MatrixExp(mat, t = 1, method=NULL, ...)mat.expm.t is
a vector of length 2 or more, an array of exponentiated matrices.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 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 $\exp(tQ)$.
If mat is a valid transition intensity matrix for a
continuous-time Markov model (i.e. diagonal entries non-positive,
off-diagonal entries non-negative, 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 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
The series approximation method was adapted from the corresponding
function in Jim Lindsey's R package rmutil (
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