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NAG Toolbox: nag_matop_complex_gen_matrix_cond_std (f01ka)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_matop_complex_gen_matrix_cond_std (f01ka) 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, 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] = f01ka(fun, a, 'n', n)
[a, conda, norma, normfa, ifail] = nag_matop_complex_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_complex_gen_matrix_cond_std (f01ka) 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: – 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.

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.

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:     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 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 fA. You can investigate further by calling nag_matop_complex_gen_matrix_exp (f01fc), nag_matop_complex_gen_matrix_log (f01fj) or nag_matop_complex_gen_matrix_fun_std (f01fk) with the matrix A.
   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_complex_gen_matrix_cond_std (f01ka) 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 routines nag_matop_complex_gen_matrix_exp (f01fc), nag_matop_complex_gen_matrix_log (f01fj) or nag_matop_complex_gen_matrix_fun_std (f01fk).
nag_matop_complex_gen_matrix_cond_std (f01ka) returns the matrix function fA. This is computed using nag_matop_complex_gen_matrix_exp (f01fc) if fun='exp', nag_matop_complex_gen_matrix_log (f01fj) if fun='log' and nag_matop_complex_gen_matrix_fun_std (f01fk) otherwise. 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_exp (f01fc), nag_matop_complex_gen_matrix_log (f01fj) or nag_matop_complex_gen_matrix_fun_std (f01fk) directly.
nag_matop_real_gen_matrix_cond_std (f01ja) can be used to find the condition number of the exponential, logarithm, sine, cosine, sinh or cosh at a real matrix.

Example

This example estimates the absolute and relative condition numbers of the matrix sinh function for
A = 0.0+1.0i -1.0+0.0i 1.0+0.0i 2.0+0.0i 2.0+1.0i 0.0-1.0i 0.0+0.0i 1.0+0.0i 0.0+1.0i 0.0+0.0i 1.0+1.0i 0.0+2.0i 1.0+0.0i 2.0+0.0i -2.0+3.0i 0.0+1.0i .  
function f01ka_example


fprintf('f01ka example results\n\n');

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

% Find absolute condition number estimate
[a, conda, norma, normfa, ifail] = ...
f01ka(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


f01ka example results


f(A) = sinh(A)
Estimated absolute condition number is:    7.33
Estimated relative condition number is:    4.94

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