NAG Library Routine Document

e04fyf (lsq_uncon_mod_func_easy)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{m}$ – IntegerInput

2:     $\mathbf{n}$ – IntegerInput

On entry: the number $m$ of residuals, $f_i\left(x\right)$, and the number $n$ of variables, $x_j$.

Constraint: $1\le \mathbf{n}\le \mathbf{m}$.

3:     $\mathbf{lsfun1}$ – Subroutine, supplied by the user.External Procedure

You must supply this routine to calculate the vector of values $f_i\left(x\right)$ at any point $x$. It should be tested separately before being used in conjunction with e04fyf (see the E04 Chapter Introduction).

The specification of lsfun1 is:

Fortran Interface

Subroutine lsfun1 ( m, n, xc, fvec, iuser, ruser) Integer, Intent (In) :: m, 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) :: fvec(m)

C Header Interface

#include <nagmk26.h> void  lsfun1 (const Integer *m, const Integer *n, const double xc[], double fvec[], Integer iuser[], double ruser[])

1:     $\mathbf{m}$ – IntegerInput

On entry: $m$, the numbers of residuals.

2:     $\mathbf{n}$ – IntegerInput

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

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

On entry: the point $x$ at which the values of the $f_i$ are required.

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

On exit: $\mathbf{fvec}\left(\mathit{i}\right)$ must contain the value of $f_{\mathit{i}}$ at the point $x$, for $\mathit{i}=1,2,\dots ,m$.

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

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

lsfun1 is called with the arguments iuser and ruser as supplied to e04fyf. You should use the arrays iuser and ruser to supply information to lsfun1.

lsfun1 must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which e04fyf is called. Arguments denoted as Input must not be changed by this procedure.

Note: lsfun1 should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by e04fyf. If your code inadvertently does return any NaNs or infinities, e04fyf is likely to produce unexpected results.

4:     $\mathbf{x}\left(\mathbf{n}\right)$ – 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 $\mathit{j}=1,2,\dots ,n$.

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.

5:     $\mathbf{fsumsq}$ – Real (Kind=nag_wp)Output

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

6:     $\mathbf{w}\left(\mathbf{lw}\right)$ – Real (Kind=nag_wp) arrayCommunication Array

7:     $\mathbf{lw}$ – IntegerInput

On entry: the dimension of the array w as declared in the (sub)program from which e04fyf is called.

Constraints:

• if $\mathbf{n}>1$, $\mathbf{lw}\ge 7\times \mathbf{n}+\mathbf{n}\times \mathbf{n}+2\times \mathbf{m}\times \mathbf{n}+3\times \mathbf{m}+\mathbf{n}\times \left(\mathbf{n}-1\right)/2$;
• if $\mathbf{n}=1$, $\mathbf{lw}\ge 9+5\times \mathbf{m}$.

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

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

iuser and ruser are not used by e04fyf, but are passed directly to lsfun1 and may be used to pass information to this routine.

10:   $\mathbf{ifail}$ – 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	$\mathbf{m}<\mathbf{n}$,
or	$\mathbf{lw}<7\times \mathbf{n}+\mathbf{n}\times \mathbf{n}+2\times \mathbf{m}\times \mathbf{n}+3\times \mathbf{m}+\mathbf{n}\times \left(\mathbf{n}-1\right)/2$, when $\mathbf{n}>1$,
or	$\mathbf{lw}<9+5\times \mathbf{m}$, when $\mathbf{n}=1$.

$\mathbf{ifail}=2$

There have been $400\times n$ calls of lsfun1, yet the algorithm does not seem to have converged. This may be due to an awkward function or to a poor starting point, so it is worth restarting e04fyf from the final point held in x.

$\mathbf{ifail}=3$

The final point does not satisfy the conditions for acceptance as a minimum, but no lower point could be found.

$\mathbf{ifail}=4$

An auxiliary routine has been unable to complete a singular value decomposition in a reasonable number of sub-iterations.

$\mathbf{ifail}=5$

$\mathbf{ifail}=6$

$\mathbf{ifail}=7$

$\mathbf{ifail}=8$

There is some doubt about whether the point x$x$ found by e04fyf is a minimum of $F\left(x\right)$. 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.

$\mathbf{ifail}=-99$

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 (e04fyfe.f90)

10.2

Program Data

Program Data (e04fyfe.d)

10.3

Program Results

Program Results (e04fyfe.r)