NAG FL Interface
e04hcf (check_deriv)
1
Purpose
e04hcf checks that a subroutine for evaluating an objective function and its first derivatives produces derivative values which are consistent with the function values calculated.
2
Specification
Fortran Interface
Integer, Intent (In) 
:: 
n, liw, lw 
Integer, Intent (Inout) 
:: 
iw(liw), ifail 
Real (Kind=nag_wp), Intent (In) 
:: 
x(n) 
Real (Kind=nag_wp), Intent (Inout) 
:: 
w(lw) 
Real (Kind=nag_wp), Intent (Out) 
:: 
f, g(n) 
External 
:: 
funct 

C Header Interface
#include <nag.h>
void 
e04hcf_ (const Integer *n, void (NAG_CALL *funct)(Integer *iflag, const Integer *n, const double xc[], double *fc, double gc[], Integer iw[], const Integer *liw, double w[], const Integer *lw), const double x[], double *f, double g[], Integer iw[], const Integer *liw, double w[], const Integer *lw, Integer *ifail) 

C++ Header Interface
#include <nag.h> extern "C" {
void 
e04hcf_ (const Integer &n, void (NAG_CALL *funct)(Integer &iflag, const Integer &n, const double xc[], double &fc, double gc[], Integer iw[], const Integer &liw, double w[], const Integer &lw), const double x[], double &f, double g[], Integer iw[], const Integer &liw, double w[], const Integer &lw, Integer &ifail) 
}

The routine may be called by the names e04hcf or nagf_opt_check_deriv.
3
Description
Routines for minimizing a function of several variables may require you to supply a subroutine to evaluate the objective function
$F\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)$ and its first derivatives.
e04hcf is designed to check the derivatives calculated by such usersupplied subroutines. As well as the routine to be checked (
funct), you must supply a point
$x={\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)}^{\mathrm{T}}$ at which the check will be made. Note that
e04hcf checks routines of the form required for
e04kdf and
e04lbf.
e04hcf first calls
funct to evaluate
$F$ and its first derivatives
${g}_{\mathit{j}}=\frac{\partial F}{\partial {x}_{\mathit{j}}}$, for
$\mathit{j}=1,2,\dots ,n$ at
$x$. The components of the usersupplied derivatives along two orthogonal directions (defined by unit vectors
${p}_{1}$ and
${p}_{2}$, say) are then calculated; these will be
${g}^{\mathrm{T}}{p}_{1}$ and
${g}^{\mathrm{T}}{p}_{2}$ respectively. The same components are also estimated by finite differences, giving quantities
where
$h$ is a small positive scalar. If the relative difference between
${v}_{1}$ and
${g}^{\mathrm{T}}{p}_{1}$ or between
${v}_{2}$ and
${g}^{\mathrm{T}}{p}_{2}$ is judged too large, an error indicator is set.
4
References
None.
5
Arguments

1:
$\mathbf{n}$ – Integer
Input

On entry: the number $n$ of independent variables in the objective function.
Constraint:
${\mathbf{n}}\ge 1$.

2:
$\mathbf{funct}$ – Subroutine, supplied by the user.
External Procedure

funct must evaluate the function and its first derivatives at a given point. (The minimization routines mentioned in
Section 3 gives you the option of resetting arguments of
funct to cause the minimization process to terminate immediately.
e04hcf will also terminate immediately, without finishing the checking process, if the argument in question is reset.)
The specification of
funct is:
Fortran Interface
Integer, Intent (In) 
:: 
n, liw, lw 
Integer, Intent (Inout) 
:: 
iflag, iw(liw) 
Real (Kind=nag_wp), Intent (In) 
:: 
xc(n) 
Real (Kind=nag_wp), Intent (Inout) 
:: 
w(lw) 
Real (Kind=nag_wp), Intent (Out) 
:: 
fc, gc(n) 

C Header Interface
void 
funct_ (Integer *iflag, const Integer *n, const double xc[], double *fc, double gc[], Integer iw[], const Integer *liw, double w[], const Integer *lw) 

C++ Header Interface
#include <nag.h> extern "C" {
void 
funct_ (Integer &iflag, const Integer &n, const double xc[], double &fc, double gc[], Integer iw[], const Integer &liw, double w[], const Integer &lw) 
}


1:
$\mathbf{iflag}$ – Integer
Input/Output

On entry: will be set to $2$.
On exit: if you reset
iflag to a negative number in
funct and return control to
e04hcf,
e04hcf will terminate immediately with
ifail set to your setting of
iflag.

2:
$\mathbf{n}$ – Integer
Input

On entry: the number $n$ of variables.

3:
$\mathbf{xc}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) array
Input

On entry: the point $x$ at which $F$ and its derivatives are required.

4:
$\mathbf{fc}$ – Real (Kind=nag_wp)
Output

On exit: unless
funct resets
iflag,
fc must be set to the value of the function
$F$ at the current point
$x$.

5:
$\mathbf{gc}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) array
Output

On exit: unless
funct resets
iflag,
${\mathbf{gc}}\left(\mathit{j}\right)$ must be set to the value of the first derivative
$\frac{\partial F}{\partial {x}_{\mathit{j}}}$ at the point
$x$, for
$\mathit{j}=1,2,\dots ,n$.

