# expm: Matrix Exponential In expm: Matrix Exponential, Log, 'etc'

## 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 Padé approximation are very importantly combined with scaling and squaring.

## Usage

 ```1 2 3 4 5 6``` ```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` `"Higham08.b"`, `"Ward77"`, `"Pade"` or `"Taylor"`, etc; The default is now `"Higham08.b"` which uses Higham's 2008 algorithm with additional balancing preconditioning, see `expm.Higham08`. The versions with "*O" call the original Fortran code, whereas the first ones call the BLAS-using and partly simplified newer code. `"R_Pade"` uses an R-code version of `"Pade"` for didactical reasons, and `"R_Ward77"` uses an R version of `"Ward77"`, still based on LAPACK's `dgebal`, see R interface `dgebal`. This has enabled us to diagnose and fix the bug in the original octave implementation of `"Ward77"`. `"R_Eigen"` tries to diagonalise the matrix `x`, if not possible, `"R_Eigen"` raises an error. `"hybrid_Eigen_Ward"` method also tries to diagonalise the matrix `x`, if not possible, it uses `"Ward77"` algorithm. `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 `x`) but for the current double precision arithmetic, one recommendation (and the Matlab implementations) uses `order = 6` unconditionally; our default, `8`, is from Ward(1977, p.606)'s recommendation, but also used for `"AlMohy-Hi09"` where a high order `order=12` may be more appropriate (and slightly more expensive). `trySym` logical indicating if `method = "R_Eigen"` should use `isSymmetric(x)` and take advantage for (almost) symmetric matrices. `tol` a given tolerance used to check if `x` is computationally singular when `method = "hybrid_Eigen_Ward"`. `do.sparseMsg` logical allowing a message about sparse to dense coercion; setting it `FALSE` suppresses that message. `preconditioning` a string specifying which implementation of Ward(1977) should be used when `method = "Ward77"`.

## Details

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"`.

## 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 [email protected], 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 [email protected].

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 http://epubs.siam.org/doi/pdf/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}.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106``` ```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 ```

### Example output

```Loading required package: Matrix

Attaching package: 'expm'

The following object is masked from 'package:Matrix':

expm

[,1]      [,2]
[1,] -0.7357588 0.5518191
[2,] -1.4715176 1.1036382
[,1]      [,2]
[1,] -0.7357588 0.5518191
[2,] -1.4715176 1.1036382
[,1]     [,2]      [,3]
[1,] 147.8666 183.7651  71.79703
[2,] 127.7811 183.7651  91.88257
[3,] 127.7811 163.6796 111.96811
attr(,"accuracy")
[1] 9.942847e-12
[,1]     [,2]      [,3]
[1,] 147.8666 183.7651  71.79703
[2,] 127.7811 183.7651  91.88257
[3,] 127.7811 163.6796 111.96811
[,1]          [,2]          [,3]
[1,]  5.496314e+15 -1.823188e+16 -3.047577e+16
[2,] -1.823188e+16  6.060523e+16  1.012918e+17
[3,] -3.047577e+16  1.012918e+17  1.692944e+17
attr(,"accuracy")
[1] 53010
[,1]          [,2]          [,3]
[1,]  5.496314e+15 -1.823188e+16 -3.047577e+16
[2,] -1.823188e+16  6.060523e+16  1.012918e+17
[3,] -3.047577e+16  1.012918e+17  1.692944e+17
[,1]      [,2]      [,3]
[1,] -1.509644 0.3678794 0.1353353
[2,] -5.632571 1.4715178 0.4060058
[3,] -4.934938 1.1036383 0.5413411
attr(,"accuracy")
[1] 4.375779e-11
[,1]      [,2]      [,3]
[1,] -1.509644 0.3678794 0.1353353
[2,] -5.632571 1.4715178 0.4060058
[3,] -4.934938 1.1036383 0.5413411
\$dim
[1] 10 10

\$accuracy
[1] 1.440257e-16

[1] 1.391826e-08
Error in solve.default(V) :
system is computationally singular: reciprocal condition number = 1.00208e-292
[,1] [,2]
[1,]    1    0
[2,]    1    1
```

expm documentation built on May 2, 2019, 5:25 p.m.