NAG Library Routine Document

e04kyf  (bounds_quasi_deriv_easy)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{n}$ – IntegerInput

On entry: the number $n$ of independent variables.

Constraint: $\mathbf{n}\ge 1$.

2:     $\mathbf{ibound}$ – IntegerInput

On entry: indicates whether the facility for dealing with bounds of special forms is to be used. It must be set to one of the following values:

$\mathbf{ibound}=0$

If you are supplying all the $l_j$ and $u_j$ individually.

$\mathbf{ibound}=1$

If there are no bounds on any $x_j$.

$\mathbf{ibound}=2$

If all the bounds are of the form $0\le x_j$.

$\mathbf{ibound}=3$

If $l_1=l_2=\cdots =l_n$ and $u_1=u_2=\cdots =u_n$.

Constraint: $0\le \mathbf{ibound}\le 3$.

3:     $\mathbf{funct2}$ – Subroutine, supplied by the user.External Procedure

You must supply funct2 to calculate the values of the function $F\left(x\right)$ and its first derivative $\frac{\partial F}{\partial x_j}$ at any point $x$. It should be tested separately before being used in conjunction with e04kyf (see the E04 Chapter Introduction).

The specification of funct2 is:

Fortran Interface

Subroutine funct2 ( n, xc, fc, gc, iuser, ruser) Integer, Intent (In) :: n Integer, Intent (Inout) :: iuser(*) Real (Kind=nag_wp), Intent (In) :: xc(n) Real (Kind=nag_wp), Intent (Inout) :: ruser(*) Real (Kind=nag_wp), Intent (Out) :: fc, gc(n)

C Header Interface

#include nagmk26.h void  funct2 ( const Integer *n, const double xc[], double *fc, double gc[], Integer iuser[], double ruser[])

1:     $\mathbf{n}$ – IntegerInput

On entry: the number $n$ of variables.

2:     $\mathbf{xc}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayInput

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

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

On exit: the value of the function $F$ at the current point $x$.

4:     $\mathbf{gc}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayOutput

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

5:     $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace

6:     $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace

funct2 is called with the arguments iuser and ruser as supplied to e04kyf. You should use the arrays iuser and ruser to supply information to funct2.

funct2 must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which e04kyf is called. Arguments denoted as Input must not be changed by this procedure.

Note: funct2 should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by e04kyf. If your code inadvertently does return any NaNs or infinities, e04kyf is likely to produce unexpected results.

4:     $\mathbf{bl}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayInput/Output

On entry: the lower bounds $l_j$.

If ibound is set to $0$, you must set $\mathbf{bl}\left(\mathit{j}\right)$ to $l_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,n$. (If a lower bound is not specified for a particular $x_{\mathit{j}}$, the corresponding $\mathbf{bl}\left(\mathit{j}\right)$ should be set to $-{10}^6$.)

If ibound is set to $3$, you must set $\mathbf{bl}\left(1\right)$ to $l_1$; e04kyf will then set the remaining elements of bl equal to $\mathbf{bl}\left(1\right)$.

On exit: the lower bounds actually used by e04kyf.

5:     $\mathbf{bu}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayInput/Output

On entry: the upper bounds $u_j$.

If ibound is set to $0$, you must set $\mathbf{bu}\left(\mathit{j}\right)$ to $u_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,n$. (If an upper bound is not specified for a particular $x_j$, the corresponding $\mathbf{bu}\left(j\right)$ should be set to ${10}^6$.)

If ibound is set to $3$, you must set $\mathbf{bu}\left(1\right)$ to $u_1$; e04kyf will then set the remaining elements of bu equal to $\mathbf{bu}\left(1\right)$.

On exit: the upper bounds actually used by e04kyf.

6:     $\mathbf{x}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayInput/Output

On entry: $\mathbf{x}\left(\mathit{j}\right)$ must be set to a guess at the $\mathit{j}$th component of the position of the minimum, for $\mathit{j}=1,2,\dots ,n$. The routine checks the gradient at the starting point, and is more likely to detect any error in your programming if the initial $\mathbf{x}\left(j\right)$ are nonzero and mutually distinct.

On exit: the lowest point found during the calculations. Thus, if $\mathbf{ifail}=\mathbf{0}$ on exit, $\mathbf{x}\left(j\right)$ is the $j$th component of the position of the minimum.

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

On exit: the value of $F\left(x\right)$ corresponding to the final point stored in x.

8:     $\mathbf{g}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: the value of $\frac{\partial F}{\partial x_{\mathit{j}}}$ corresponding to the final point stored in x, for $\mathit{j}=1,2,\dots ,n$; the value of $\mathbf{g}\left(j\right)$ for variables not on a bound should normally be close to zero.

