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NAG Toolbox: nag_matop_complex_gen_matrix_fun_num (f01fl)

Purpose

nag_matop_complex_gen_matrix_fun_num (f01fl) computes the matrix function, f(A)f(A), of a complex nn by nn matrix AA. Numerical differentiation is used to evaluate the derivatives of ff when they are required.

Syntax

[a, user, iflag, ifail] = f01fl(a, f, 'n', n, 'user', user)
[a, user, iflag, ifail] = nag_matop_complex_gen_matrix_fun_num(a, f, 'n', n, 'user', user)

Description

f(A)f(A) is computed using the Schur–Parlett algorithm described in Higham (2008) and Davies and Higham (2003). The coefficients of the Taylor series used in the algorithm are evaluated using the numerical differentiation algorithm of Lyness and Moler (1967).
The scalar function ff is supplied via function f which evaluates f(zi)f(zi) at a number of points zizi.

References

Davies P I and Higham N J (2003) A Schur–Parlett algorithm for computing matrix functions. SIAM J. Matrix Anal. Appl. 25(2) 464–485
Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA
Lyness J N and Moler C B (1967) Numerical differentiation of analytic functions SIAM J. Numer. Anal. 4(2) 202–210

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
The function f evaluates f(zi)f(zi) at a number of points zizi.
[iflag, fz, user] = f(iflag, nz, z, user)

Input Parameters

1:     iflag – int64int32nag_int scalar
iflag will be zero.
2:     nz – int64int32nag_int scalar
nznz, the number of function values required.
3:     z(nz) – complex array
The nznz points z1,z2,,znzz1,z2,,znz at which the function ff is to be evaluated.
4:     user – Any MATLAB object
f is called from nag_matop_complex_gen_matrix_fun_num (f01fl) with the object supplied to nag_matop_complex_gen_matrix_fun_num (f01fl).

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(zi)f(zi); for instance f(zi)f(zi) may not be defined. If iflag is returned as nonzero then nag_matop_complex_gen_matrix_fun_num (f01fl) will terminate the computation, with ifail = 2ifail=2.
2:     fz(nz) – complex array
The nznz function values. fz(i)fzi should return the value f(zi)f(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_fun_num (f01fl), 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 = 2ifail=2.
4:     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
A Taylor series failed to converge after 4040 terms. Further Taylor series coefficients can no longer reliably be obtained by numerical differentiation.
  ifail = 2ifail=2
iflag has been set nonzero by the user.
  ifail = 3ifail=3
The function was unable to compute the Schur decomposition of AA.
Note:  this failure should not occur and suggests that the function has been called incorrectly.
There was an error whilst reordering the Schur form of AA.
Note:  this failure should not occur and suggests that the function has been called incorrectly.
  ifail = 5ifail=5
An unexpected internal error occurred. Please contact NAG.
  ifail = 1ifail=-1
Input argument number __ is invalid.
  ifail = 3ifail=-3
On entry, parameter lda is invalid.
Constraint: ldanldan.
  ifail = 999ifail=-999
Allocation of memory failed. Up to 6 × N26×N2 of complex allocatable memory may be required.

Accuracy

For a normal matrix AA (for which AHA = AAHAHA=AAH) Schur decomposition is diagonal and the algorithm reduces to evaluating ff at the eigenvalues of AA and then constructing f(A)f(A) using the Schur vectors. See Section 9.4 of Higham (2008) for further discussion of the Schur–Parlett algorithm, and Lyness and Moler (1967) for discussion of the numerical differentiation subroutine.

Further Comments

The integer allocatable memory required is nn, and up to 6n26n2 of complex allocatable memory is required.
The cost of the Schur–Parlett algorithm depends on the spectrum of AA, but is roughly between 28n328n3 and n4 / 3n4/3 floating point operations. There is an additional cost in numerically differentiating ff, in order to obtain the Taylor series coefficients. If the derivatives of ff are known analytically, then nag_matop_complex_gen_matrix_fun_usd (f01fm) can be used to evaluate f(A)f(A) more accurately. If AA is complex Hermitian then it is recommended that nag_matop_complex_herm_matrix_fun (f01ff) be used as it is more efficient and, in general, more accurate than nag_matop_complex_gen_matrix_fun_num (f01fl).
Note that ff must be analytic in the region of the complex plane containing the spectrum of AA.
For further information on matrix functions, see Higham (2008).
If estimates of the condition number of the matrix function are required then nag_matop_complex_gen_matrix_cond_num (f01kb) should be used.
nag_matop_real_gen_matrix_fun_num (f01el) can be used to find the matrix function f(A)f(A) for a real matrix AA.

Example

function nag_matop_complex_gen_matrix_fun_num_example
a = [1.0+0.0i, 0.0+1.0i,  1.0+0.0i, 0.0+1.0i;
    -1.0+0.0i, 0.0+0.0i,  2.0+1.0i, 0.0+0.0i;
     0.0+0.0i, 2.0+1.0i,  0.0+2.0i, 0.0+1.0i;
     1.0+0.0i, 1.0+1.0i, -1.0+0.0i, 2.0+1.0i];
% Compute f(a)
[a, user, iflag, ifail] = nag_matop_complex_gen_matrix_fun_num(a, @f)


function [iflag, fz, user] = f(iflag, nz, z, user)
  fz = sin(2*z);
 

a =

   1.1960 - 3.2270i -21.0733 - 9.6441i -15.4159 -14.1977i -12.4279 -11.9638i
   3.2957 - 3.6334i -14.6084 -21.4846i  -6.7764 -24.1726i  -5.1338 -17.0926i
   5.0928 - 3.7806i -14.6839 -34.5063i  -0.9231 -35.4729i  -2.0715 -26.3460i
  -1.8349 + 0.0808i  -8.2484 - 0.4014i  -6.0093 - 1.6831i  -7.1318 - 1.9396i


user =

     0


iflag =

                    0


ifail =

                    0


function f01fl_example
a = [1.0+0.0i, 0.0+1.0i,  1.0+0.0i, 0.0+1.0i;
    -1.0+0.0i, 0.0+0.0i,  2.0+1.0i, 0.0+0.0i;
     0.0+0.0i, 2.0+1.0i,  0.0+2.0i, 0.0+1.0i;
     1.0+0.0i, 1.0+1.0i, -1.0+0.0i, 2.0+1.0i];
% Compute f(a)
[a, user, iflag, ifail] = f01fl(a, @f)


function [iflag, fz, user] = f(iflag, nz, z, user)
  fz = sin(2*z);
 

a =

   1.1960 - 3.2270i -21.0733 - 9.6441i -15.4159 -14.1977i -12.4279 -11.9638i
   3.2957 - 3.6334i -14.6084 -21.4846i  -6.7764 -24.1726i  -5.1338 -17.0926i
   5.0928 - 3.7806i -14.6839 -34.5063i  -0.9231 -35.4729i  -2.0715 -26.3460i
  -1.8349 + 0.0808i  -8.2484 - 0.4014i  -6.0093 - 1.6831i  -7.1318 - 1.9396i


user =

     0


iflag =

                    0


ifail =

                    0



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