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NAG Toolbox: nag_matop_real_gen_matrix_cond_std (f01ja)

Purpose

nag_matop_real_gen_matrix_cond_std (f01ja) computes an estimate of the absolute condition number of a matrix function ff at a real nn by nn matrix AA in the 11-norm, where ff is either the exponential, logarithm, sine, cosine, hyperbolic sine (sinh) or hyperbolic cosine (cosh). The evaluation of the matrix function, f(A)f(A), 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 ff at AA, condabs(f,A)condabs(f,A) is given by the norm of the Fréchet derivative of ff, L(A)L(A), which is defined by
L(X) := maxE0 (L(X,E))/(E) ,
L(X) := maxE0 L(X,E) E ,
where L(X,E)L(X,E) is the Fréchet derivative in the direction EE. L(X,E)L(X,E) is linear in EE and can therefore be written as
vec (L(X,E)) = K(X) vec(E) ,
vec ( L(X,E) ) = K(X) vec(E) ,
where the vecvec operator stacks the columns of a matrix into one vector, so that K(X)K(X) is n2 × n2n2×n2. nag_matop_real_gen_matrix_cond_std (f01ja) computes an estimate γγ such that γ K(X)1 γ K(X) 1 , where K(X)1 [ n1 L(X)1 , n L(X)1 ] K(X) 1 [ n-1 L(X) 1 , n L(X) 1 ] . The relative condition number can then be computed via
condrel (f,A) = ( condabs (f,A) A1 )/(f(A)1) .
cond rel (f,A) = cond abs (f,A) A1 f(A) 1 .
The algorithm used to find γγ 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'fun='exp'
The matrix exponential, eAeA, will be used.
fun = 'sin'fun='sin'
The matrix sine, sin(A)sin(A), will be used.
fun = 'cos'fun='cos'
The matrix cosine, cos(A)cos(A), will be used.
fun = 'sinh'fun='sinh'
The hyperbolic matrix sine, sinh(A)sinh(A), will be used.
fun = 'cosh'fun='cosh'
The hyperbolic matrix cosine, cosh(A)cosh(A), will be used.
fun = 'log'fun='log'
The matrix logarithm, log(A)log(A), will be used.
Constraint: fun = 'exp'fun='exp', 'sin''sin', 'cos''cos', 'sinh''sinh', 'cosh''cosh' or 'log''log'.
2:     a(lda, : :) – double array
The first dimension of the array a must be at least max (1,n)max(1,n)
The second dimension of the array must be at least nn
The nn by nn matrix AA.

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The second dimension of the array a.
nn, the order of the matrix AA.
Constraint: n0n0.

Input Parameters Omitted from the MATLAB Interface

lda

Output Parameters

1:     a(lda, : :) – double array
The first dimension of the array a will be max (1,n)max(1,n)
The second dimension of the array will be nn
ldamax (1,n)ldamax(1,n).
The nn by nn matrix, f(A)f(A).
2:     conda – double scalar
An estimate of the absolute condition number of ff at AA.
3:     norma – double scalar
The 11-norm of AA.
4:     normfa – double scalar
The 11-norm of f(A)f(A).
5:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:
  ifail = 1ifail=1
An internal error occurred when evaluating the matrix function f(A)f(A). Please contact NAG.
  ifail = 2ifail=2
An internal error occurred when estimating the norm of the Fréchet derivative of ff at AA. Please contact NAG.
  ifail = 1ifail=-1
On entry, fun = 'exp'fun='exp', 'sin''sin', 'cos''cos', 'sinh''sinh', 'cosh''cosh' or 'log''log'.
Input parameter number __ is invalid.
  ifail = 2ifail=-2
On entry, n < 0n<0.
Input parameter number __ is invalid.
  ifail = 4ifail=-4
On entry, parameter lda is invalid.
Constraint: ldanldan.
  ifail = 999ifail=-999
Allocation of memory 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 γγ, where γ K(X)1 γ K(X) 1  and K(X)1 [ n1 L(X)1 , n L(X)1 ] K(X) 1 [ 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 one of three underlying matrix function routines:
Approximately 6n26n2 of real allocatable memory is required by the routine, in addition to the memory used by these underlying matrix function routines.
If only f(A)f(A) 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

function nag_matop_real_gen_matrix_cond_std_example
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] = nag_matop_real_gen_matrix_cond_std(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 = nag_machine_precision;
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 and so the\nrelative condition number is undefined.\n');
end
 

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

function f01ja_example
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 and so the\nrelative condition number is undefined.\n');
end
 

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


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