9:     $\mathbf{iw}\left(\mathbf{liw}\right)$ – Integer arrayOutput

On exit: if $\mathbf{ifail}=\mathbf{0}$, $\mathbf{3}$ or $\mathbf{5}$, the first n elements of iw contain information about which variables are currently on their bounds and which are free. Specifically, if $x_i$ is:

–	fixed on its upper bound, $\mathbf{iw}\left(i\right)$ is $-1$;
–	fixed on its lower bound, $\mathbf{iw}\left(i\right)$ is $-2$;
–	effectively a constant (i.e., $l_j=u_j$), $\mathbf{iw}\left(i\right)$ is $-3$;
–	free, $\mathbf{iw}\left(i\right)$ gives its position in the sequence of free variables.

In addition, $\mathbf{iw}\left(\mathbf{n}+1\right)$ contains the number of free variables (i.e., $n_z$). The rest of the array is used as workspace.

10:   $\mathbf{liw}$ – IntegerInput

On entry: the dimension of the array iw as declared in the (sub)program from which e04kyf is called.

Constraint: $\mathbf{liw}\ge \mathbf{n}+2$.

11:   $\mathbf{w}\left(\mathbf{lw}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: if $\mathbf{ifail}=\mathbf{0}$, $\mathbf{3}$ or $\mathbf{5}$, $\mathbf{w}\left(\mathit{i}\right)$ contains the $\mathit{i}$th element of the projected gradient vector $g_z$, for $\mathit{i}=1,2,\dots ,\mathbf{n}$. In addition, $\mathbf{w}\left(\mathbf{n}+1\right)$ contains an estimate of the condition number of the projected Hessian matrix (i.e., $k$). The rest of the array is used as workspace.

12:   $\mathbf{lw}$ – IntegerInput

On entry: the dimension of the array w as declared in the (sub)program from which e04kyf is called.

Constraint: $\mathbf{lw}\ge \max \left(10\times \mathbf{n}+\mathbf{n}\times \left(\mathbf{n}-1\right)/2,11\right)$.

13:   $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace

14:   $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace

iuser and ruser are not used by e04kyf, but are passed directly to funct2 and may be used to pass information to this routine.

15:   $\mathbf{ifail}$ – IntegerInput/Output

On entry: ifail must be set to $0$, $-1\text{ or }1$. 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 $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, because for this routine the values of the output arguments may be useful even if $\mathbf{ifail}\ne \mathbf{0}$ on exit, the recommended value is $-1$. When the value $-\mathbf{1}\text{ or }1$ 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

$\mathbf{ifail}=1$

On entry,	$\mathbf{n}<1$,
or	$\mathbf{ibound}<0$,
or	$\mathbf{ibound}>3$,
or	$\mathbf{ibound}=0$ and $\mathbf{bl}\left(j\right)>\mathbf{bu}\left(j\right)$ for some $j$,
or	$\mathbf{ibound}=3$ and $\mathbf{bl}\left(1\right)>\mathbf{bu}\left(1\right)$,
or	$\mathbf{liw}<\mathbf{n}+2$,
or	$\mathbf{lw}<\max \left(11,10\times \mathbf{n}+\mathbf{n}\times \left(\mathbf{n}-1\right)/2\right)$.

$\mathbf{ifail}=2$

There have been $100\times n$ function evaluations, yet the algorithm does not seem to be converging. The calculations can be restarted from the final point held in x. The error may also indicate that $F\left(x\right)$ has no minimum.

$\mathbf{ifail}=3$

The conditions for a minimum have not all been met but a lower point could not be found and the algorithm has failed.

$\mathbf{ifail}=4$

An overflow has occurred during the computation. This is an unlikely failure, but if it occurs you should restart at the latest point given in x.

$\mathbf{ifail}=5$

$\mathbf{ifail}=6$

$\mathbf{ifail}=7$

$\mathbf{ifail}=8$

There is some doubt about whether the point $x$ found by e04kyf is a minimum. The degree of confidence in the result decreases as ifail increases. Thus, when $\mathbf{ifail}=\mathbf{5}$ it is probable that the final $x$ gives a good estimate of the position of a minimum, but when $\mathbf{ifail}=\mathbf{8}$ it is very unlikely that the routine has found a minimum.

$\mathbf{ifail}=9$

In the search for a minimum, the modulus of one of the variables has become very large $\left(\sim {10}^6\right)$. This indicates that there is a mistake in funct2, that your problem has no finite solution, or that the problem needs rescaling (see Section 9).

$\mathbf{ifail}=10$

It is very likely that you have made an error in forming the gradient.

$\mathbf{ifail}=-99$

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.

$\mathbf{ifail}=-399$

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.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (e04kyfe.f90)

10.2

Program Data

10.3

Program Results

Program Results (e04kyfe.r)