Calculates the exponential of a square matrix.
Usage
1 
Arguments
mat 
A square matrix 
t 
An optional scaling factor for 
method 
Under the default of Otherwise, for backwards compatibility, the following options, which
use code in the msm package, are available:

... 
Arguments to pass to 
Details
See the expm
documentation for details of the
algorithms it uses.
Generally the exponential E of a square matrix M can often be 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 "pade"
or "series"
is supplied but M has distinct
eigenvalues. I If M has repeated eigenvalues, then its
eigenvector matrix may be noninvertible. 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 continuoustime 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).
If mat
is a valid transition intensity matrix for a
continuoustime Markov model (i.e. diagonal entries nonpositive,
offdiagonal entries nonnegative, rows sum to zero), then for certain
simpler model structures, there are
analytic formulae for the individual entries of the exponential
of mat
. These structures are
listed in the PDF manual and the formulae are coded in the msm
source file src/analyticp.c
. These formulae are only used if
method="analytic"
. This is more efficient, but it is not the
default in MatrixExp
because the code is not robust to extreme
values. However it is the default when calculating likelihoods for models fitted
by msm
.
The implementation of the Pade approximation used by
method="pade"
was taken from
JAGS by Martyn Plummer (http://mcmcjags.sourceforge.net).
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, twentyfive years later. SIAM Review 45, 3–49.