NAG CL Interface
f01elc (real_​gen_​matrix_​fun_​num)

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1 Purpose

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

2 Specification

#include <nag.h>
void  f01elc (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)
The function may be called by the names: f01elc or nag_matop_real_gen_matrix_fun_num.

3 Description

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 f is supplied via function f which evaluates f(zi) 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: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
2: a[dim] double Input/Output
Note: the dimension, dim, of the array a must be at least pda×n.
The (i,j)th element of the matrix A is stored in a[(j-1)×pda+i-1].
On entry: the n×n matrix A.
On exit: the n×n matrix, f(A).
3: pda Integer Input
On entry: the stride separating matrix row elements in the array a.
Constraint: pdan.
4: f function, supplied by the user External Function
The function f evaluates f(zi) 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: iflag Integer * 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 f(zi); for instance f(zi) may not be defined. If iflag is returned as nonzero then f01elc will terminate the computation, with fail.code= NE_INT, NE_INT_2 or NE_USER_STOP.
2: nz Integer Input
On entry: nz, the number of function values required.
3: z[dim] const Complex Input
On entry: the nz points z1,z2,,znz at which the function f is to be evaluated.
4: fz[dim] Complex Output
On exit: the nz function values. fz[i-1] should return the value f(zi), for i=1,2,,nz. If zi lies on the real line, then so must f(zi).
5: comm Nag_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 f01elc you may allocate memory and initialize these pointers with various quantities for use by f when called from f01elc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by f01elc. If your code inadvertently does return any NaNs or infinities, f01elc is likely to produce unexpected results.
5: comm Nag_Comm *
The NAG communication argument (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
6: iflag Integer * 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_INT, NE_INT_2 or NE_USER_STOP.
7: imnorm double * Output
On exit: if A has complex eigenvalues, f01elc will use complex arithmetic to compute f(A). 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, f01elc uses real arithmetic and imnorm=0.
8: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
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.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
Termination requested in f.

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 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 a discussion of numerical differentiation.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f01elc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library. In these implementations, this function may make calls to the user-supplied functions from within an OpenMP parallel region. Thus OpenMP pragmas within the user functions can only be used if you are compiling the user-supplied function and linking the executable in accordance with the instructions in the Users' Note for your implementation. 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.
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.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also 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 f01emc can be used to evaluate f(A) more accurately. If A is real symmetric then it is recommended that f01efc be used as it is more efficient and, in general, more accurate than f01elc.
For any z on the real line, f(z) must be real. f must also be complex analytic on the spectrum of A. These conditions ensure that f(A) 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 f01jbc should be used.
f01flc can be used to find the matrix function f(A) 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)