expm {expm} | R Documentation |

## Matrix Exponential

### Description

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 PadÃ©
approximation are very importantly combined with scaling and squaring.

### Usage

```
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"))
```

### Arguments

`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 |

### Details

The exponential of a matrix is defined as the infinite Taylor series

`e^M = \sum_{k = 1}^\infty \frac{M^k}{k!}.`

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"`

.
### Value

The matrix exponential of `x`

.

### Note

For a good general discussion of the matrix exponential problem, see Moler and van Loan (2003).

### Author(s)

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.

### References

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.

### See Also

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}`

.

### Examples

```
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 method
```

*expm*version 0.999-9 Index]