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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_matop_complex_gen_matrix_actexp (f01ha)

## Purpose

nag_matop_complex_gen_matrix_actexp (f01ha) computes the action of the matrix exponential ${e}^{tA}$, on the matrix $B$, where $A$ is a complex $n$ by $n$ matrix, $B$ is a complex $n$ by $m$ matrix and $t$ is a complex scalar.

## Syntax

[a, b, ifail] = f01ha(m, a, b, t, 'n', n)
[a, b, ifail] = nag_matop_complex_gen_matrix_actexp(m, a, b, t, 'n', n)
Note: the interface to this routine has changed since earlier releases of the toolbox:
 At Mark 25: m was made optional

## Description

${e}^{tA}B$ is computed using the algorithm described in Al–Mohy and Higham (2011) which uses a truncated Taylor series to compute the product ${e}^{tA}B$ without explicitly forming ${e}^{tA}$.

## References

Al–Mohy A H and Higham N J (2011) Computing the action of the matrix exponential, with an application to exponential integrators SIAM J. Sci. Statist. Comput. 33(2) 488-511
Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{m}$int64int32nag_int scalar
$m$, the number of columns of the matrix $B$.
Constraint: ${\mathbf{m}}\ge 0$.
2:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array
The first dimension of the array a must be at least ${\mathbf{n}}$.
The second dimension of the array a must be at least ${\mathbf{n}}$.
The $n$ by $n$ matrix $A$.
3:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – complex array
The first dimension of the array b must be at least ${\mathbf{n}}$.
The second dimension of the array b must be at least ${\mathbf{m}}$.
The $n$ by $m$ matrix $B$.
4:     $\mathrm{t}$ – complex scalar
The scalar $t$.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the first dimension of the arrays a, b. (An error is raised if these dimensions are not equal.)
$n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.

### Output Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array
The first dimension of the array a will be ${\mathbf{n}}$.
The second dimension of the array a will be ${\mathbf{n}}$.
$A$ is overwritten during the computation.
2:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – complex array
The first dimension of the array b will be ${\mathbf{n}}$.
The second dimension of the array b will be ${\mathbf{m}}$.
The $n$ by $m$ matrix ${e}^{tA}B$.
3:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W  ${\mathbf{ifail}}=2$
${e}^{tA}B$ has been computed using an IEEE double precision Taylor series, although the arithmetic precision is higher than IEEE double precision.
${\mathbf{ifail}}=-1$
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-2$
Constraint: ${\mathbf{m}}\ge 0$.
${\mathbf{ifail}}=-4$
Constraint: $\mathit{lda}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=-6$
Constraint: $\mathit{ldb}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

For a Hermitian matrix $A$ (for which ${A}^{\mathrm{H}}=A$) the computed matrix ${e}^{tA}B$ is guaranteed to be close to the exact matrix, that is, the method is forward stable. No such guarantee can be given for non-Hermitian matrices. See Section 4 of Al–Mohy and Higham (2011) for details and further discussion.

The matrix ${e}^{tA}B$ could be computed by explicitly forming ${e}^{tA}$ using nag_matop_complex_gen_matrix_exp (f01fc) and multiplying $B$ by the result. However, experiments show that it is usually both more accurate and quicker to use nag_matop_complex_gen_matrix_actexp (f01ha).
The cost of the algorithm is $\mathit{O}\left({n}^{2}m\right)$. The precise cost depends on $A$ since a combination of balancing, shifting and scaling is used prior to the Taylor series evaluation.
Approximately ${n}^{2}+\left(2m+8\right)n$ of complex allocatable memory is required by nag_matop_complex_gen_matrix_actexp (f01ha).
nag_matop_real_gen_matrix_actexp (f01ga) can be used to compute ${e}^{tA}B$ for real $A$, $B$, and $t$. nag_matop_complex_gen_matrix_actexp_rcomm (f01hb) provides an implementation of the algorithm with a reverse communication interface, which returns control to the user when matrix multiplications are required. This should be used if $A$ is large and sparse.

## Example

This example computes ${e}^{tA}B$, where
 $A = 0.5+0.0i -0.2+0.0i 1.0+0.1i 0.0+0.4i 0.3+0.0i 0.5+1.2i 3.1+0.0i 1.0+0.2i 0.0+2.0i 0.1+0.0i 1.2+0.2i 0.5+0.0i 1.0+0.3i 0.0+0.2i 0.0+0.9i 0.5+0.0i ,$
 $B = 0.4+0.0i 1.2+0.0i 1.3+0.0i -0.2+0.1i 0.0+0.3i 2.1+0.0i 0.4+0.0i -0.9+0.0i$
and
 $t=-0.5+0.0i .$
```function f01ha_example

fprintf('f01ha example results\n\n');

a = [0.5+0.0i, -0.2+0.0i, 1.0+0.1i, 0.0+0.4i;
0.3+0.0i,  0.5+1.2i, 3.1+0.0i, 1.0+0.2i;
0.0+2.0i,  0.1+0.0i, 1.2+0.2i, 0.5+0.0i;
1.0+0.3i,  0.0+0.2i, 0.0+0.9i, 0.5+0.0i];
b = [0.4+0.0i,  1.2+0.0i;
1.3+0.0i, -0.2+0.1i;
0.0+0.3i,  2.1+0.0i;
0.4+0.0i, -0.9+0.0i];

t = complex(-0.5);

% Compute exp(ta)b

[a, exptab, ifail] = f01ha(a, b, t);

disp('exp(tA)B');
disp(exptab);

```
```f01ha example results

exp(tA)B
0.4251 - 0.1061i  -0.0220 + 0.3289i
0.7229 - 0.5940i  -1.7931 + 1.4952i
-0.1394 - 0.1151i   1.4781 - 0.4514i
0.1054 - 0.0786i  -1.0059 - 0.7079i

```