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

Fortran Interface

Subroutine c05zdf (

mode, m, n, x, fvec, fjac, xp, fvecp, err, ifail)

Integer, Intent (In)	::	mode, m, n
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	x(n), fvec(m), fjac(m,n), fvecp(m)
Real (Kind=nag_wp), Intent (Out)	::	xp(n), err(m)

C Header Interface

#include <nag.h>

void	c05zdf_ (const Integer mode, const Integer m, const Integer n, const double x[], const double fvec[], const double fjac[], double xp[], const double fvecp[], double err[], Integer ifail)

The routine may be called by the names c05zdf or nagf_roots_sys_deriv_check.

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 Arguments

1: $mode$ – Integer Input: 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: $m$ – Integer Input: On entry: $m$ , the number of functions.

Constraint: $m \geq 1$ .
3: $n$ – Integer Input: 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) array Input: 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) array Input: 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) array Input: 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) array Output: 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) array Input: 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) array Output: 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), 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: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $- 1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $- 1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $- 1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. 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, $mode = ⟨ value ⟩$ .
Constraint: $mode = 1$ or $2$ .

$ifail = 2$: On entry, $m = ⟨ value ⟩$ .
Constraint: $m \geq 1$ .

$ifail = 3$: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 1$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

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.