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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_matop_real_gen_matrix_fun_num (f01el)

## Purpose

nag_matop_real_gen_matrix_fun_num (f01el) computes the matrix function, f(A)$f\left(A\right)$, of a real n$n$ by n$n$ matrix A$A$. Numerical differentiation is used to evaluate the derivatives of f$f$ when they are required.

## Syntax

[a, user, iflag, imnorm, ifail] = f01el(a, f, 'n', n, 'user', user)
[a, user, iflag, imnorm, ifail] = nag_matop_real_gen_matrix_fun_num(a, f, 'n', n, 'user', user)

## Description

f(A)$f\left(A\right)$ is computed using the Schur–Parlett algorithm described in Higham (2008) and Davies and Higham (2003). The coefficients of the Taylor series used in the algorithm are evaluated using the numerical differentiation algorithm of Lyness and Moler (1967).
The scalar function f$f$ is supplied via function f which evaluates f(zi)$f\left({z}_{i}\right)$ at a number of points zi${z}_{i}$.

## 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
Lyness J N and Moler C B (1967) Numerical differentiation of analytic functions SIAM J. Numer. Anal. 4(2) 202–210

## Parameters

### Compulsory Input Parameters

1:     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$A$.
2:     f – function handle or string containing name of m-file
The function f evaluates f(zi)$f\left({z}_{i}\right)$ at a number of points zi${z}_{i}$.
[iflag, fz, user] = f(iflag, nz, z, user)

Input Parameters

1:     iflag – int64int32nag_int scalar
iflag will be zero.
2:     nz – int64int32nag_int scalar
nz${n}_{z}$, the number of function values required.
3:     z(nz) – complex array
The nz${n}_{z}$ points z1,z2,,znz${z}_{1},{z}_{2},\dots ,{z}_{{n}_{z}}$ at which the function f$f$ is to be evaluated.
4:     user – Any MATLAB object
f is called from nag_matop_real_gen_matrix_fun_num (f01el) with the object supplied to nag_matop_real_gen_matrix_fun_num (f01el).

Output Parameters

1:     iflag – int64int32nag_int scalar
iflag should either be unchanged from its entry value of zero, or may be set nonzero to indicate that there is a problem in evaluating the function f(zi)$f\left({z}_{i}\right)$; for instance f(zi)$f\left({z}_{i}\right)$ may not be defined. If iflag is returned as nonzero then nag_matop_real_gen_matrix_fun_num (f01el) will terminate the computation, with ${\mathbf{ifail}}={\mathbf{2}}$.
2:     fz(nz) – complex array
The nz${n}_{z}$ function values. fz(i)${\mathbf{fz}}\left(\mathit{i}\right)$ should return the value f(zi)$f\left({z}_{\mathit{i}}\right)$, for i = 1,2,,nz$\mathit{i}=1,2,\dots ,{n}_{z}$. If zi${z}_{i}$ lies on the real line, then so must f(zi)$f\left({z}_{i}\right)$.
3:     user – Any MATLAB object

### 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${\mathbf{n}}\ge 0$.
2:     user – Any MATLAB object
user is not used by nag_matop_real_gen_matrix_fun_num (f01el), but is passed to f. Note that for large objects it may be more efficient to use a global variable which is accessible from the m-files than to use user.

lda iuser ruser

### 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:     user – Any MATLAB object
3:     iflag – int64int32nag_int scalar
iflag = 0${\mathbf{iflag}}=0$, unless iflag has been set nonzero inside f, in which case iflag will be the value set and ifail will be set to ${\mathbf{ifail}}={\mathbf{2}}$.
4:     imnorm – double scalar
If A$A$ has complex eigenvalues, nag_matop_real_gen_matrix_fun_num (f01el) will use complex arithmetic to compute f(A)$f\left(A\right)$. 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 has real eigenvalues, nag_matop_real_gen_matrix_fun_num (f01el) uses real arithmetic and imnorm = 0${\mathbf{imnorm}}=0$.
5:     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 after 40$40$ terms. Further Taylor series coefficients can no longer reliably be obtained by numerical differentiation.
ifail = 2${\mathbf{ifail}}=2$
iflag has been set nonzero by the user.
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 = 1${\mathbf{ifail}}=-1$
Input argument number _$_$ is invalid.
ifail = 3${\mathbf{ifail}}=-3$
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. See Section 9.4 of Higham (2008) for further discussion of the Schur–Parlett algorithm, and Lyness and Moler (1967) for discussion of the numerical differentiation subroutine.

The integer allocatable memory required is n$n$. If A$A$ has real eigenvalues then up to 6n2$6{n}^{2}$ of double allocatable memory may be required. If A$A$ has complex eigenvalues then up to 6n2$6{n}^{2}$ 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. There is an additional cost in numerically differentiating f$f$, in order to obtain the Taylor series coefficients. If the derivatives of f$f$ are known analytically, then nag_matop_real_gen_matrix_fun_usd (f01em) can be used to evaluate f(A)$f\left(A\right)$ more accurately. If A$A$ is real symmetric then it is recommended that nag_matop_real_symm_matrix_fun (f01ef) be used as it is more efficient and, in general, more accurate than nag_matop_real_gen_matrix_fun_num (f01el).
For any z$z$ on the real line, f(z)$f\left(z\right)$ must be real. f$f$ must also be complex analytic ont he spectrum of A$A$. These conditions ensure that f(A)$f\left(A\right)$ is real for real A$A$.
For further information on matrix functions, see Higham (2008).
If estimates of the condition number of the matrix function are required then nag_matop_real_gen_matrix_cond_num (f01jb) should be used.
nag_matop_complex_gen_matrix_fun_num (f01fl) can be used to find the matrix function f(A)$f\left(A\right)$ for a complex matrix A$A$.

## Example

```function nag_matop_real_gen_matrix_fun_num_example
a =  [3,  0,  1,  2;
-1,  1,  3,  1;
0,  2,  2,  1;
2,  1, -1,  1];
% Compute cos(2a)
[a, user, iflag, imnorm, ifail] = nag_matop_real_gen_matrix_fun_num(a, @f)

function [iflag, fz, user] = f(iflag, nz, z, user)
fz = complex(cos(2*z));
```
```

a =

-0.1704   -1.1597   -0.1878   -0.7307
-0.3950   -0.4410    0.7606    0.0655
-0.0950   -0.0717    0.0619   -0.4351
-0.1034    0.6424   -1.3964    0.1042

user =

0

iflag =

0

imnorm =

0

ifail =

0

```
```function f01el_example
a =  [3,  0,  1,  2;
-1,  1,  3,  1;
0,  2,  2,  1;
2,  1, -1,  1];
% Compute cos(2a)
[a, user, iflag, imnorm, ifail] = f01el(a, @f)

function [iflag, fz, user] = f(iflag, nz, z, user)
fz = complex(cos(2*z));
```
```

a =

-0.1704   -1.1597   -0.1878   -0.7307
-0.3950   -0.4410    0.7606    0.0655
-0.0950   -0.0717    0.0619   -0.4351
-0.1034    0.6424   -1.3964    0.1042

user =

0

iflag =

0

imnorm =

0

ifail =

0

```