NAG Library Routine Document

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 <nagmk26.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)

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: $mode$ – 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), 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 argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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, $mode = 〈value〉$ .
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 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation 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.

8

Parallelism and Performance

c05zdf is not threaded in any implementation.

9

Further Comments

The time required by c05zdf increases with m and n.

10

Example

This example checks the Jacobian matrix for a problem with

15

functions of

3

variables (sometimes referred to as the Bard problem).

10.1

Program Text

Program Text (c05zdfe.f90)

10.2

Program Data

None.

10.3

Program Results

Program Results (c05zdfe.r)

NAG Library Manual, Mark 26

NAG AD Library Manual, Mark 26

NAG C Library Manual, Mark 26

C05 (roots) Chapter Contents

C05 (roots) Chapter Introduction

NAG Library Routine Document

c05zdf (sys_deriv_check)

▸▿ 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