expAtv: Compute Matrix Exponential exp(A t) * v directly

Description

Compute $\exp(A t) * v$ directly, without evaluating $\exp(A)$.

Usage

expAtv(A, v, t = 1,
       method = "Sidje98",
       rescaleBelow = 1e-6,
       tol = 1e-07, btol = 1e-07, m.max = 30, mxrej = 10,
       verbose = getOption("verbose"))

Arguments

n x n matrix

n - vector

number (scalar);

method

a string indicating the method to be used; there's only one currently; we would like to add newer ones.

rescaleBelow

if norm(A,"I") is smaller than rescaleBelow, rescale A to norm 1 and t such that $A t$ remains unchanged. This step is in addition to Sidje's original algorithm

tol, btol

tolerances; these are tuning constants of the "Sidje1998" method which the user should typically not change.

m.max, mxrej

integer constants you should only change if you know what you're doing

verbose

flag indicating if the algorithm should be verbose..

Value

a list with components
eAtv.....fixme...

References

Roger B. Sidje (1998) EXPOKIT: Software Package for Computing Matrix Exponentials. ACM - Transactions On Mathematical Software 24(1), 130--156.

((NOT yet available!)) Al-Mohy, A. and Higham, N. (2011). Computing the Action of the Matrix Exponential, with an Application to Exponential Integrators. SIAM Journal on Scientific Computing, 33(2), 488--511.

Examples

Run this code

source(system.file("demo", "exact-fn.R", package = "expm"))
##-> rnilMat() ; xct10
set.seed(1)
(s5 <- Matrix(m5 <- rnilMat(5))); v <- c(1,6:9)
(em5 <- expm(m5))
r5  <- expAtv(m5, v)
r5. <- expAtv(s5, v)
stopifnot(all.equal(r5, r5., tol = 1e-14),
          all.equal(c(em5 %*% v), r5$eAtv))

v <- 10:1
with(xct10, all.equal(expm(m), expm))
all.equal(c(xct10$expm %*% v),
          expAtv(xct10$m, v)$eAtv)