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NAG Toolbox: nag_matop_real_gen_matrix_cond_std (f01ja)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_matop_real_gen_matrix_cond_std (f01ja) 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, where f is either the exponential, logarithm, sine, cosine, hyperbolic sine (sinh) or hyperbolic cosine (cosh). The evaluation of the matrix function, fA, is also returned.

Syntax

[a, conda, norma, normfa, ifail] = f01ja(fun, a, 'n', n)
[a, conda, norma, normfa, ifail] = nag_matop_real_gen_matrix_cond_std(fun, a, 'n', n)

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_real_gen_matrix_cond_std (f01ja) 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:     fun – string
Indicates which matrix function will be used.
fun='exp'
The matrix exponential, eA, will be used.
fun='sin'
The matrix sine, sinA, will be used.
fun='cos'
The matrix cosine, cosA, will be used.
fun='sinh'
The hyperbolic matrix sine, sinhA, will be used.
fun='cosh'
The hyperbolic matrix cosine, coshA, will be used.
fun='log'
The matrix logarithm, logA, will be used.
Constraint: fun='exp', 'sin', 'cos', 'sinh', 'cosh' or 'log'.
2:     alda: – 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.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the first dimension of the array a and the second dimension of the array a. (An error is raised if these dimensions are not equal.)
n, the order of the matrix A.
Constraint: n0.

Output Parameters

1:     alda: – 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, fA.
2:     conda – double scalar
An estimate of the absolute condition number of f at A.
3:     norma – double scalar
The 1-norm of A.
4:     normfa – double scalar
The 1-norm of fA.
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
An internal error occurred when evaluating the matrix function fA. Please contact NAG.
   ifail=2
An internal error occurred when estimating the norm of the Fréchet derivative of f at A. Please contact NAG.
   ifail=-1
On entry, fun=_ was an illegal value.
   ifail=-2
On entry, n<0.
Input argument number _ is invalid.
   ifail=-4
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_real_gen_matrix_cond_std (f01ja) uses the norm estimation function nag_linsys_real_gen_norm_rcomm (f04yd) 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_real_gen_norm_rcomm (f04yd).

Further Comments

The matrix function is computed using one of three underlying matrix function routines:
Approximately 6n2 of real allocatable memory is required by the routine, in addition to the memory used by these underlying matrix function routines.
If only fA is required, without an estimate of the condition number, then it is far more efficient to use the appropriate matrix function routine listed above.
nag_matop_complex_gen_matrix_cond_std (f01ka) can be used to find the condition number of the exponential, logarithm, sine, cosine, sinh or cosh matrix functions at a complex matrix.

Example

This example estimates the absolute and relative condition numbers of the matrix sinh function where
A = 2 1 3 1 3 -1 0 2 1 0 3 1 1 2 0 3 .  
function f01ja_example


fprintf('f01ja example results\n\n');

a = [2,   1,   3,   1;
     3,  -1,   0,   2;
     1,   0,   3,   1;
     1,   2,   0,   3];
fun = 'sinh';

% Find absolute condition number estimate
[a, conda, norma, normfa, ifail] = f01ja( ...
                                          fun, a);

fprintf('\nf(A) = %s(A)\n', fun);
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


f01ja example results


f(A) = sinh(A)
Estimated absolute condition number is:  204.45
Estimated relative condition number is:    7.90

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