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# NAG Toolbox: nag_roots_sys_deriv_check (c05zd)

## Purpose

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.

## Syntax

[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)

## Description

nag_roots_sys_deriv_check (c05zd) 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_roots_sys_deriv_check (c05zd) is its invariance under changes in scale of the variables or functions.

## 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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{mode}$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: ${\mathbf{mode}}=1$ or $2$.
2:     $\mathrm{x}\left({\mathbf{n}}\right)$ – double array
The components of a point $x$, at which the consistency check is to be made. (See Accuracy.)
3:     $\mathrm{fvec}\left({\mathbf{m}}\right)$ – double array
If ${\mathbf{mode}}=2$, fvec must contain the value of the functions evaluated at $x$. If ${\mathbf{mode}}=1$, fvec is not referenced.
4:     $\mathrm{fjac}\left({\mathbf{m}},{\mathbf{n}}\right)$ – double array
If ${\mathbf{mode}}=2$, fjac must contain the value of $\frac{\partial {f}_{i}}{\partial {x}_{j}}$ at the point $x$, for $\mathit{i}=1,2,\dots ,m$ and $\mathit{j}=1,2,\dots ,n$. If ${\mathbf{mode}}=1$, fjac is not referenced.
5:     $\mathrm{fvecp}\left({\mathbf{m}}\right)$ – double array
If ${\mathbf{mode}}=2$, 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 ${\mathbf{mode}}=1$). If ${\mathbf{mode}}=1$, fvecp is not referenced.

### Optional Input Parameters

1:     $\mathrm{m}$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: ${\mathbf{m}}\ge 1$.
2:     $\mathrm{n}$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.)
$n$, the number of variables. For use with nag_roots_sys_deriv_easy (c05rb), nag_roots_sys_deriv_expert (c05rc) and nag_roots_sys_deriv_rcomm (c05rd), ${\mathbf{m}}={\mathbf{n}}$.
Constraint: ${\mathbf{n}}\ge 1$.

### Output Parameters

1:     $\mathrm{xp}\left({\mathbf{n}}\right)$ – double array
If ${\mathbf{mode}}=1$, xp is set to a point neighbouring x. If ${\mathbf{mode}}=2$, xp is undefined.
2:     $\mathrm{err}\left({\mathbf{m}}\right)$ – double array
If ${\mathbf{mode}}=2$, err contains measures of correctness of the respective gradients. If ${\mathbf{mode}}=1$, err is undefined. If there is no loss of significance (see Accuracy), then if ${\mathbf{err}}\left(i\right)$ is $1.0$ the $i$th user-supplied gradient $\frac{\partial {f}_{i}}{\partial {x}_{j}}$, for $\mathit{j}=1,2,\dots ,n$ is correct, whilst if ${\mathbf{err}}\left(i\right)$ is $0.0$ the $i$th gradient is incorrect. For values of ${\mathbf{err}}\left(i\right)$ between $0.0$ and $1.0$ the categorisation is less certain. In general, a value of ${\mathbf{err}}\left(i\right)>0.5$ indicates that the $i$th gradient is probably correct.
3:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
Constraint: ${\mathbf{mode}}=1$ or $2$.
${\mathbf{ifail}}=2$
Constraint: ${\mathbf{m}}\ge 1$.
${\mathbf{ifail}}=3$
Constraint: ${\mathbf{n}}\ge 1$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

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).

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

## Example

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];
end

[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));
end
end
else
end

```
```c05zd example results

At point       0.9200       0.1300       0.5400
```

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