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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:

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;

`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
```

`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
```

`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
```

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
```

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
```

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
```

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