expm (version 0.999-4)

expm.Higham08: 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).

Value

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

Details

The algorithm comprises the following steps

1. 0.Balancing

2. 1.Scaling

3. 2.Pad<U+00E9>-Approximation

4. 3.Squaring

5. 4.Reverse Balancing

References

Higham, N.~J. (2008). Functions of Matrices: Theory and Computation; Society for Industrial and Applied Mathematics, Philadelphia, PA, USA.

Michael Stadelmann (2009). Matrixfunktionen; Analyse und Implementierung. [in German] Master's thesis and Research Report 2009-12, SAM, ETH Zurich; http://www.sam.math.ethz.ch/reports/2009, or ftp://ftp.sam.math.ethz.ch/pub/sam-reports/reports/reports2009/2009-12.pdf.

See Also

For now, the other algorithms expm. This will change there will be one function with optional arguments to chose the method !.

expmCond, to compute the exponential-condition number.

Examples

# NOT RUN {
## 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 ...)
# }

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