Documentation

This is machine translation

Translated by
Mouseover text to see original. Click the button below to return to the English version of the page.

Matrix Exponentials

This example shows three of the 19 ways to compute the exponential of a matrix.

For background on the computation of matrix exponentials, see "Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later," SIAM Review 45, 3-49, 2003.

The Pseudospectra Gateway is also highly recommended. The web site is:

http://web.comlab.ox.ac.uk/projects/pseudospectra/

`A = [0 1 2; 0.5 0 1; 2 1 0]`
```A = 0 1.0000 2.0000 0.5000 0 1.0000 2.0000 1.0000 0 ```
`Asave = A;`

Scaling and Squaring

`expmdemo1` is an implementation of algorithm 11.3.1 in Golub and Van Loan, Matrix Computations, 3rd edition.

```% Scale A by power of 2 so that its norm is < 1/2 . [f,e] = log2(norm(A,'inf')); s = max(0,e+1); A = A/2^s; % Pade approximation for exp(A) X = A; c = 1/2; E = eye(size(A)) + c*A; D = eye(size(A)) - c*A; q = 6; p = 1; for k = 2:q c = c * (q-k+1) / (k*(2*q-k+1)); X = A*X; cX = c*X; E = E + cX; if p D = D + cX; else D = D - cX; end p = ~p; end E = D\E; % Undo scaling by repeated squaring for k = 1:s E = E*E; end E1 = E```
```E1 = 5.3091 4.0012 5.5778 2.8088 2.8845 3.1930 5.1737 4.0012 5.7132 ```

Matrix Exponential via Taylor Series

`expmdemo2` uses the classic definition for the matrix exponential. As a practical numerical method, this is slow and inaccurate if `norm(A)` is too large.

```A = Asave; % Taylor series for exp(A) E = zeros(size(A)); F = eye(size(A)); k = 1; while norm(E+F-E,1) > 0 E = E + F; F = A*F/k; k = k+1; end E2 = E```
```E2 = 5.3091 4.0012 5.5778 2.8088 2.8845 3.1930 5.1737 4.0012 5.7132 ```

Matrix Exponential via Eigenvalues and Eigenvectors

`expmdemo3` assumes that the matrix has a full set of eigenvectors. As a practical numerical method, the accuracy is determined by the condition of the eigenvector matrix.

```A = Asave; [V,D] = eig(A); E = V * diag(exp(diag(D))) / V; E3 = E```
```E3 = 5.3091 4.0012 5.5778 2.8088 2.8845 3.1930 5.1737 4.0012 5.7132 ```

Compare the Results

For this matrix, they all do equally well.

```E = expm(Asave); err1 = E - E1```
```err1 = 1.0e-14 * 0.3553 0.1776 0.0888 0.0888 0.1332 -0.0444 0 0 -0.2665 ```
`err2 = E - E2`
```err2 = 1.0e-14 * 0 0 -0.1776 -0.0444 0 -0.0888 0.1776 0 0.0888 ```
`err3 = E - E3`
```err3 = 1.0e-13 * -0.0799 -0.0444 -0.0622 -0.0622 -0.0488 -0.0933 -0.0711 -0.0533 -0.1066 ```

Taylor Series Fails

Here is a matrix where the terms in the Taylor series become very large before they go to zero. Consequently, `expmdemo2` fails.

```A = [-147 72; -192 93]; E1 = expmdemo1(A)```
```E1 = -0.0996 0.0747 -0.1991 0.1494 ```
`E2 = expmdemo2(A)`
```E2 = 1.0e+06 * -1.1985 -0.5908 -2.7438 -2.0442 ```
`E3 = expmdemo3(A)`
```E3 = -0.0996 0.0747 -0.1991 0.1494 ```

Eigenvalues and Vectors Fails

Here is a matrix that does not have a full set of eigenvectors. Consequently, `expmdemo3` fails.

```A = [-1 1; 0 -1]; E1 = expmdemo1(A)```
```E1 = 0.3679 0.3679 0 0.3679 ```
`E2 = expmdemo2(A)`
```E2 = 0.3679 0.3679 0 0.3679 ```
`E3 = expmdemo3(A)`
```E3 = 0.3679 0 0 0.3679 ```