NAG Library Routine Document

C05ZDF

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

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

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

SUBROUTINE C05ZDF ( MODE, M, N, X, FVEC, FJAC, XP, FVECP, ERR, IFAIL)

INTEGER MODE, M, N, IFAIL

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

i

th gradient for consistency with the

i

th 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: $M \geq 1$ .
3: N – INTEGERInput: On entry: $n$ , the number of variables. For use with C05RBF, C05RCF and C05RDF, $M = N$ .
Constraint: $N \geq 1$ .
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 $\frac{\partial f_{i}}{\partial x_{j}}$ at the point $x$ , for $i = 1, 2, \dots, m$ and $j = 1, 2, \dots, n$ . 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 $ERR (i)$ is $1.0$ the $i$ th user-supplied gradient $\frac{\partial f_{i}}{\partial x_{j}}$ , for $j = 1, 2, \dots, n$ is correct, whilst if $ERR (i)$ is $0.0$ the $i$ th gradient is incorrect. For values of $ERR (i)$ between $0.0$ and $1.0$ the categorisation is less certain. In general, a value of $ERR (i) > 0.5$ indicates that the $i$ th 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

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$: On entry, $MODE \neq 1$ or $2$ .

$IFAIL = 2$: On entry, $M < 1$ .

$IFAIL = 3$: 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.