Matrix Exponential
This function computes the exponential of a square matrix A, defined as the sum from r=0 to infinity of A^r/r!. Several methods are provided. The Taylor series and Padé approximation are very importantly combined with scaling and squaring.
expm(x, method = c("Higham08.b", "Higham08",
"AlMohy-Hi09",
"Ward77", "PadeRBS", "Pade", "Taylor", "PadeO", "TaylorO",
"R_Eigen", "R_Pade", "R_Ward77", "hybrid_Eigen_Ward"),
order = 8, trySym = TRUE, tol = .Machine$double.eps, do.sparseMsg = TRUE,
preconditioning = c("2bal", "1bal", "buggy"))x |
a square matrix. |
method |
The versions with "*O" call the
original Fortran code, whereas the first ones call the BLAS-using
and partly simplified newer code. |
order |
an integer, the order of approximation to be used, for
the "Pade" and "Taylor" methods. The best value for this depends on
machine precision (and slightly on |
trySym |
logical indicating if |
tol |
a given tolerance used to check if |
do.sparseMsg |
logical allowing a message about sparse to dense
coercion; setting it |
preconditioning |
a string specifying which implementation of
Ward(1977) should be used when |
The exponential of a matrix is defined as the infinite Taylor series
exp(M) = I + M + M^2/2! + M^3/3! + …
For the "Pade" and "Taylor" methods, there is an "accuracy"
attribute of the result. It is an upper bound for the L2 norm of the
Cauchy error expm(x, *, order + 10) - expm(x, *, order).
Currently, only algorithms which are “R-code only” accept sparse
matrices (see the
sparseMatrix class in package
Matrix), i.e., currently only "R_Eigen" and
"Higham08".
The matrix exponential of x.
For a good general discussion of the matrix exponential problem, see Moler and van Loan (2003).
The "Ward77" method:
Vincent Goulet vincent.goulet@act.ulaval.ca, and Christophe
Dutang, based on code translated by Doug Bates and Martin Maechler
from the implementation of the corresponding Octave function
contributed by A. Scottedward Hodel A.S.Hodel@eng.auburn.edu.
The "PadeRBS" method:
Roger B. Sidje, see the EXPOKIT reference.
The "PadeO" and "TaylorO" methods:
Marina Shapira (U Oxford, UK) and David Firth (U Warwick, UK);
The "Pade" and "Taylor" methods are slight
modifications of the "*O" ([O]riginal versions) methods,
by Martin Maechler, using BLAS and LINPACK where possible.
The "hybrid_Eigen_Ward" method by Christophe Dutang is a C
translation of "R_Eigen" method by Martin Maechler.
The "Higham08" and "Higham08.b" (current default) were
written by Michael Stadelmann, see expm.Higham08.
The "AlMohy-Hi09" implementation (R code interfacing to
stand-alone C) was provided and donated by Drew Schmidt, U. Tennesse.
Ward, R. C. (1977). Numerical computation of the matrix exponential with accuracy estimate. SIAM J. Num. Anal. 14, 600–610.
Roger B. Sidje (1998). EXPOKIT: Software package for computing matrix exponentials. ACM - Transactions on Mathematical Software 24(1), 130–156.
Moler, C and van Loan, C (2003). Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Review 45, 3–49. At doi: 10.1137/S00361445024180
Awad H. Al-Mohy and Nicholas J. Higham (2009) A New Scaling and Squaring Algorithm for the Matrix Exponential. SIAM. J. Matrix Anal. & Appl., 31(3), 970–989.
The package vignette for details on the algorithms and calling the function from external packages.
expm.Higham08 for "Higham08".
expAtv(A,v,t) computes e^{At} v (for scalar
t and n-vector v) directly and more
efficiently than computing e^{At}.
x <- matrix(c(-49, -64, 24, 31), 2, 2)
expm(x)
expm(x, method = "AlMohy-Hi09")
## ----------------------------
## Test case 1 from Ward (1977)
## ----------------------------
test1 <- t(matrix(c(
4, 2, 0,
1, 4, 1,
1, 1, 4), 3, 3))
expm(test1, method="Pade")
## Results on Power Mac G3 under Mac OS 10.2.8
## [,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)
## Compare with the naive "R_Eigen" method:
try(
expm(test1, method="R_Eigen")
) ## platform depently, sometimes gives an error from solve
## or is accurate or one older result was
## [,1] [,2] [,3]
##[1,] 147.86662244637003 88.500223574029647 103.39983337000028
##[2,] 127.78108552318220 117.345806155250600 90.70416537273444
##[3,] 127.78108552318226 90.384173332156763 117.66579819582827
## -- hopelessly inaccurate in all but the first column.
##
## ----------------------------
## 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(test2, method="Pade")
## [,1] [,2] [,3]
##[1,] 5496313853692357 -18231880972009844 -30475770808580828
##[2,] -18231880972009852 60605228702227024 101291842930256144
##[3,] -30475770808580840 101291842930256144 169294411240859072
## -- which agrees with Ward (1977) to 13 significant figures
expm(test2, method="R_Eigen")
## [,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(test3, method="Pade")
## [,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.
expm(test3, method="R_Eigen")
## [,1] [,2] [,3]
##[1,] -1.509644158796182 0.3678794391103086 0.13533528117547022
##[2,] -5.632570799902948 1.4715177585023838 0.40600584352641989
##[3,] -4.934938326098410 1.1036383173309319 0.54134112676302582
## -- in this case, a similar level of agreement with Ward (1977).
##
## ----------------------------
## 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))
attributes(expm(test4, method="Pade"))
max(abs(expm(test4, method="Pade") - expm(test4, method="R_Eigen")))
##[1] 8.746826694186494e-08
## -- here mexp2 is accurate only to 7 d.p., whereas mexp
## is correct to at least 14 d.p.
##
## Note that these results are achieved with the default
## settings order=8, method="Pade" -- accuracy could
## presumably be improved still further by some tuning
## of these settings.
##
## example of computationally singular matrix
##
m <- matrix(c(0,1,0,0), 2,2)
try(
expm(m, m="R_Eigen")
)
## error since m is computationally singular
expm(m, m="hybrid")
## hybrid use the Ward77 methodPlease choose more modern alternatives, such as Google Chrome or Mozilla Firefox.