Matrix Exponential [Higham 2008]
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.
expm.Higham08(A, balancing = TRUE)
A |
square matrix, may be a |
balancing |
logical indicating if balancing should happen (before and after scaling and squaring). |
The algorithm comprises the following steps
0.Balancing
1.Scaling
2.Padé-Approximation
3.Squaring
4.Reverse Balancing
a matrix of the same dimension as A, the matrix exponential of A.
Michael Stadelmann (final polish by Martin Maechler).
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; 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.
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.
## 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 ...)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.