# expm: Matrix Exponential

### Description

Compute the exponential of a matrix.

### Usage

 `1` ```expm(x) ```

### Arguments

 `x` a matrix, typically inheriting from the `dMatrix` class.

### Details

The exponential of a matrix is defined as the infinite Taylor series `expm(A) = I + A + A^2/2! + A^3/3! + ...` (although this is definitely not the way to compute it). The method for the `dgeMatrix` class uses Ward's diagonal Pade' approximation with three step preconditioning.

### Value

The matrix exponential of `x`.

### Note

The expm package contains newer (partly faster and more accurate) algorithms for `expm()` and includes `logm` and `sqrtm`.

### Author(s)

This is a translation of the implementation of the corresponding Octave function contributed to the Octave project by A. Scottedward Hodel A.S.Hodel@Eng.Auburn.EDU. A bug in there has been fixed by Martin Maechler.

### References

Cleve Moler and Charles Van Loan (2003) Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Review 45, 1, 3–49.

Eric W. Weisstein et al. (1999) Matrix Exponential. From MathWorld, http://mathworld.wolfram.com/MatrixExponential.html

`Schur`; additionally, `expm`, `logm`, etc in package expm.

### Examples

 ```1 2 3 4 5 6 7``` ```(m1 <- Matrix(c(1,0,1,1), nc = 2)) (e1 <- expm(m1)) ; e <- exp(1) stopifnot(all.equal(e1@x, c(e,0,e,e), tolerance = 1e-15)) (m2 <- Matrix(c(-49, -64, 24, 31), nc = 2)) (e2 <- expm(m2)) (m3 <- Matrix(cbind(0,rbind(6*diag(3),0))))# sparse! (e3 <- expm(m3)) # upper triangular ```

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