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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_roots_sys_deriv_check (c05zd)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_roots_sys_deriv_check (c05zd) 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.


[xp, err, ifail] = c05zd(mode, x, fvec, fjac, fvecp, 'm', m, 'n', n)
[xp, err, ifail] = nag_roots_sys_deriv_check(mode, x, fvec, fjac, fvecp, 'm', m, 'n', n)


nag_roots_sys_deriv_check (c05zd) is based on the MINPACK routine CHKDER (see Moré et al. (1980)). It checks the ith gradient for consistency with the ith 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_roots_sys_deriv_check (c05zd) is its invariance under changes in scale of the variables or functions.


Moré J J, Garbow B S and Hillstrom K E (1980) User guide for MINPACK-1 Technical Report ANL-80-74 Argonne National Laboratory


Compulsory Input Parameters

1:     mode int64int32nag_int scalar
The value 1 on the first call and the value 2 on the second call of nag_roots_sys_deriv_check (c05zd).
Constraint: mode=1 or 2.
2:     xn – double array
The components of a point x, at which the consistency check is to be made. (See Accuracy.)
3:     fvecm – double array
If mode=2, fvec must contain the value of the functions evaluated at x. If mode=1, fvec is not referenced.
4:     fjacmn – double array
If mode=2, fjac must contain the value of fi xj  at the point x, for i=1,2,,m and j=1,2,,n. If mode=1, fjac is not referenced.
5:     fvecpm – double array
If mode=2, fvecp must contain the value of the functions evaluated at xp (as output by a preceding call to nag_roots_sys_deriv_check (c05zd) with mode=1). If mode=1, fvecp is not referenced.

Optional Input Parameters

1:     m int64int32nag_int scalar
Default: the dimension of the arrays fvec, fvecp and the first dimension of the array fjac. (An error is raised if these dimensions are not equal.)
m, the number of functions.
Constraint: m1.
2:     n int64int32nag_int scalar
Default: the dimension of the array x and the second dimension of the array fjac. (An error is raised if these dimensions are not equal.)
Constraint: n1.

Output Parameters

1:     xpn – double array
If mode=1, xp is set to a point neighbouring x. If mode=2, xp is undefined.
2:     errm – double array
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 Accuracy), then if erri is 1.0 the ith user-supplied gradient fi xj , for j=1,2,,n is correct, whilst if erri is 0.0 the ith gradient is incorrect. For values of erri between 0.0 and 1.0 the categorisation is less certain. In general, a value of erri>0.5 indicates that the ith gradient is probably correct.
3:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
Constraint: mode=1 or 2.
Constraint: m1.
Constraint: n1.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


nag_roots_sys_deriv_check (c05zd) 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 (x02aj).

Further Comments

The time required by nag_roots_sys_deriv_check (c05zd) increases with m and n.


This example checks the Jacobian matrix for a problem with 15 functions of 3 variables (sometimes referred to as the Bard problem).
function c05zd_example

fprintf('c05zd example results\n\n');

% Point at which to check gradients:
x = [0.92, 0.13, 0.54];

fvec  = zeros(15, 1);
fjac  = zeros(15, 3);
fvecp = zeros(15, 1);

y = 0.01*[14, 18, 22, 25, 29, 32, 35, 39, 47, 58, 73, 96, 134, 210, 439];

[xp, err, ifail] = c05zd(int64(1), x, fvec, fjac, fvecp);

for i=1:15
  u = i;
  v = 16 - i;
  w = min(u, v);
  fvec(i)  = y(i) - (x(1)+u/(v*x(2)+w*x(3)));
  fvecp(i) = y(i) - (xp(1)+u/(v*xp(2)+w*xp(3)));
  denom = (v*x(2)+w*x(3))^(-2);
  fjac(i,:) = [-1, u*v*denom, u*w*denom];

[xp, err, ifail] = c05zd(int64(2), x, fvec, fjac, fvecp);

fprintf('\nAt point %12.4f %12.4f %12.4f\n', x);
if any(err <= 0.5)
  for i=1:15
    if err(i) <= 0.5
      fprintf('Suspicious gradient number %d with error measure %12.4f\n', ...
              i, err(i));
  fprintf('Gradients appear correct\n');

c05zd example results

At point       0.9200       0.1300       0.5400
Gradients appear correct

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