expm.Higham08 {expm}

Matrix Exponential [Higham 2008]

Description

Calculation of matrix exponential e^A with the ‘Scaling & Squaring’ method with balancing.

Implementation of Higham's Algorithm from his book (see references), Chapter 10, Algorithm 10.20.

The balancing option is an extra from Michael Stadelmann's Masters thesis.

Usage

expm.Higham08(A, balancing = TRUE)

Arguments

A

square matrix, may be a "sparseMatrix", currently only if balancing is false.

balancing

logical indicating if balancing should happen (before and after scaling and squaring).

Details

The algorithm comprises the following steps

0.

Balancing

1.

Scaling

2.

Padé-Approximation

3.

Squaring

4.

Reverse Balancing

Value

a matrix of the same dimension as A, the matrix exponential of A.

Note

expm.Higham8() no longer needs to be called directly; rather expm(A, "Higham8b") and expm(A, "Higham8") correspond to the two options of balancing = TRUE || FALSE.

Author(s)

Michael Stadelmann (final polish by Martin Maechler).

References

Higham, Nicholas J. (2008). Functions of Matrices: Theory and Computation; SIAM (Society for Industrial and Applied Mathematics), Philadelphia, USA; doi:10.1137/1.9780898717778

Michael Stadelmann (2009). Matrixfunktionen; Analyse und Implementierung. [in German] Master's thesis and Research Report 2009-12, SAM, ETH Zurich; https://math.ethz.ch/sam/research/reports.html?year=2009, or the pdf directly at https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2009/2009-12.pdf.

See Also

The other algorithms expm(x, method = *).

expmCond, to compute the exponential-condition number.

Examples

## The *same* examples as in ../expm.Rd  {FIXME} --
x <- matrix(c(-49, -64, 24, 31), 2, 2)
expm.Higham08(x)

## ----------------------------
## Test case 1 from Ward (1977)
## ----------------------------
test1 <- t(matrix(c(
    4, 2, 0,
    1, 4, 1,
    1, 1, 4), 3, 3))
expm.Higham08(test1)
##                    [,1]               [,2]               [,3]
## [1,] 147.86662244637000 183.76513864636857  71.79703239999643
## [2,] 127.78108552318250 183.76513864636877  91.88256932318409
## [3,] 127.78108552318204 163.67960172318047 111.96810624637124
## -- these agree with ward (1977, p608)


## ----------------------------
## Test case 2 from Ward (1977)
## ----------------------------
test2 <- t(matrix(c(
    29.87942128909879, .7815750847907159, -2.289519314033932,
    .7815750847907159, 25.72656945571064,  8.680737820540137,
   -2.289519314033932, 8.680737820540137,  34.39400925519054),
           3, 3))
expm.Higham08(test2)
expm.Higham08(test2, balancing = FALSE)
##                   [,1]               [,2]               [,3]
##[1,]   5496313853692405 -18231880972009100 -30475770808580196
##[2,] -18231880972009160  60605228702221760 101291842930249376
##[3,] -30475770808580244 101291842930249200 169294411240850880
## -- in this case a very similar degree of accuracy.

## ----------------------------
## Test case 3 from Ward (1977)
## ----------------------------
test3 <- t(matrix(c(
    -131, 19, 18,
    -390, 56, 54,
    -387, 57, 52), 3, 3))
expm.Higham08(test3)
expm.Higham08(test3, balancing = FALSE)
##                    [,1]                [,2]                [,3]
##[1,] -1.5096441587713636 0.36787943910439874 0.13533528117301735
##[2,] -5.6325707997970271 1.47151775847745725 0.40600584351567010
##[3,] -4.9349383260294299 1.10363831731417195 0.54134112675653534
## -- agrees to 10dp with Ward (1977), p608. ??? (FIXME)

## ----------------------------
## Test case 4 from Ward (1977)
## ----------------------------
test4 <-
    structure(c(0, 0, 0, 0, 0, 0, 0, 0, 0, 1e-10,
                1, 0, 0, 0, 0, 0, 0, 0, 0, 0,
                0, 1, 0, 0, 0, 0, 0, 0, 0, 0,
                0, 0, 1, 0, 0, 0, 0, 0, 0, 0,
                0, 0, 0, 1, 0, 0, 0, 0, 0, 0,
                0, 0, 0, 0, 1, 0, 0, 0, 0, 0,
                0, 0, 0, 0, 0, 1, 0, 0, 0, 0,
                0, 0, 0, 0, 0, 0, 1, 0, 0, 0,
                0, 0, 0, 0, 0, 0, 0, 1, 0, 0,
                0, 0, 0, 0, 0, 0, 0, 0, 1, 0),
    .Dim = c(10, 10))

E4 <- expm.Higham08(test4)
Matrix(zapsmall(E4))

S4 <- as(test4, "sparseMatrix") # some R based expm() methods work for sparse:
ES4 <- expm.Higham08(S4, bal=FALSE)
stopifnot(all.equal(E4, unname(as.matrix(ES4))))
## NOTE: Need much larger sparse matrices for sparse arith to be faster!

##
## example of computationally singular matrix
##
m <- matrix(c(0,1,0,0), 2,2)
eS <- expm.Higham08(m)  # "works"  (hmm ...)