https://github.com/cran/pracma
Tip revision: e970b8e92676e0d1f2c84b116f0bad93b03b89c6 authored by Hans W. Borchers on 15 December 2019, 21:30:02 UTC
version 2.2.9
version 2.2.9
Tip revision: e970b8e
expm.Rd
\name{expm}
\alias{expm}
\alias{logm}
\title{
Matrix Exponential
}
\description{
Computes the exponential of a matrix.
}
\usage{
expm(A, np = 128)
logm(A)
}
\arguments{
\item{A}{numeric square matrix.}
\item{np}{number of points to use on the unit circle.}
}
\details{
For an analytic function \eqn{f} and a matrix \eqn{A} the expression
\eqn{f(A)} can be computed by the Cauchy integral
\deqn{f(A) = (2 \pi i)^{-1} \int_G (zI-A)^{-1} f(z) dz}
where \eqn{G} is a closed contour around the eigenvalues of \eqn{A}.
Here this is achieved by taking G to be a circle and approximating the
integral by the trapezoid rule.
\code{logm} is a fake at the moment as it computes the matrix logarithm
through taking the logarithm of its eigenvalues; will be replaced by an
approach using Pade interpolation.
Another more accurate and more reliable approach for computing these
functions can be found in the R package `expm'.
}
\value{
Matrix of the same size as \code{A}.
}
\references{
Moler, C., and Ch. Van Loan (2003). Nineteen Dubious Ways to Compute
the Exponential of a Matrix, Twenty-Five Years Later.
SIAM Review, Vol. 1, No. 1, pp. 1--46.
N. J. Higham (2008). Matrix Functions: Theory and Computation. SIAM
Society for Industrial and Applied Mathematics.
}
\author{
Idea and Matlab code for a cubic root by Nick Trefethen in his
``10 digits 1 page'' project.
}
\note{
This approach could be used for other analytic functions, but a point to
consider is which branch to take (e.g., for the \code{logm} function).
}
\seealso{
\code{expm::expm}
}
\examples{
## The Ward test cases described in the help for expm::expm agree up to
## 10 digits with the values here and with results from Matlab's expm !
A <- matrix(c(-49, -64, 24, 31), 2, 2)
expm(A)
# -0.7357588 0.5518191
# -1.4715176 1.1036382
A1 <- matrix(c(10, 7, 8, 7,
7, 5, 6, 5,
8, 6, 10, 9,
7, 5, 9, 10), nrow = 4, ncol = 4, byrow = TRUE)
expm(logm(A1))
logm(expm(A1))
## System of linear differential equations: y' = M y (y = c(y1, y2, y3))
M <- matrix(c(2,-1,1, 0,3,-1, 2,1,3), 3, 3, byrow=TRUE)
M
C1 <- 0.5; C2 <- 1.0; C3 <- 1.5
t <- 2.0; Mt <- expm(t * M)
yt <- Mt %*% c(C1, C2, C3) # [y1,y2,y3](t) = [C1, C2, C3] %*% t(Mt)
}
\keyword{ math }