NAG Library Function Document

nag_check_derivs (c05zdc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_check_derivs (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>

#include <nagc05.h>

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

3 Description

nag_check_derivs (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 nag_check_derivs (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 – IntegerInput: On entry: the value $1$ on the first call and the value $2$ on the second call of nag_check_derivs (c05zdc).
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 nag_zero_nonlin_eqns_deriv_easy (c05rbc), nag_zero_nonlin_eqns_deriv_expert (c05rcc) and nag_zero_nonlin_eqns_deriv_rcomm (c05rdc), $m = n$ .
Constraint: $n \geq 1$ .
4: x[n] – const doubleInput: On entry: the components of a point $x$ , at which the consistency check is to be made. (See Section 7.)
5: fvec[m] – const doubleInput: 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 doubleInput: 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] – doubleOutput: On exit: if $mode = 1$ , xp is set to a point neighbouring x. If $mode = 2$ , xp is undefined.
8: fvecp[m] – const doubleInput: On entry: if $mode = 2$ , fvecp must contain the value of the functions evaluated at xp (as output by a preceding call to nag_check_derivs (c05zdc) with $mode = 1$ ). If $mode = 1$ , fvecp is not referenced.
9: err[m] – doubleOutput: 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 $err [i - 1]$ 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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

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.

7 Accuracy

nag_check_derivs (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 nag_machine_precision (X02AJC).