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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_matop_real_gen_matrix_fun_std (f01ek)

## Purpose

nag_matop_real_gen_matrix_fun_std (f01ek) computes the matrix exponential, sine, cosine, sinh or cosh, of a real n$n$ by n$n$ matrix A$A$ using the Schur–Parlett algorithm.

## Syntax

[a, imnorm, ifail] = f01ek(fun, a, 'n', n)
[a, imnorm, ifail] = nag_matop_real_gen_matrix_fun_std(fun, a, 'n', n)

## Description

f(A)$f\left(A\right)$, where f$f$ is either the exponential, sine, cosine, sinh or cosh, is computed using the Schur–Parlett algorithm described in Higham (2008) and Davies and Higham (2003).

## References

Davies P I and Higham N J (2003) A Schur–Parlett algorithm for computing matrix functions. SIAM J. Matrix Anal. Appl. 25(2) 464–485
Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA

## Parameters

### Compulsory Input Parameters

1:     fun – string
Indicates which matrix function will be computed.
fun = 'exp'${\mathbf{fun}}=\text{'exp'}$
The matrix exponential, eA${e}^{A}$, will be computed.
fun = 'sin'${\mathbf{fun}}=\text{'sin'}$
The matrix sine, sin(A)$\mathrm{sin}\left(A\right)$, will be computed.
fun = 'cos'${\mathbf{fun}}=\text{'cos'}$
The matrix cosine, cos(A)$\mathrm{cos}\left(A\right)$, will be computed.
fun = 'sinh'${\mathbf{fun}}=\text{'sinh'}$
The hyperbolic matrix sine, sinh(A)$\mathrm{sinh}\left(A\right)$, will be computed.
fun = 'cosh'${\mathbf{fun}}=\text{'cosh'}$
The hyperbolic matrix cosine, cosh(A)$\mathrm{cosh}\left(A\right)$, will be computed.
Constraint: fun = 'exp'${\mathbf{fun}}=\text{'exp'}$, 'sin'$\text{'sin'}$, 'cos'$\text{'cos'}$, 'sinh'$\text{'sinh'}$ or 'cosh'$\text{'cosh'}$.
2:     a(lda, : $:$) – double array
The first dimension of the array a must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array must be at least n${\mathbf{n}}$
The n$n$ by n$n$ matrix A$A$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array a.
n$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.

lda

### Output Parameters

1:     a(lda, : $:$) – double array
The first dimension of the array a will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array will be n${\mathbf{n}}$
ldamax (1,n)$\mathit{lda}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The n$n$ by n$n$ matrix, f(A)$f\left(A\right)$.
2:     imnorm – double scalar
If A$A$ has complex eigenvalues, nag_matop_real_gen_matrix_fun_std (f01ek) will use complex arithmetic to compute the matrix function. The imaginary part is discarded at the end of the computation, because it will theoretically vanish. imnorm contains the 1$1$-norm of the imaginary part, which should be used to check that the routine has given a reliable answer.
If A$A$ has real eigenvalues, nag_matop_real_gen_matrix_fun_std (f01ek) uses real arithmetic and imnorm = 0${\mathbf{imnorm}}=0$.
3:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
A Taylor series failed to converge.
ifail = 2${\mathbf{ifail}}=2$
ifail = 3${\mathbf{ifail}}=3$
There was an error whilst reordering the Schur form of A$A$.
Note:  this failure should not occur and suggests that the function has been called incorrectly.
ifail = 4${\mathbf{ifail}}=4$
The function was unable to compute the Schur decomposition of A$A$.
Note:  this failure should not occur and suggests that the function has been called incorrectly.
ifail = 5${\mathbf{ifail}}=5$
ifail = 6${\mathbf{ifail}}=6$
The linear equations to be solved are nearly singular and the Padé approximant used to compute the exponential may have no correct figures.
Note:  this failure should not occur and suggests that the function has been called incorrectly.
ifail = 1${\mathbf{ifail}}=-1$
Input parameter number _$_$ is invalid.
ifail = 2${\mathbf{ifail}}=-2$
Input parameter number _$_$ is invalid.
ifail = 4${\mathbf{ifail}}=-4$
On entry, parameter lda is invalid.
Constraint: ldan$\mathit{lda}\ge {\mathbf{n}}$.
ifail = 999${\mathbf{ifail}}=-999$
Allocation of memory failed.

## Accuracy

For a normal matrix A$A$ (for which ATA = AAT${A}^{\mathrm{T}}A=A{A}^{\mathrm{T}}$), the Schur decomposition is diagonal and the algorithm reduces to evaluating f$f$ at the eigenvalues of A$A$ and then constructing f(A)$f\left(A\right)$ using the Schur vectors. This should give a very accurate result. In general, however, no error bounds are available for the algorithm.
For further discussion of the Schur–Parlett algorithm see Section 9.4 of Higham (2008).

The integer allocatable memory required is n$n$. If A$A$ has real eigenvalues then up to 9n2$9{n}^{2}$ of double allocatable memory may be required. If A$A$ has complex eigenvalues then up to 9n2$9{n}^{2}$ of complex allocatable memory may be required.
The cost of the Schur–Parlett algorithm depends on the spectrum of A$A$, but is roughly between 28n3$28{n}^{3}$ and n4 / 3${n}^{4}/3$ floating point operations; see Algorithm 9.6 of Higham (2008).
If the matrix exponential is required then it is recommended that nag_matop_real_gen_matrix_exp (f01ec) be used. nag_matop_real_gen_matrix_exp (f01ec) uses an algorithm which is, in general, more accurate than the Schur–Parlett algorithm used by nag_matop_real_gen_matrix_fun_std (f01ek).
If estimates of the condition number of the matrix function are required then nag_matop_real_gen_matrix_cond_std (f01ja) should be used.
nag_matop_complex_gen_matrix_fun_std (f01fk) can be used to find the matrix exponential, sin, cos, sinh or cosh of a complex matrix.

## Example

```function nag_matop_real_gen_matrix_fun_std_example
a =  [2,  0,  1,  0
0,  2, -2,  1
0,  2,  3,  1
1,  4,  0,  0];
fun = 'cos';
% Compute cos(a)
[a, imnorm, ifail] = nag_matop_real_gen_matrix_fun_std(fun, a)
```
```

a =

-0.2998    1.5003   -0.7849    0.4677
-0.2385   -3.2657    0.5812   -1.1460
0.4677    0.3008   -4.0853   -0.2200
-0.2107   -2.8199   -1.2964   -0.8325

imnorm =

3.3307e-16

ifail =

0

```
```function f01ek_example
a =  [2,  0,  1,  0
0,  2, -2,  1
0,  2,  3,  1
1,  4,  0,  0];
fun = 'cos';
% Compute cos(a)
[a, imnorm, ifail] = f01ek(fun, a)
```
```

a =

-0.2998    1.5003   -0.7849    0.4677
-0.2385   -3.2657    0.5812   -1.1460
0.4677    0.3008   -4.0853   -0.2200
-0.2107   -2.8199   -1.2964   -0.8325

imnorm =

3.3307e-16

ifail =

0

```