NAG Library Routine Document

e04hyf (lsq_uncon_mod_deriv2_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:     $\mathbf{lsfun2}$ – Subroutine, supplied by the user.External Procedure

You must supply this routine to calculate the vector of values $f_i\left(x\right)$ and the Jacobian matrix of first derivatives $\frac{\partial f_i}{\partial x_j}$ at any point $x$. It should be tested separately before being used in conjunction with e04hyf (see the E04 Chapter Introduction).

The specification of lsfun2 is:

Fortran Interface

Subroutine lsfun2 ( m, n, xc, fvec, fjac, ldfjac, iuser, ruser) Integer, Intent (In) :: m, n, ldfjac Integer, Intent (Inout) :: iuser(*) Real (Kind=nag_wp), Intent (In) :: xc(n) Real (Kind=nag_wp), Intent (Inout) :: fjac(ldfjac,n), ruser(*) Real (Kind=nag_wp), Intent (Out) :: fvec(m)

C Header Interface

#include <nagmk26.h> void  lsfun2 (const Integer *m, const Integer *n, const double xc[], double fvec[], double fjac[], const Integer *ldfjac, Integer iuser[], double ruser[])

Important: the dimension declaration for fjac must contain the variable ldfjac, not an integer constant.

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 $f_i$ and the $\frac{\partial f_i}{\partial x_j}$ are required.

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

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

5:     $\mathbf{fjac}\left(\mathbf{ldfjac},\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: $\mathbf{fjac}\left(\mathit{i},\mathit{j}\right)$ must be set to the value of $\frac{\partial f_{\mathit{i}}}{\partial x_{\mathit{j}}}$ at the point $x$, for $\mathit{i}=1,2,\dots ,m$ and $\mathit{j}=1,2,\dots ,n$.

6:     $\mathbf{ldfjac}$ – IntegerInput

On entry: the first dimension of the array fjac as declared in the (sub)program from which e04hyf is called.

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

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

lsfun2 is called with the arguments iuser and ruser as supplied to e04hyf. You should use the arrays iuser and ruser to supply information to lsfun2.

lsfun2 must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which e04hyf is called. Arguments denoted as Input must not be changed by this procedure.

Note: lsfun2 should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by e04hyf. If your code inadvertently does return any NaNs or infinities, e04hyf is likely to produce unexpected results.

4:     $\mathbf{lshes2}$ – Subroutine, supplied by the user.External Procedure

You must supply this routine to calculate the elements of the symmetric matrix

$$B\left(x\right)=\sum _{i=1}^m{f}_i\left(x\right)G_i\left(x\right)\text{,}$$

at any point $x$, where $G_i\left(x\right)$ is the Hessian matrix of $f_i\left(x\right)$. It should be tested separately before being used in conjunction with e04hyf (see the E04 Chapter Introduction).

The specification of lshes2 is:

Fortran Interface

Subroutine lshes2 ( m, n, fvec, xc, b, lb, iuser, ruser) Integer, Intent (In) :: m, n, lb Integer, Intent (Inout) :: iuser(*) Real (Kind=nag_wp), Intent (In) :: fvec(m), xc(n) Real (Kind=nag_wp), Intent (Inout) :: ruser(*) Real (Kind=nag_wp), Intent (Out) :: b(lb)

C Header Interface

#include <nagmk26.h> void  lshes2 (const Integer *m, const Integer *n, const double fvec[], const double xc[], double b[], const Integer *lb, Integer iuser[], double ruser[])

1:     $\mathbf{m}$ – IntegerInput

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

2:     $\mathbf{n}$ – IntegerInput

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

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

On entry: the value of the residual $f_{\mathit{i}}$ at the point $x$, for $\mathit{i}=1,2,\dots ,m$, so that the values of the $f_{\mathit{i}}$ can be used in the calculation of the elements of b.

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

On entry: the point $x$ at which the elements of b are to be evaluated.

5:     $\mathbf{b}\left(\mathbf{lb}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: must contain the lower triangle of the matrix $B\left(x\right)$, evaluated at the point $x$, stored by rows. (The upper triangle is not required because the matrix is symmetric.) More precisely, $\mathbf{b}\left(\mathit{j}\left(\mathit{j}-1\right)/2+\mathit{k}\right)$ must contain ${\displaystyle \sum _{\mathit{i}=1}^m}{f}_{\mathit{i}}\frac{{\partial }^2{f}_{\mathit{i}}}{\partial x_{\mathit{j}}\partial x_{\mathit{k}}}$ evaluated at the point $x$, for $\mathit{j}=1,2,\dots ,n$ and $\mathit{k}=1,2,\dots ,\mathit{j}$.

6:     $\mathbf{lb}$ – IntegerInput

On entry: the length of the array b.

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

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

lshes2 is called with the arguments iuser and ruser as supplied to e04hyf. You should use the arrays iuser and ruser to supply information to lshes2.

lshes2 must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which e04hyf is called. Arguments denoted as Input must not be changed by this procedure.

Note: lshes2 should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by e04hyf. If your code inadvertently does return any NaNs or infinities, e04hyf is likely to produce unexpected results.

5:     $\mathbf{x}\left(\mathbf{n}\right)$ – 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 lsfun2 and lshes2 at the starting point and so is more likely to detect any error in your routines 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.

6:     $\mathbf{fsumsq}$ – 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.

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

8:     $\mathbf{lw}$ – IntegerInput

On entry: the dimension of the array w as declared in the (sub)program from which e04hyf is called.

Constraints:

• if $\mathbf{n}>1$, $\mathbf{lw}\ge 8\times \mathbf{n}+2\times \mathbf{n}\times \mathbf{n}+2\times \mathbf{m}\times \mathbf{n}+3\times \mathbf{m}$;
• if $\mathbf{n}=1$, $\mathbf{lw}\ge 11+5\times \mathbf{m}$.

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

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

iuser and ruser are not used by e04hyf, but are passed directly to lsfun2 and lshes2 and may be used to pass information to these routines.

11:   $\mathbf{ifail}$ – 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	$\mathbf{lw}<8\times \mathbf{n}+2\times \mathbf{n}\times \mathbf{n}+2\times \mathbf{m}\times \mathbf{n}+3\times \mathbf{m}$, when $\mathbf{n}>1$,
or	$\mathbf{lw}<11+5\times \mathbf{m}$, when $\mathbf{n}=1$.

$\mathbf{ifail}=2$

There have been $50\times n$ calls of lsfun2, 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 e04hyf 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 e04hyf 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.

$\mathbf{ifail}=9$

It is very likely that you have made an error in forming the derivatives $\frac{\partial f_i}{\partial x_j}$ in lsfun2.

$\mathbf{ifail}=10$

It is very likely that you have made an error in forming the quantities $B_{jk}$ in lshes2.

$\mathbf{ifail}=-99$

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 (e04hyfe.f90)

10.2

Program Data

Program Data (e04hyfe.d)

10.3

Program Results

Program Results (e04hyfe.r)