NAG FL Interface
e04cbf (uncon_​simplex)

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

1: $\mathbf{n}$ – Integer Input

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

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

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

On entry: a guess at the position of the minimum. Note that the problem should be scaled so that the values of the $\mathbf{x}\left(i\right)$ are of order unity.

On exit: the value of $\mathbf{x}$ corresponding to the function value in f.

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

On exit: the lowest function value found.

4: $\mathbf{tolf}$ – Real (Kind=nag_wp) Input

On entry: the error tolerable in the function values, in the following sense. If $f_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n+1$, are the individual function values at the vertices of the current simplex, and if $f_m$ is the mean of these values, then you can request that e04cbf should terminate if

$$\sqrt{\frac{1}{n+1}\sum _{i=1}^{n+1}{(f_i-f_m)}^2}<\mathbf{tolf}\text{.}$$

(1)

You may specify $\mathbf{tolf}=0$ if you wish to use only the termination criterion (2) on the spatial values: see the description of tolx.

Constraint: $\mathbf{tolf}$ must be greater than or equal to the machine precision (see Chapter X02), or if tolf equals zero, then tolx must be greater than or equal to the machine precision.

5: $\mathbf{tolx}$ – Real (Kind=nag_wp) Input

On entry: the error tolerable in the spatial values, in the following sense. If $LV$ denotes the ‘linearized’ volume of the current simplex, and if ${LV}_{\mathrm{init}}$ denotes the ‘linearized’ volume of the initial simplex, then you can request that e04cbf should terminate if

$$\frac{LV}{{LV}_{\mathrm{init}}}<\mathbf{tolx}\text{.}$$

(2)

You may specify $\mathbf{tolx}=0$ if you wish to use only the termination criterion (1) on function values: see the description of tolf.

Constraint: $\mathbf{tolx}$ must be greater than or equal to the machine precision (see Chapter X02), or if tolx equals zero, then tolf must be greater than or equal to the machine precision.

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

funct must evaluate the function $F$ at a specified point. It should be tested separately before being used in conjunction with e04cbf.

The specification of funct is:

Fortran Interface copy

Subroutine funct ( n, xc, fc, 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

C Header Interface copy

void  funct (const Integer *n, const double xc[], double *fc, Integer iuser[], double ruser[])

C++ Header Interface copy

#include <nag.h> extern "C" { void  funct (const Integer &n, const double xc[], double &fc, Integer iuser[], double ruser[]) }

1: $\mathbf{n}$ – Integer Input

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

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

On entry: the point at which the function value is required.

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

On exit: the value of the function $F$ at the current point $\mathbf{x}$.

4: $\mathbf{iuser}\left(*\right)$ – Integer array User Workspace

5: $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) array User Workspace

funct is called with the arguments iuser and ruser as supplied to e04cbf. You should use the arrays iuser and ruser to supply information to funct.

funct must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which e04cbf is called. Arguments denoted as Input must not be changed by this procedure.

Note: funct should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by e04cbf. If your code inadvertently does return any NaNs or infinities, e04cbf is likely to produce unexpected results.

7: $\mathbf{monit}$ – Subroutine, supplied by the NAG Library or the user. External Procedure

monit may be used to monitor the optimization process. It is invoked once every iteration.

If no monitoring is required, monit may be the dummy monitoring routine e04cbk supplied by the NAG Library.

The specification of monit is:

Fortran Interface copy

Subroutine monit ( fmin, fmax, sim, n, ncall, serror, vratio, iuser, ruser) Integer, Intent (In) :: n, ncall Integer, Intent (Inout) :: iuser(*) Real (Kind=nag_wp), Intent (In) :: fmin, fmax, sim(n+1,n), serror, vratio Real (Kind=nag_wp), Intent (Inout) :: ruser(*)

C Header Interface copy

void  monit (const double *fmin, const double *fmax, const double sim[], const Integer *n, const Integer *ncall, const double *serror, const double *vratio, Integer iuser[], double ruser[])

C++ Header Interface copy

#include <nag.h> extern "C" { void  monit (const double &fmin, const double &fmax, const double sim[], const Integer &n, const Integer &ncall, const double &serror, const double &vratio, Integer iuser[], double ruser[]) }

1: $\mathbf{fmin}$ – Real (Kind=nag_wp) Input

On entry: the smallest function value in the current simplex.

2: $\mathbf{fmax}$ – Real (Kind=nag_wp) Input

On entry: the largest function value in the current simplex.

3: $\mathbf{sim}(\mathbf{n}+1,\mathbf{n})$ – Real (Kind=nag_wp) array Input

On entry: the $n+1$ position vectors of the current simplex.

4: $\mathbf{n}$ – Integer Input

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

5: $\mathbf{ncall}$ – Integer Input

On entry: the number of times that funct has been called so far.

6: $\mathbf{serror}$ – Real (Kind=nag_wp) Input

On entry: the current value of the standard deviation in function values used in termination test (1).

7: $\mathbf{vratio}$ – Real (Kind=nag_wp) Input

On entry: the current value of the linearized volume ratio used in termination test (2).

8: $\mathbf{iuser}\left(*\right)$ – Integer array User Workspace

9: $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) array User Workspace

monit is called with the arguments iuser and ruser as supplied to e04cbf. You should use the arrays iuser and ruser to supply information to monit.

monit must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which e04cbf is called. Arguments denoted as Input must not be changed by this procedure.

8: $\mathbf{maxcal}$ – Integer Input

On entry: the maximum number of function evaluations to be allowed.

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

9: $\mathbf{iuser}\left(*\right)$ – Integer array User Workspace

10: $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) array User Workspace

iuser and ruser are not used by e04cbf, but are passed directly to funct and monit and may be used to pass information to these routines.

11: $\mathbf{ifail}$ – 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 $0$ is recommended. 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

$\mathbf{ifail}=1$

On entry, $\mathbf{maxcal}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{maxcal}\ge 1$.

On entry, $\mathbf{n}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{n}\ge 1$.

On entry, $\mathbf{tolf}=0.0$ and $\mathbf{tolx}=⟨\mathit{\text{value}}⟩$.
Constraint: if $\mathbf{tolf}=0.0$ then $\mathbf{tolx}$ is greater than or equal to the machine precision.

On entry, $\mathbf{tolf}=⟨\mathit{\text{value}}⟩$ and $\mathbf{tolx}=⟨\mathit{\text{value}}⟩$.
Constraint: if $\mathbf{tolf}\ne 0.0$ and $\mathbf{tolx}\ne 0.0$ then both should be greater than or equal to the machine precision.

On entry, $\mathbf{tolx}=0.0$ and $\mathbf{tolf}=⟨\mathit{\text{value}}⟩$.
Constraint: if $\mathbf{tolx}=0.0$ then $\mathbf{tolf}$ is greater than or equal to the machine precision.

$\mathbf{ifail}=2$

maxcal function evaluations have been completed without any other termination test passing. Check the coding of funct before increasing the value of maxcal.

$\mathbf{ifail}=-99$

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

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results