c05zdc checks the user-supplied gradients of a set of nonlinear functions in several variables, for consistency with the functions themselves. The function must be called twice.

2 Specification

#include <nag.h>

void	c05zdc (Integer mode, Integer m, Integer n, const double x[], const double fvec[], const double fjac[], double xp[], const double fvecp[], double err[], NagError *fail)

The function may be called by the names: c05zdc, nag_roots_sys_deriv_check or nag_check_derivs.

3 Description

c05zdc 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 c05zdc 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 c05zdc.

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 c05rbc, c05rcc and c05rdc, $m = n$ .

Constraint: $n \geq 1$ .
4: $x [n]$ – const double Input: On entry: the components of a point $x$ , at which the consistency check is to be made. (See Section 7.)
5: $fvec [m]$ – const double 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: $fjac [m \times n]$ – const double Input: Note: the $(i, j)$ th element of the matrix is stored in $fjac [(j - 1) \times m + i - 1]$ .

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: $xp [n]$ – double Output: On exit: if $mode = 1$ , xp is set to a point neighbouring x. If $mode = 2$ , xp is undefined.
8: $fvecp [m]$ – const double Input: On entry: if $mode = 2$ , fvecp must contain the value of the functions evaluated at xp (as output by a preceding call to c05zdc with $mode = 1$ ). If $mode = 1$ , fvecp is not referenced.
9: $err [m]$ – double 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 $err [i - 1]$ 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 $err [i - 1]$ is $0.0$ the $i$ th gradient is incorrect. For values of $err [i - 1]$ between $0.0$ and $1.0$ the categorisation is less certain. In general, a value of $err [i - 1] > 0.5$ indicates that the $i$ th gradient is probably correct.
10: $fail$ – NagError * Input/Output: The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_INT: On entry, $m = ⟨ value ⟩$ .
Constraint: $m \geq 1$ .

On entry, $mode = ⟨ value ⟩$ .
Constraint: $mode = 1$ or $2$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 1$ .
NE_INTERNAL_ERROR: 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.
NE_NO_LICENCE: 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.

7 Accuracy

c05zdc 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 X02AJC.