hide long namesshow long names
hide short namesshow short names
Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_matop_complex_gen_matrix_cond_usd (f01kc)

Purpose

nag_matop_complex_gen_matrix_cond_usd (f01kc) computes an estimate of the absolute condition number of a matrix function ff at a complex nn by nn matrix AA in the 11-norm, using analytical derivatives of ff you have supplied.

Syntax

[a, user, iflag, conda, norma, normfa, ifail] = f01kc(a, f, 'n', n, 'user', user)
[a, user, iflag, conda, norma, normfa, ifail] = nag_matop_complex_gen_matrix_cond_usd(a, f, 'n', n, 'user', user)

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,E)L(A,E), which is defined by
L(X) := maxE0 (L(X,E))/(E) .
L(X) := maxE0 L(X,E) E .
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_complex_gen_matrix_cond_usd (f01kc) 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).
The function ff, and the derivatives of ff, are returned by function f which, given an integer mm, evaluates f(m)(zi)f(m)(zi) at a number of points zizi, for i = 1,2,,nzi=1,2,,nz, on the complex plane. nag_matop_complex_gen_matrix_cond_usd (f01kc) 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, : :) – complex 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.
2:     f – function handle or string containing name of m-file
Given an integer mm, the function f evaluates f(m)(zi)f(m)(zi) at a number of points zizi.
[iflag, fz, user] = f(m, iflag, nz, z, user)

Input Parameters

1:     m – int64int32nag_int scalar
The order, mm, of the derivative required.
If m = 0m=0, f(zi)f(zi) should be returned. For m > 0m>0, f(m)(zi)f(m)(zi) should be returned.
2:     iflag – int64int32nag_int scalar
iflag will be zero.
3:     nz – int64int32nag_int scalar
nznz, the number of function or derivative values required.
4:     z(nz) – complex array
The nznz points z1,z2,,znzz1,z2,,znz at which the function ff is to be evaluated.
5:     user – Any MATLAB object
f is called from nag_matop_complex_gen_matrix_cond_usd (f01kc) with the object supplied to nag_matop_complex_gen_matrix_cond_usd (f01kc).

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)f(z); for instance f(z)f(z) may not be defined. If iflag is returned as nonzero then nag_matop_complex_gen_matrix_cond_usd (f01kc) will terminate the computation, with ifail = 3ifail=3.
2:     fz(nz) – complex array
The nznz function or derivative values. fz(i)fzi should return the value f(m)(zi)f(m)(zi), for i = 1,2,,nzi=1,2,,nz.
3:     user – Any MATLAB object

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array a.
nn, the order of the matrix AA.
Constraint: n0n0.
2:     user – Any MATLAB object
user is not used by nag_matop_complex_gen_matrix_cond_usd (f01kc), 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.

Input Parameters Omitted from the MATLAB Interface

lda iuser ruser

Output Parameters

1:     a(lda, : :) – complex 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:     user – Any MATLAB object
3:     iflag – int64int32nag_int scalar
iflag = 0iflag=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 = 3ifail=3.
4:     conda – double scalar
An estimate of the absolute condition number of ff at AA.
5:     norma – double scalar
The 11-norm of AA.
6:     normfa – double scalar
The 11-norm of f(A)f(A).
7:     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 estimating the norm of the Fréchet derivative of ff at AA. Please contact NAG.
  ifail = 2ifail=2
An internal error occurred when evaluating the matrix function f(A)f(A). You can investigate further by calling nag_matop_complex_gen_matrix_fun_usd (f01fm) with the matrix AA and the function ff.
  ifail = 3ifail=3
iflag has been set nonzero by the user-supplied function.
  ifail = 1ifail=-1
On entry, n < 0n<0.
  ifail = 3ifail=-3
On entry, parameter lda is invalid.
Constraint: ldanldan.
  ifail = 999ifail=-999
Allocation of memory failed.

Accuracy

nag_matop_complex_gen_matrix_cond_usd (f01kc) uses the norm estimation function nag_linsys_complex_gen_norm_rcomm (f04zd) 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_complex_gen_norm_rcomm (f04zd).

Further Comments

Approximately 6n26n2 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_usd (f01fm).
nag_matop_complex_gen_matrix_cond_usd (f01kc) returns the matrix function f(A)f(A). This is computed using nag_matop_complex_gen_matrix_fun_usd (f01fm). If only f(A)f(A) is required, without an estimate of the condition number, then it is far more efficient to use nag_matop_complex_gen_matrix_fun_usd (f01fm) directly.
The real analogue of this function is nag_matop_real_gen_matrix_cond_usd (f01jc).

Example

function nag_matop_complex_gen_matrix_cond_usd_example
a = [1+1i, 0+1i, 1+0i, 2+0i;
     0+0i, 2+0i, 0+2i, 1+0i;
     0+1i, 0+1i, 0+0i, 2+0i;
     1+0i, 0+1i, 1+0i, 0+1i];
% Find absolute condition number estimate
[a, user, iflag, conda, norma, normfa, ifail] = nag_matop_complex_gen_matrix_cond_usd(a, @fexp3);

fprintf('\nF(a) = exp(3a)\n');
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


function [iflag, fz, user] = fexp3(m, iflag, nz, z, user)
  fz = 3^double(m)*exp(3*z);
 

F(a) = exp(3a)
Estimated absolute condition number is: 9474.43
Estimated relative condition number is:   13.74

function f01kc_example
a = [1+1i, 0+1i, 1+0i, 2+0i;
     0+0i, 2+0i, 0+2i, 1+0i;
     0+1i, 0+1i, 0+0i, 2+0i;
     1+0i, 0+1i, 1+0i, 0+1i];
% Find absolute condition number estimate
[a, user, iflag, conda, norma, normfa, ifail] = f01kc(a, @fexp3);

fprintf('\nF(a) = exp(3a)\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 and so the\nrelative condition number is undefined.\n');
end


function [iflag, fz, user] = fexp3(m, iflag, nz, z, user)
  fz = 3^double(m)*exp(3*z);
 

F(a) = exp(3a)
Estimated absolute condition number is: 9474.43
Estimated relative condition number is:   13.74


PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2013