NAG Library Routine Document

c05qcf  (sys_func_expert)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

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

fcn must return the values of the functions $f_i$ at a point $x$, unless $\mathbf{iflag}=0$ on entry to c05qcf.

The specification of fcn is:

Fortran Interface

Subroutine fcn ( n, x, fvec, iuser, ruser, iflag) Integer, Intent (In) :: n Integer, Intent (Inout) :: iuser(*), iflag Real (Kind=nag_wp), Intent (In) :: x(n) Real (Kind=nag_wp), Intent (Inout) :: fvec(n), ruser(*)

C Header Interface

#include nagmk26.h void  fcn ( const Integer *n, const double x[], double fvec[], Integer iuser[], double ruser[], Integer *iflag)

1:     $\mathbf{n}$ – IntegerInput

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

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

On entry: the components of the point $x$ at which the functions must be evaluated.

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

On entry: if $\mathbf{iflag}=0$, fvec contains the function values $f_i\left(x\right)$ and must not be changed.

On exit: if $\mathbf{iflag}>0$ on entry, fvec must contain the function values $f_i\left(x\right)$ (unless iflag is set to a negative value by fcn).

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

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

fcn is called with the arguments iuser and ruser as supplied to c05qcf. You should use the arrays iuser and ruser to supply information to fcn.

6:     $\mathbf{iflag}$ – IntegerInput/Output

On entry: $\mathbf{iflag}\ge 0$.

$\mathbf{iflag}=0$

x and fvec are available for printing (see nprint).

$\mathbf{iflag}>0$

fvec must be updated.

On exit: in general, iflag should not be reset by fcn. If, however, you wish to terminate execution (perhaps because some illegal point x has been reached), iflag should be set to a negative integer.

fcn must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which c05qcf is called. Arguments denoted as Input must not be changed by this procedure.

Note: fcn should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by c05qcf. If your code inadvertently does return any NaNs or infinities, c05qcf is likely to produce unexpected results.

2:     $\mathbf{n}$ – IntegerInput

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

Constraint: $\mathbf{n}>0$.

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

On entry: an initial guess at the solution vector.

On exit: the final estimate of the solution vector.

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

On exit: the function values at the final point returned in x.

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

On entry: the accuracy in x to which the solution is required.

Suggested value: $\sqrt{\epsilon }$, where $\epsilon$ is the machine precision returned by x02ajf.

Constraint: $\mathbf{xtol}\ge 0.0$.

6:     $\mathbf{maxfev}$ – IntegerInput

On entry: the maximum number of calls to fcn with $\mathbf{iflag}\ne 0$. c05qcf will exit with $\mathbf{ifail}=\mathbf{2}$, if, at the end of an iteration, the number of calls to fcn exceeds maxfev.

Suggested value: $\mathbf{maxfev}=200\times \left(\mathbf{n}+1\right)$.

Constraint: $\mathbf{maxfev}>0$.

7:     $\mathbf{ml}$ – IntegerInput

On entry: the number of subdiagonals within the band of the Jacobian matrix. (If the Jacobian is not banded, or you are unsure, set $\mathbf{ml}=\mathbf{n}-1$.)

Constraint: $\mathbf{ml}\ge 0$.

8:     $\mathbf{mu}$ – IntegerInput

On entry: the number of superdiagonals within the band of the Jacobian matrix. (If the Jacobian is not banded, or you are unsure, set $\mathbf{mu}=\mathbf{n}-1$.)

Constraint: $\mathbf{mu}\ge 0$.

9:     $\mathbf{epsfcn}$ – Real (Kind=nag_wp)Input

On entry: a rough estimate of the largest relative error in the functions. It is used in determining a suitable step for a forward difference approximation to the Jacobian. If epsfcn is less than machine precision (returned by x02ajf) then machine precision is used. Consequently a value of $0.0$ will often be suitable.

Suggested value: $\mathbf{epsfcn}=0.0$.

10:   $\mathbf{mode}$ – IntegerInput

On entry: indicates whether or not you have provided scaling factors in diag.

