C05ZDF (PDF version)
C05 Chapter Contents
C05 Chapter Introduction
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

C05ZDF checks the user-supplied gradients of a set of nonlinear functions in several variables, for consistency with the functions themselves. The routine must be called twice.

2  Specification

REAL (KIND=nag_wp)  X(N), FVEC(M), FJAC(M,N), XP(N), FVECP(M), ERR(M)

3  Description

C05ZDF is based on the MINPACK routine CHKDER (see Moré et al. (1980)). It checks the ith gradient for consistency with the ith function by computing a forward-difference approximation along a suitably chosen direction and comparing this approximation with the user-supplied gradient along the same direction. The principal characteristic of C05ZDF is its invariance under changes in scale of the variables or functions.

4  References

Moré J J, Garbow B S and Hillstrom K E (1980) User guide for MINPACK-1 Technical Report ANL-80-74 Argonne National Laboratory

5  Parameters

1:     MODE – INTEGERInput
On entry: the value 1 on the first call and the value 2 on the second call of C05ZDF.
Constraint: MODE=1 or 2.
2:     M – INTEGERInput
On entry: m, the number of functions.
Constraint: M1.
3:     N – INTEGERInput
On entry: n, the number of variables. For use with C05RBF, C05RCF and C05RDF, M=N.
Constraint: N1.
4:     X(N) – REAL (KIND=nag_wp) arrayInput
On entry: the components of a point x, at which the consistency check is to be made. (See Section 7.)
5:     FVEC(M) – REAL (KIND=nag_wp) arrayInput
On entry: if MODE=2, FVEC must contain the value of the functions evaluated at x. If MODE=1, FVEC is not referenced.
6:     FJAC(M,N) – REAL (KIND=nag_wp) arrayInput
On entry: if MODE=2, FJAC must contain the value of fi xj  at the point x, for i=1,2,,m and j=1,2,,n. If MODE=1, FJAC is not referenced.
7:     XP(N) – REAL (KIND=nag_wp) arrayOutput
On exit: if MODE=1, XP is set to a point neighbouring X. If MODE=2, XP is undefined.
8:     FVECP(M) – REAL (KIND=nag_wp) arrayInput
On entry: if MODE=2, FVECP must contain the value of the functions evaluated at XP (as output by a preceding call to C05ZDF with MODE=1). If MODE=1, FVECP is not referenced.
9:     ERR(M) – REAL (KIND=nag_wp) arrayOutput
On exit: if MODE=2, ERR contains measures of correctness of the respective gradients. If MODE=1, ERR is undefined. If there is no loss of significance (see Section 7), then if ERRi is 1.0 the ith user-supplied gradient fi xj , for j=1,2,,n is correct, whilst if ERRi is 0.0 the ith gradient is incorrect. For values of ERRi between 0.0 and 1.0 the categorisation is less certain. In general, a value of ERRi>0.5 indicates that the ith gradient is probably correct.
10:   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:
On entry, MODE1 or 2.
On entry, M<1.
On entry, N<1.

7  Accuracy

C05ZDF does not perform reliably if cancellation or rounding errors cause a severe loss of significance in the evaluation of a function. Therefore, none of the components of x should be unusually small (in particular, zero) or any other value which may cause loss of significance. The relative differences between corresponding elements of FVECP and FVEC should be at least two orders of magnitude greater than the machine precision returned by X02AJF.

8  Further Comments

The time required by C05ZDF increases with M and N.

9  Example

This example checks the Jacobian matrix for a problem with 15 functions of 3 variables (sometimes referred to as the Bard problem).

9.1  Program Text

Program Text (c05zdfe.f90)

9.2  Program Data


9.3  Program Results

Program Results (c05zdfe.r)

C05ZDF (PDF version)
C05 Chapter Contents
C05 Chapter Introduction
NAG Library Manual

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