Compute Matrix Exponential exp(A t) * v directly

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

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

A

n x n matrix

v

n - vector

t

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 and easily seen to be necessary even in simple cases (e.g., \(n = 3\)).

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 in our expm package!)
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. doi:10.1137/100788860

Author

Ravi Varadhan, Johns Hopkins University; Martin Maechler (cosmetic, generalization to sparse matrices; rescaling (see rescaleBelow).

See also

expm

Examples

source(system.file("demo", "exact-fn.R", package = "expm"))
##-> rnilMat() ; xct10
set.seed(1)
(s5 <- Matrix(m5 <- rnilMat(5))); v <- c(1,6:9)
#> 5 x 5 sparse Matrix of class "dtCMatrix"
#>               
#> [1,] . 4 4 8 9
#> [2,] . . 5 3 6
#> [3,] . . . 8 6
#> [4,] . . . . 2
#> [5,] . . . . .
(em5 <- expm(m5))
#>      [,1] [,2] [,3]     [,4]     [,5]
#> [1,]    1    4   14 56.66667 89.00000
#> [2,]    0    1    5 23.00000 37.33333
#> [3,]    0    0    1  8.00000 14.00000
#> [4,]    0    0    0  1.00000  2.00000
#> [5,]    0    0    0  0.00000  1.00000
r5  <- expAtv(m5, v)
r5. <- expAtv(s5, v)
stopifnot(all.equal(r5, r5., tolerance = 1e-14),
          all.equal(c(em5 %*% v), r5$eAtv))

v <- 10:1
with(xct10, all.equal(expm(m), expm))
#> [1] "Mean relative difference: 0.0473538"
all.equal(c(xct10$expm %*% v),
          expAtv(xct10$m, v)$eAtv)
#> [1] "Mean relative difference: 1.600917e-05"