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Chapter Contents

Chapter Introduction

NAG Toolbox: nag_matop_real_gen_matrix_fun_usd (f01em)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_matop_real_gen_matrix_fun_usd (f01em) computes the matrix function,

f (A)

, of a real

n

by

n

matrix

A

, using analytical derivatives of

f

you have supplied.

Syntax

[a, user, iflag, imnorm, ifail] = f01em(a, f, 'n', n, 'user', user)

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

Description

f (A)

is computed using the Schur–Parlett algorithm described in Higham (2008) and Davies and Higham (2003).

The scalar function

f

, and the derivatives of

f

, are returned by the function f which, given an integer

m

, should evaluate

f^{(m)} (z_{i})

at a number of (generally complex) points

z_{i}

, for

i = 1, 2, \dots, n_{z}

. For any

z

on the real line,

f (z)

must also be real. nag_matop_real_gen_matrix_fun_usd (f01em) is therefore appropriate for functions that can be evaluated on the complex plane and whose derivatives, of arbitrary order, can also be evaluated on the complex plane.

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: $a (lda :)$ – double array

The first dimension of the array a must be at least

n

.

The second dimension of the array a must be at least

n

.

The

n

by

n

matrix

A

.

2: $f$ – function handle or string containing name of m-file

Given an integer

m

, the function f evaluates

f^{(m)} (z_{i})

at a number of points

z_{i}

.

[iflag, fz, user] = f(m, iflag, nz, z, user)

Input Parameters

1: $m$ – int64int32nag_int scalar: The order, $m$ , of the derivative required.
If $m = 0$ , $f (z_{i})$ should be returned. For $m > 0$ , $f^{(m)} (z_{i})$ should be returned.
2: $iflag$ – int64int32nag_int scalar: iflag will be zero.
3: $nz$ – int64int32nag_int scalar: $n_{z}$ , the number of function or derivative values required.
4: $z (nz)$ – complex array: The $n_{z}$ points $z_{1}, z_{2}, \dots, z_{n_{z}}$ at which the function $f$ is to be evaluated.
5: $user$ – Any MATLAB object: f is called from nag_matop_real_gen_matrix_fun_usd (f01em) with the object supplied to nag_matop_real_gen_matrix_fun_usd (f01em).

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 $f (z)$ ; for instance $f (z_{i})$ may not be defined for a particular $z_{i}$ . If iflag is returned as nonzero then nag_matop_real_gen_matrix_fun_usd (f01em) will terminate the computation, with $ifail = 2$ .
2: $fz (nz)$ – complex array: The $n_{z}$ function or derivative values. $fz (i)$ should return the value $f^{(m)} (z_{i})$ , for $i = 1, 2, \dots, n_{z}$ . If $z_{i}$ lies on the real line, then so must $f^{(m)} (z_{i})$ .
3: $user$ – Any MATLAB object

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the first dimension of the array a.
$n$ , the order of the matrix $A$ .

Constraint: $n \geq 0$ .
2: $user$ – Any MATLAB object: user is not used by nag_matop_real_gen_matrix_fun_usd (f01em), 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.

Output Parameters

1: $a (lda :)$ – double array: The first dimension of the array a will be $n$ .
The second dimension of the array a will be $n$ .

The $n$ by $n$ matrix, $f (A)$ .
2: $user$ – Any MATLAB object
3: $iflag$ – int64int32nag_int scalar: $iflag = 0$ , unless iflag has been set nonzero inside f, in which case iflag will be the value set and ifail will be set to $ifail = 2$ .
4: $imnorm$ – double scalar: If $A$ has complex eigenvalues, nag_matop_real_gen_matrix_fun_usd (f01em) will use complex arithmetic to compute $f (A)$ . The imaginary part is discarded at the end of the computation, because it will theoretically vanish. imnorm contains the $1$ -norm of the imaginary part, which should be used to check that the function has given a reliable answer.
If $A$ has real eigenvalues, nag_matop_real_gen_matrix_fun_usd (f01em) uses real arithmetic and $imnorm = 0$ .
5: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$: A Taylor series failed to converge.

$ifail = 2$: iflag has been set nonzero by the user.

$ifail = 3$: There was an error whilst reordering the Schur form of $A$ .
Note: this failure should not occur and suggests that the function has been called incorrectly.

$ifail = 4$: The routine was unable to compute the Schur decomposition of $A$ .
Note: this failure should not occur and suggests that the function has been called incorrectly.

$ifail = 5$: An unexpected internal error occurred. Please contact NAG.

$ifail = - 1$: Input argument number $_$ is invalid.

$ifail = - 3$: On entry, argument lda is invalid.
Constraint: $lda \geq n$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

For a normal matrix

A

(for which

A^{T} A = A A^{T}

), the Schur decomposition is diagonal and the algorithm reduces to evaluating

f

at the eigenvalues of

A

and then constructing

f (A)

using the Schur vectors. This should give a very accurate result. In general, however, no error bounds are available for the algorithm. See Section 9.4 of Higham (2008) for further discussion of the Schur–Parlett algorithm.

Further Comments

If

A

has real eigenvalues then up to

6 n^{2}

of double allocatable memory may be required. If

A

has complex eigenvalues then up to

6 n^{2}

of complex allocatable memory may be required.

The cost of the Schur–Parlett algorithm depends on the spectrum of

A

, but is roughly between

28 n^{3}

and

n^{4} / 3

floating-point operations. There is an additional cost in evaluating

f

and its derivatives. If the derivatives of

f

are not known analytically, then nag_matop_real_gen_matrix_fun_num (f01el) can be used to evaluate

f (A)

using numerical differentiation. If

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_usd (f01em).

For any

z

on the real line,

f (z)

must be real.

f

must also be complex analytic on the spectrum of

A

. These conditions ensure that

f (A)

is real for real

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_usd (f01jc) should be used.

nag_matop_complex_gen_matrix_fun_usd (f01fm) can be used to find the matrix function

f (A)

for a complex matrix

A

.

Example

This example finds the

e^{2 A}

where

A = (\begin{array}{r} 1 & 0 & - 2 & 1 \\ - 1 & 2 & 0 & 1 \\ 2 & 0 & 1 & 0 \\ 1 & 0 & - 1 & 2 \end{array}) .

Open in the MATLAB editor: f01em_example

function f01em_example


fprintf('f01em example results\n\n');

a =  [1,  0, -2,  1;
     -1,  2,  0,  1;
      2,  0,  1,  0;
      1,  0, -1,  2];

% Compute f(a)
[exp2a, user, iflag, imnorm, ifail] = ...
  f01em(a, @f);

disp('f(A) = exp(2A)');
disp(exp2a);

fprintf('Imnorm = %6.2f\n',imnorm);



function [iflag, fz, user] = f(m, iflag, nz, z, user)
  fz = double(2^m)*exp(2*z);

f01em example results

f(A) = exp(2A)
  -12.1880         0   -3.4747    8.3697
  -13.7274   54.5982  -23.9801   82.8593
   -9.7900         0  -25.4527   26.5294
  -18.1597         0  -34.8991   49.2404

Imnorm =   0.00

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Chapter Contents

Chapter Introduction

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