If $\mathbf{mode}=2$, the scaling must have been specified in diag.

Otherwise, if $\mathbf{mode}=1$, the variables will be scaled internally.

Constraint: $\mathbf{mode}=1$ or $2$.

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

On entry: if $\mathbf{mode}=2$, diag must contain multiplicative scale factors for the variables.

If $\mathbf{mode}=1$, diag need not be set.

Constraint: if $\mathbf{mode}=2$, $\mathbf{diag}\left(\mathit{i}\right)>0.0$, for $\mathit{i}=1,2,\dots ,n$.

On exit: the scale factors actually used (computed internally if $\mathbf{mode}=1$).

12:   $\mathbf{factor}$ – Real (Kind=nag_wp)Input

On entry: a quantity to be used in determining the initial step bound. In most cases, factor should lie between $0.1$ and $100.0$. (The step bound is $\mathbf{factor}\times {‖\mathbf{diag}\times \mathbf{x}‖}_2$ if this is nonzero; otherwise the bound is factor.)

Suggested value: $\mathbf{factor}=100.0$.

Constraint: $\mathbf{factor}>0.0$.

13:   $\mathbf{nprint}$ – IntegerInput

On entry: indicates whether (and how often) special calls to fcn, with iflag set to $0$, are to be made for printing purposes.

$\mathbf{nprint}\le 0$

No calls are made.

$\mathbf{nprint}>0$

fcn is called at the beginning of the first iteration, every nprint iterations thereafter and immediately before the return from c05qcf.

14:   $\mathbf{nfev}$ – IntegerOutput

On exit: the number of calls made to fcn with $\mathbf{iflag}>0$.

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

On exit: the orthogonal matrix $Q$ produced by the $QR$ factorization of the final approximate Jacobian.

16:   $\mathbf{r}\left(\mathbf{n}\times \left(\mathbf{n}+1\right)/2\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: the upper triangular matrix $R$ produced by the $QR$ factorization of the final approximate Jacobian, stored row-wise.

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

On exit: the vector $Q^{\mathrm{T}}f$.

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

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

iuser and ruser are not used by c05qcf, but are passed directly to fcn and may be used to pass information to this routine.

20:   $\mathbf{ifail}$ – 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, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{ or }\mathbf{1}$ 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

$\mathbf{ifail}=2$

There have been at least maxfev calls to fcn: $\mathbf{maxfev}=〈\mathit{\text{value}}〉$. Consider restarting the calculation from the final point held in x.

$\mathbf{ifail}=3$

No further improvement in the solution is possible. xtol is too small: $\mathbf{xtol}=〈\mathit{\text{value}}〉$.

$\mathbf{ifail}=4$

The iteration is not making good progress, as measured by the improvement from the last $〈\mathit{\text{value}}〉$ Jacobian evaluations.

$\mathbf{ifail}=5$

The iteration is not making good progress, as measured by the improvement from the last $〈\mathit{\text{value}}〉$ iterations.

$\mathbf{ifail}=6$

iflag was set negative in fcn. $\mathbf{iflag}=〈\mathit{\text{value}}〉$.

$\mathbf{ifail}=11$

On entry, $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{n}>0$.

$\mathbf{ifail}=12$

On entry, $\mathbf{xtol}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{xtol}\ge 0.0$.

$\mathbf{ifail}=13$

On entry, $\mathbf{mode}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{mode}=1$ or $2$.

$\mathbf{ifail}=14$

On entry, $\mathbf{factor}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{factor}>0.0$.

$\mathbf{ifail}=15$

On entry, $\mathbf{mode}=2$ and diag contained a non-positive element.

$\mathbf{ifail}=16$

On entry, $\mathbf{ml}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{ml}\ge 0$.

$\mathbf{ifail}=17$

On entry, $\mathbf{mu}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{mu}\ge 0$.

$\mathbf{ifail}=18$

On entry, $\mathbf{maxfev}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{maxfev}>0$.

$\mathbf{ifail}=-99$

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 (c05qcfe.f90)

10.2

Program Data

10.3

Program Results

Program Results (c05qcfe.r)