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Chapter Contents

Chapter Introduction

NAG Toolbox: nag_matop_real_gen_matrix_cond_usd (f01jc)

▸▿ 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_cond_usd (f01jc) computes an estimate of the absolute condition number of a matrix function

f

at a real

n

by

n

matrix

A

in the

1

-norm, using analytical derivatives of

f

you have supplied.

Syntax

[a, user, iflag, conda, norma, normfa, ifail] = f01jc(a, f, 'n', n, 'user', user)

[a, user, iflag, conda, norma, normfa, ifail] = nag_matop_real_gen_matrix_cond_usd(a, f, 'n', n, 'user', user)

Description

The absolute condition number of

f

at

A

,

{cond}_{abs} (f, A)

is given by the norm of the Fréchet derivative of

f

,

L (A)

, which is defined by

‖L (X)‖ := \max_{E \neq 0} \frac{‖L (X, E)‖}{‖E‖},

where

L (X, E)

is the Fréchet derivative in the direction

E

.

L (X, E)

is linear in

E

and can therefore be written as

vec (L (X, E)) = K (X) vec (E),

where the

vec

operator stacks the columns of a matrix into one vector, so that

K (X)

is

n^{2} \times n^{2}

. nag_matop_real_gen_matrix_cond_usd (f01jc) computes an estimate

γ

such that

γ \leq {‖K (X)‖}_{1}

, where

{‖K (X)‖}_{1} \in [n^{- 1} {‖L (X)‖}_{1}, n {‖L (X)‖}_{1}]

. The relative condition number can then be computed via

{cond}_{rel} (f, A) = \frac{{cond}_{abs} (f, A) {‖A‖}_{1}}{{‖f (A)‖}_{1}} .

The algorithm used to find

γ

is detailed in Section 3.4 of Higham (2008).

The function

f

, and the derivatives of

f

, are returned by function f which, given an integer

m

, evaluates

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_cond_usd (f01jc) 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

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_cond_usd (f01jc) with the object supplied to nag_matop_real_gen_matrix_cond_usd (f01jc).

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 $f (z)$ may not be defined. If iflag is returned as nonzero then nag_matop_real_gen_matrix_cond_usd (f01jc) will terminate the computation, with $ifail = 3$ .
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_cond_usd (f01jc), 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 $ifail = 3$ .
4: $conda$ – double scalar: An estimate of the absolute condition number of $f$ at $A$ .
5: $norma$ – double scalar: The $1$ -norm of $A$ .
6: $normfa$ – double scalar: The $1$ -norm of $f (A)$ .
7: $ifail$ – 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:

$ifail = 1$: An internal error occurred when estimating the norm of the Fréchet derivative of $f$ at $A$ . Please contact NAG.

$ifail = 2$: An internal error occurred when evaluating the matrix function $f (A)$ . You can investigate further by calling nag_matop_real_gen_matrix_fun_usd (f01em) with the matrix $A$ and the function $f$ .

$ifail = 3$: iflag has been set nonzero by the user-supplied function.

$ifail = - 1$: On entry, $n < 0$ .
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

nag_matop_real_gen_matrix_cond_usd (f01jc) uses the norm estimation routine nag_linsys_real_gen_norm_rcomm (f04yd) to estimate a quantity

γ

, where

γ \leq {‖K (X)‖}_{1}

and

{‖K (X)‖}_{1} \in [n^{- 1} {‖L (X)‖}_{1}, n {‖L (X)‖}_{1}]

. For further details on the accuracy of norm estimation, see the documentation for nag_linsys_real_gen_norm_rcomm (f04yd).

Further Comments

The matrix function is computed using the underlying matrix function routine nag_matop_real_gen_matrix_fun_usd (f01em). Approximately

6 n^{2}

of real allocatable memory is required by the routine, in addition to the memory used by the underlying matrix function routine.

If only

f (A)

is required, without an estimate of the condition number, then it is far more efficient to use the underlying matrix function routine directly.

The complex analogue of this function is nag_matop_complex_gen_matrix_cond_usd (f01kc).

Example

This example estimates the absolute and relative condition numbers of the matrix function

e^{2 A}

where

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

Open in the MATLAB editor: f01jc_example

function f01jc_example


fprintf('f01jc example results\n\n');

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

% Find absolute condition number estimate
[a, user, iflag, conda, norma, normfa, ifail] = ...
f01jc(a, @fexp2);

fprintf('\nf(A) = exp(2A)\n');
fprintf('Estimated absolute condition number is: %7.2f\n', conda);

%  Find relative condition number estimate
eps = x02aj;
if normfa > eps
   cond_rel = conda*norma/normfa;
   fprintf('Estimated relative condition number is: %7.2f\n', cond_rel);
else
  fprintf('The estimated norm of f(A) is effectively zero;\n');
  fprintf('the relative condition number is therefore undefined.\n');
end



function [iflag, fz, user] = fexp2(m, iflag, nz, z, user)
  fz = 2^double(m)*exp(2*z);

f01jc example results


f(A) = exp(2A)
Estimated absolute condition number is:  183.90
Estimated relative condition number is:   13.90

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Chapter Contents

Chapter Introduction

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