Compute the Frechet derivative of the matrix exponential operator
Usage
expmFrechet(A, E, method = c("SPS", "blockEnlarge"), expm = TRUE)
Arguments
A
square matrix (n x n).
E
the “small Error” matrix,
used in L(A,E) = f(A + E, A)
method
string specifying the method / algorithm; the default
"SPS" is “Scaling + Pade + Squaring” as in the
algorithm 6.4 below; otherwise see the ‘Details’ section.
expm
logical indicating if the matrix exponential itself, which
is computed anyway, should be returned as well.
Details
Calculation of e^A and the Exponential Frechet-Derivative
L(A,E).
method = "blockEnlarge" uses the matrix identity of
f([A E ; 0 A ]) = [f(A) Df(A); 0 f(A)]
for the (2n) x (2n) block matrices where f(A) := expm(A) and
Df(A) := L(A,E). Note that "blockEnlarge" is much
simpler to implement but slower (CPU time is doubled for n = 100).
Value
a list with components
expm
if expm is true, the matrix exponential
(n x n matrix).
Lexpm
the Exponential-Frechet-Derivative L(A,E), a matrix
of the same dimension.
Author(s)
Michael Stadelmann (final polish by Martin Maechler).
References
see expmCond.
See Also
expm.Higham08 for the matrix exponential.
expmCond for exponential condition number computations
which are based on expmFrechet.
Examples
(A <- cbind(1, 2:3, 5:8, c(9,1,5,3)))
E <- matrix(1e-3, 4,4)
(L.AE <- expmFrechet(A, E))
all.equal(L.AE, expmFrechet(A, E, "block"), tolerance = 1e-14) ## TRUE