6:
$\mathbf{iw}\left({\mathbf{liw}}\right)$ – Integer array
Workspace

7:
$\mathbf{liw}$ – Integer
Input

8:
$\mathbf{w}\left({\mathbf{lw}}\right)$ – Real (Kind=nag_wp) array
Workspace

9:
$\mathbf{lw}$ – Integer
Input

These arguments are present so that
funct will be of the form required by the minimization routines mentioned in
Section 3.
funct is called with
e04hcf's arguments
iw,
liw,
w,
lw as these arguments. If the advice given in the minimization routine documents is being followed, you will have no reason to examine or change any elements of
iw or
w. In any case,
funct must not change the first
$3\times {\mathbf{n}}$ elements of
w.
funct must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which
e04hcf is called. Arguments denoted as
Input must
not be changed by this procedure.
Note: funct should not return floatingpoint NaN (Not a Number) or infinity values, since these are not handled by
e04hcf. If your code inadvertently
does return any NaNs or infinities,
e04hcf is likely to produce unexpected results.

3:
$\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) array
Input

On entry:
${\mathbf{x}}\left(\mathit{j}\right)$, for
$\mathit{j}=1,2,\dots ,n$, must be set to the coordinates of a suitable point at which to check the derivatives calculated by
funct. ‘Obvious’ settings, such as
$0.0$ or
$1.0$, should not be used since, at such particular points, incorrect terms may take correct values (particularly zero), so that errors could go undetected. Similarly, it is preferable that no two elements of
x should be the same.

4:
$\mathbf{f}$ – Real (Kind=nag_wp)
Output

On exit: unless you set
iflag negative in the first call of
funct,
f contains the value of the objective function
$F\left(x\right)$ at the point given by you in
x.

5:
$\mathbf{g}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) array
Output

On exit: unless you set
iflag negative in the first call of
funct,
${\mathbf{g}}\left(\mathit{j}\right)$ contains the value of the derivative
$\frac{\partial F}{\partial {x}_{\mathit{j}}}$ at the point given in
x, as calculated by
funct, for
$\mathit{j}=1,2,\dots ,n$.

6:
$\mathbf{iw}\left({\mathbf{liw}}\right)$ – Integer array
Workspace

7:
$\mathbf{liw}$ – Integer
Input

This array is in the argument list so that it can be used by other library routines for passing integer quantities to
funct. It is not examined or changed by
e04hcf. Generally, you must provide an array
iw but are advised not to use it.
Constraint:
${\mathbf{liw}}\ge 1$.

8:
$\mathbf{w}\left({\mathbf{lw}}\right)$ – Real (Kind=nag_wp) array
Workspace

9:
$\mathbf{lw}$ – Integer
Input

On entry: the dimension of the array
w as declared in the (sub)program from which
e04hcf is called.
Constraint:
${\mathbf{lw}}\ge 3\times {\mathbf{n}}$.

10:
$\mathbf{ifail}$ – Integer
Input/Output

On entry:
ifail must be set to
$0$,
$1$ or
$1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value
$1$ or
$1$ is recommended. If message printing is undesirable, then the value
$1$ is recommended. Otherwise, the value
$1$ is recommended since useful values can be provided in some output arguments even when
${\mathbf{ifail}}\ne {\mathbf{0}}$ on exit.
When the value $\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit:
${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6
Error Indicators and Warnings
If on entry
${\mathbf{ifail}}=0$ or
$1$, explanatory error messages are output on the current error message unit (as defined by
x04aaf).
Errors or warnings detected by the routine:
Note: in some cases e04hcf may return useful information.
 ${\mathbf{ifail}}=1$

On entry, ${\mathbf{liw}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{liw}}\ge 1$.
On entry, ${\mathbf{lw}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{lw}}\ge 3\times {\mathbf{n}}$.
On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}\ge 1$.
 ${\mathbf{ifail}}=2$

Large errors were found in the derivatives of
$F$ computed by
funct.
 ${\mathbf{ifail}}<0$

User requested termination by setting
iflag negative in
funct.
 ${\mathbf{ifail}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 7 in the Introduction to the NAG Library FL Interface for further information.
 ${\mathbf{ifail}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library FL Interface for further information.
 ${\mathbf{ifail}}=999$
Dynamic memory allocation failed.
See
Section 9 in the Introduction to the NAG Library FL Interface for further information.
7
Accuracy
ifail is set to
$2$ if
for
$k=1$ or
$2$. (See
Section 3 for definitions of the quantities involved.) The scalar
$h$ is set equal to
$\sqrt{\epsilon}$, where
$\epsilon $ is the
machine precision as given by
x02ajf.
8
Parallelism and Performance
e04hcf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
funct is called
$3$ times.
Before using
e04hcf to check the calculation of first derivatives, you should be confident that
funct is calculating
$F$ correctly. The usual way of checking the calculation of the function is to compare values of
$F\left(x\right)$ calculated by
funct at nontrivial points
$x$ with values calculated independently. (‘Nontrivial’ means that, as when setting
$x$ before calling
e04hcf, coordinates such as
$0.0$ or
$1.0$ should be avoided.)
e04hcf only checks the derivatives calculated when
${\mathbf{iflag}}=2$. So, if
funct is intended for use in conjunction with a minimization routine which may set
iflag to
$1$, you must check that, for given settings of the
${\mathbf{xc}}\left(j\right)$,
funct produces the same values for the
${\mathbf{gc}}\left(j\right)$ when
iflag is set to
$1$ as when
iflag is set to
$2$.
10
Example
Suppose that it is intended to use
e04kdf to minimize
The following program could be used to check the first derivatives calculated by
funct. (The tests of whether
${\mathbf{iflag}}=0$ or
$1$ in
funct are present ready for when
funct is called by
e04kdf.
e04hcf will always call
funct with
iflag set to 2.)
10.1
Program Text
10.2
Program Data
None.
10.3
Program Results