nag_matop_real_gen_matrix_fun_num (f01elc) (PDF version)
f01 Chapter Contents
f01 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_matop_real_gen_matrix_fun_num (f01elc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_matop_real_gen_matrix_fun_num (f01elc) computes the matrix function, fA, of a real n by n matrix A. Numerical differentiation is used to evaluate the derivatives of f when they are required.

2  Specification

#include <nag.h>
#include <nagf01.h>
void  nag_matop_real_gen_matrix_fun_num (Integer n, double a[], Integer pda,
void (*f)(Integer *iflag, Integer nz, const Complex z[], Complex fz[], Nag_Comm *comm),
Nag_Comm *comm, Integer *iflag, double *imnorm, NagError *fail)

3  Description

fA 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 f is supplied via function f which evaluates fzi at a number of points zi.

4  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

5  Arguments

1:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
2:     a[dim]doubleInput/Output
Note: the dimension, dim, of the array a must be at least pda×n.
The i,jth element of the matrix A is stored in a[j-1×pda+i-1].
On entry: the n by n matrix A.
On exit: the n by n matrix, fA.
3:     pdaIntegerInput
On entry: the stride separating matrix row elements in the array a.
Constraint: pdan.
4:     ffunction, supplied by the userExternal Function
The function f evaluates fzi at a number of points zi.
The specification of f is:
void  f (Integer *iflag, Integer nz, const Complex z[], Complex fz[], Nag_Comm *comm)
1:     iflagInteger *Input/Output
On entry: iflag will be zero.
On exit: 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 fzi; for instance fzi may not be defined. If iflag is returned as nonzero then nag_matop_real_gen_matrix_fun_num (f01elc) will terminate the computation, with fail.code= NE_USER_STOP.
2:     nzIntegerInput
On entry: nz, the number of function values required.
3:     z[nz]const ComplexInput
On entry: the nz points z1,z2,,znz at which the function f is to be evaluated.
4:     fz[nz]ComplexOutput
On exit: the nz function values. fz[i-1] should return the value fzi, for i=1,2,,nz. If zi lies on the real line, then so must fzi.
5:     commNag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to f.
userdouble *
iuserInteger *
The type Pointer will be void *. Before calling nag_matop_real_gen_matrix_fun_num (f01elc) you may allocate memory and initialize these pointers with various quantities for use by f when called from nag_matop_real_gen_matrix_fun_num (f01elc) (see Section in the Essential Introduction).
5:     commNag_Comm *Communication Structure
The NAG communication argument (see Section in the Essential Introduction).
6:     iflagInteger *Output
On exit: iflag=0, unless iflag has been set nonzero inside f, in which case iflag will be the value set and fail will be set to fail.code= NE_USER_STOP.
7:     imnormdouble *Output
On exit: if A has complex eigenvalues, nag_matop_real_gen_matrix_fun_num (f01elc) will use complex arithmetic to compute fA. The imaginary part is discarded at the end of the computation, because it will theoretically vanish. imnorm contains the 1-norm of the imaginary part, which should be used to check that the routine has given a reliable answer.
If a has real eigenvalues, nag_matop_real_gen_matrix_fun_num (f01elc) uses real arithmetic and imnorm=0.
8:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

Allocation of memory failed.
On entry, argument value had an illegal value.
A Taylor series failed to converge after 40 terms. Further Taylor series coefficients can no longer reliably be obtained by numerical differentiation.
On entry, n=value.
Constraint: n0.
On entry, pda=value and n=value.
Constraint: pdan.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
An unexpected internal error occurred when ordering the eigenvalues of A. Please contact NAG.
The function was unable to compute the Schur decomposition of A.
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 A.
Note:  this failure should not occur and suggests that the function has been called incorrectly.
iflag has been set nonzero by the user.

7  Accuracy

For a normal matrix A (for which ATA=AAT) the Schur decomposition is diagonal and the algorithm reduces to evaluating f at the eigenvalues of A and then constructing fA 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 a discussion of numerical differentiation.

8  Parallelism and Performance

nag_matop_real_gen_matrix_fun_num (f01elc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_matop_real_gen_matrix_fun_num (f01elc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
In these implementations, this may make calls to the user supplied functions from within an OpenMP parallel region. Thus OpenMP directives within the user functions should be avoided, unless you are using the same OpenMP runtime library (which normally means using the same compiler) as that used to build your NAG Library implementation, as listed in the Installers' Note. You must also ensure that you use the NAG communication argument comm in a thread safe manner, which is best achieved by only using it to supply read-only data to the user functions.
Please consult the Users' Note for your implementation for any additional implementation-specific information.

9  Further Comments

The Integer allocatable memory required is n. If A has real eigenvalues then up to 6n2 of double allocatable memory may be required. If A has complex eigenvalues then up to 6n2 of Complex allocatable memory may be required.
The cost of the Schur–Parlett algorithm depends on the spectrum of A, but is roughly between 28n3 and n4/3 floating-point operations. There is an additional cost in numerically differentiating f, in order to obtain the Taylor series coefficients. If the derivatives of f are known analytically, then nag_matop_real_gen_matrix_fun_usd (f01emc) can be used to evaluate fA more accurately. If A is real symmetric then it is recommended that nag_matop_real_symm_matrix_fun (f01efc) be used as it is more efficient and, in general, more accurate than nag_matop_real_gen_matrix_fun_num (f01elc).
For any z on the real line, fz must be real. f must also be complex analytic on the spectrum of A. These conditions ensure that fA is real for real A.
For further information on matrix functions, see Higham (2008).
If estimates of the condition number of the matrix function are required then nag_matop_real_gen_matrix_cond_num (f01jbc) should be used.
nag_matop_complex_gen_matrix_fun_num (f01flc) can be used to find the matrix function fA for a complex matrix A.

10  Example

This example finds cos2A where
A= 3 0 1 2 -1 1 3 1 0 2 2 1 2 1 -1 1 .

10.1  Program Text

Program Text (f01elce.c)

10.2  Program Data

Program Data (f01elce.d)

10.3  Program Results

Program Results (f01elce.r)

nag_matop_real_gen_matrix_fun_num (f01elc) (PDF version)
f01 Chapter Contents
f01 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014