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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_matop_complex_gen_matrix_cond_num (f01kb)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_matop_complex_gen_matrix_cond_num (f01kb) computes an estimate of the absolute condition number of a matrix function f of a complex n by n matrix A in the 1-norm. Numerical differentiation is used to evaluate the derivatives of f when they are required.

Syntax

[a, user, iflag, conda, norma, normfa, ifail] = f01kb(a, f, 'n', n, 'user', user)
[a, user, iflag, conda, norma, normfa, ifail] = nag_matop_complex_gen_matrix_cond_num(a, f, 'n', n, 'user', user)

Description

The absolute condition number of f at A, condabsf,A is given by the norm of the Fréchet derivative of f, LA, which is defined by
LX := maxE0 LX,E E ,  
where LX,E is the Fréchet derivative in the direction E. LX,E is linear in E and can therefore be written as
vec LX,E = KX vecE ,  
where the vec operator stacks the columns of a matrix into one vector, so that KX is n2×n2. nag_matop_complex_gen_matrix_cond_num (f01kb) computes an estimate γ such that γ KX 1 , where KX 1 n-1 LX 1 , n LX 1 . The relative condition number can then be computed via
cond rel f,A = cond abs f,A A1 fA 1 .  
The algorithm used to find γ is detailed in Section 3.4 of Higham (2008).

References

Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA

Parameters

Compulsory Input Parameters

1:     alda: – complex 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
The function f evaluates fzi at a number of points zi.
[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, the number of function values required.
3:     znz – complex array
The nz points z1,z2,,znz at which the function f is to be evaluated.
4:     user – Any MATLAB object
f is called from nag_matop_complex_gen_matrix_cond_num (f01kb) with the object supplied to nag_matop_complex_gen_matrix_cond_num (f01kb).

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 fz; for instance fz may not be defined. If iflag is returned as nonzero then nag_matop_complex_gen_matrix_cond_num (f01kb) will terminate the computation, with ifail=3.
2:     fznz – complex array
The nz function values. fzi should return the value fzi, for i=1,2,,nz.
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: n0.
2:     user – Any MATLAB object
user is not used by nag_matop_complex_gen_matrix_cond_num (f01kb), 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:     alda: – complex 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, fA.
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 fA.
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 while evaluating the matrix function fA. You can investigate further by calling nag_matop_complex_gen_matrix_fun_num (f01fl) 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.
   ifail=-3
On entry, argument lda is invalid.
Constraint: ldan.
   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_complex_gen_matrix_cond_num (f01kb) uses the norm estimation function nag_linsys_complex_gen_norm_rcomm (f04zd) to estimate a quantity γ, where γ KX 1  and KX 1 n-1 LX 1 , n LX 1 . For further details on the accuracy of norm estimation, see the documentation for nag_linsys_complex_gen_norm_rcomm (f04zd).

Further Comments

Approximately 6n2 of complex allocatable memory is required by the routine, in addition to the memory used by the underlying matrix function routine nag_matop_complex_gen_matrix_fun_num (f01fl).
nag_matop_complex_gen_matrix_cond_num (f01kb) returns the matrix function fA. This is computed using nag_matop_complex_gen_matrix_fun_num (f01fl). If only fA is required, without an estimate of the condition number, then it is far more efficient to use nag_matop_complex_gen_matrix_fun_num (f01fl) directly.
The real analogue of this function is nag_matop_real_gen_matrix_cond_num (f01jb).

Example

This example estimates the absolute and relative condition numbers of the matrix function sin2A where
A = 2.0+0.0i 0.0+1.0i 1.0+1.0i 0.0+3.0i 1.0+1.0i 0.0+2.0i 2.0+2.0i 0.0+0.0i 0.0+0.0i 2.0+0.0i 1.0+2.0i 1.0+0.0i 1.0+1.0i 3.0+0.0i 0.0+0.0i 1.0+2.0i .  
function f01kb_example


fprintf('f01kb example results\n\n');

a = [2+0i, 0+1i, 1+1i, 0+3i;
     1+1i, 0+2i, 2+2i, 0+0i;
     0+0i, 2+0i, 1+2i, 1+0i;
     1+1i, 3+0i, 0+0i, 1+2i];

% Find absolute condition number estimate
[a, user, iflag, conda, norma, normfa, ifail] = ...
f01kb(a, @fsin2);

fprintf('\nf(A) = sin(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] = fsin2(iflag, nz, z, user)
  fz = sin(2*z);
f01kb example results


f(A) = sin(2A)
Estimated absolute condition number is: 2016.99
Estimated relative condition number is:   12.86

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