Description Usage Arguments Details Value Note Author(s) References See Also Examples

Computes the exponential of a matrix.

1 2 3 |

`A` |
numeric square matrix. |

`np` |
number of points to use on the unit circle. |

For an analytic function *f* and a matrix *A* the expression
*f(A)* can be computed by the Cauchy integral

*f(A) = (2 π i)^{-1} \int_G (zI-A)^{-1} f(z) dz*

where *G* is a closed contour around the eigenvalues of *A*.

Here this is achieved by taking G to be a circle and approximating the integral by the trapezoid rule.

`logm`

is a fake at the moment as it computes the matrix logarithm
through taking the logarithm of its eigenvalues; will be replaced by an
approach using Pade interpolation.

Another more accurate and more reliable approach for computing these functions can be found in the R package ‘expm’.

Matrix of the same size as `A`

.

This approach could be used for other analytic functions, but a point to
consider is which branch to take (e.g., for the `logm`

function).

Idea and Matlab code for a cubic root by Nick Trefethen in his “10 digits 1 page” project.

Moler, C., and Ch. Van Loan (2003). Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later. SIAM Review, Vol. 1, No. 1, pp. 1–46.

N. J. Higham (2008). Matrix Functions: Theory and Computation. SIAM Society for Industrial and Applied Mathematics.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 | ```
## The Ward test cases described in the help for expm::expm agree up to
## 10 digits with the values here and with results from Matlab's expm !
A <- matrix(c(-49, -64, 24, 31), 2, 2)
expm(A)
# -0.7357588 0.5518191
# -1.4715176 1.1036382
A1 <- matrix(c(10, 7, 8, 7,
7, 5, 6, 5,
8, 6, 10, 9,
7, 5, 9, 10), nrow = 4, ncol = 4, byrow = TRUE)
expm(logm(A1))
logm(expm(A1))
## System of linear differential equations: y' = M y (y = c(y1, y2, y3))
M <- matrix(c(2,-1,1, 0,3,-1, 2,1,3), 3, 3, byrow=TRUE)
M
C1 <- 0.5; C2 <- 1.0; C3 <- 1.5
t <- 2.0; Mt <- expm(t * M)
yt <- Mt
``` |

```
[,1] [,2]
[1,] -0.7357588 0.5518191
[2,] -1.4715176 1.1036382
[,1] [,2] [,3] [,4]
[1,] 10 7 8 7
[2,] 7 5 6 5
[3,] 8 6 10 9
[4,] 7 5 9 10
[,1] [,2] [,3] [,4]
[1,] 9.999724 7.073307 7.844668 7.111353
[2,] 7.073307 4.785193 6.225976 4.842985
[3,] 7.844668 6.225976 10.042868 8.947233
[4,] 7.111353 4.842985 8.947233 10.057541
[,1] [,2] [,3]
[1,] 2 -1 1
[2,] 0 3 -1
[3,] 2 1 3
```

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