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NAG Library Manual

NAG Library Routine Document


Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

F01EFF computes the matrix function, fA, of a real symmetric n by n matrix A. fA must also be a real symmetric matrix.

2  Specification

REAL (KIND=nag_wp)  A(LDA,*), RUSER(*)

3  Description

fA is computed using a spectral factorization of A 
A = Q D QT ,
where D is the diagonal matrix whose diagonal elements, di, are the eigenvalues of A, and Q is an orthogonal matrix whose columns are the eigenvectors of A. fA is then given by
fA = Q fD QT ,
where fD is the diagonal matrix whose ith diagonal element is fdi. See for example Section 4.5 of Higham (2008). fdi is assumed to be real.

4  References

Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA

5  Parameters

1:     UPLO – CHARACTER(1)Input
On entry: if UPLO='U', the upper triangle of the matrix A is stored.
If UPLO='L', the lower triangle of the matrix A is stored.
Constraint: UPLO='U' or 'L'.
2:     N – INTEGERInput
On entry: n, the order of the matrix A.
Constraint: N0.
3:     A(LDA,*) – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array A must be at least N.
On entry: the n by n symmetric matrix A.
  • If UPLO='U', the upper triangular part of A must be stored and the elements of the array below the diagonal are not referenced.
  • If UPLO='L', the lower triangular part of A must be stored and the elements of the array above the diagonal are not referenced.
On exit: if IFAIL=0, the upper or lower triangular part of the n by n matrix function, fA.
4:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F01EFF is called.
Constraint: LDAmax1,N.
5:     F – SUBROUTINE, supplied by the user.External Procedure
The subroutine F evaluates fzi at a number of points zi.
The specification of F is:
REAL (KIND=nag_wp)  X(N), FX(N), RUSER(*)
1:     IFLAG – INTEGERInput/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 fx; for instance fx may not be defined, or may be complex. If IFLAG is returned as nonzero then F01EFF will terminate the computation, with IFAIL=-6.
2:     N – INTEGERInput
On entry: n, the number of function values required.
3:     X(N) – REAL (KIND=nag_wp) arrayInput
On entry: the n points x1,x2,,xn at which the function f is to be evaluated.
4:     FX(N) – REAL (KIND=nag_wp) arrayOutput
On exit: the n function values. FXi should return the value fxi, for i=1,2,,n.
5:     IUSER(*) – INTEGER arrayUser Workspace
6:     RUSER(*) – REAL (KIND=nag_wp) arrayUser Workspace
F is called with the parameters IUSER and RUSER as supplied to F01EFF. You are free to use the arrays IUSER and RUSER to supply information to F as an alternative to using COMMON global variables.
F must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which F01EFF is called. Parameters denoted as Input must not be changed by this procedure.
6:     IUSER(*) – INTEGER arrayUser Workspace
7:     RUSER(*) – REAL (KIND=nag_wp) arrayUser Workspace
IUSER and RUSER are not used by F01EFF, but are passed directly to F and may be used to pass information to this routine as an alternative to using COMMON global variables.
8:     IFLAG – INTEGEROutput
On exit: IFLAG=0, unless you have set IFLAG nonzero inside F, in which case IFLAG will be the value you set and IFAIL will be set to IFAIL=-6.
9:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to 0, -1​ or ​1. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ or ​1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is 0. When the value -1​ or ​1 is used it is essential to test the value of IFAIL on exit.
On exit: IFAIL=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

If on entry IFAIL=0 or -1, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
IFAIL<0 and IFAIL-999 or -6
If IFAIL=-i, the ith argument had an illegal value.
IFLAG has been set nonzero by the user.
Internal memory allocation failed.
The integer allocatable memory required is N, and the real allocatable memory required is approximately N+nb+4×N, where nb is the block size required by F08FAF (DSYEV).
The algorithm to compute the spectral factorization failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero (see F08FAF (DSYEV)).
Note:  this failure is unlikely to occur.

7  Accuracy

Provided that fD can be computed accurately then the computed matrix function will be close to the exact matrix function. See Section 10.2 of Higham (2008) for details and further discussion.

8  Further Comments

The cost of the algorithm is On3 plus the cost of evaluating fD. If λ^i is the ith computed eigenvalue of A, then the user-supplied subroutine F will be asked to evaluate the function f at fλ^i, i=1,2,,n.
For further information on matrix functions, see Higham (2008).
F01FFF can be used to find the matrix function fA for a complex Hermitian matrix A.

9  Example

This example finds the matrix cosine, cosA, of the symmetric matrix
A= 1 2 3 4 2 1 2 3 3 2 1 2 4 3 2 1 .

9.1  Program Text

Program Text (f01effe.f90)

9.2  Program Data

Program Data (f01effe.d)

9.3  Program Results

Program Results (f01effe.r)

F01EFF (PDF version)
F01 Chapter Contents
F01 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012