# Documentation

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# expm

Matrix exponential

## Syntax

• ``Y = expm(X)``
example

## Description

example

````Y = expm(X)` computes the matrix exponential of `X`. Although it is not computed this way, if `X` has a full set of eigenvectors `V` with corresponding eigenvalues `D`, then ```[V,D] = eig(X)``` andexpm(X) = V*diag(exp(diag(D)))/VUse `exp` for the element-by-element exponential.```

## Examples

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Compute and compare the exponential of `A` with the matrix exponential of `A`.

```A = [1 1 0; 0 0 2; 0 0 -1]; exp(A) ```
```ans = 2.7183 2.7183 1.0000 1.0000 1.0000 7.3891 1.0000 1.0000 0.3679 ```
```expm(A) ```
```ans = 2.7183 1.7183 1.0862 0 1.0000 1.2642 0 0 0.3679 ```

Notice that the diagonal elements of the two results are equal, which is true for any triangular matrix. The off-diagonal elements, including those below the diagonal, are different.

## Input Arguments

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Input matrix, specified as a square matrix.

Data Types: `single` | `double`
Complex Number Support: Yes

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### Algorithms

The algorithm `expm` uses is described in [1] and [2].

 Note   The files, `expmdemo1.m`, `expmdemo2.m`, and `expmdemo3.m` illustrate the use of Padé approximation, Taylor series approximation, and eigenvalues and eigenvectors, respectively, to compute the matrix exponential. References [3] and [4] describe and compare many algorithms for computing a matrix exponential.

## References

[1] Higham, N. J., "The Scaling and Squaring Method for the Matrix Exponential Revisited," SIAM J. Matrix Anal. Appl., 26(4) (2005), pp. 1179–1193.

[2] Al-Mohy, A. H. and N. J. Higham, "A new scaling and squaring algorithm for the matrix exponential," SIAM J. Matrix Anal. Appl., 31(3) (2009), pp. 970–989.

[3] Golub, G. H. and C. F. Van Loan, Matrix Computation, p. 384, Johns Hopkins University Press, 1983.

[4] Moler, C. B. and C. F. Van Loan, "Nineteen Dubious Ways to Compute the Exponential of a Matrix," SIAM Review 20, 1978, pp. 801–836. Reprinted and updated as "Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later," SIAM Review 45, 2003, pp. 3–49.