NAG Library Routine Document

c05rcf (sys_deriv_expert)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

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

Depending upon the value of iflag, fcn must either return the values of the functions $f_i$ at a point $x$ or return the Jacobian at $x$.

The specification of fcn is:

Fortran Interface

Subroutine fcn ( n, x, fvec, fjac, 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), fjac(n,n), ruser(*)

C Header Interface

#include <nagmk26.h> void  fcn (const Integer *n, const double x[], double fvec[], double fjac[], 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 or the Jacobian must be evaluated.

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

On entry: if $\mathbf{iflag}=0$ or $2$, fvec contains the function values $f_i\left(x\right)$ and must not be changed.

On exit: if $\mathbf{iflag}=1$ on entry, fvec must contain the function values $f_i\left(x\right)$ (unless iflag is set to a negative value by fcn).

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

On entry: if $\mathbf{iflag}=0$, $\mathbf{fjac}\left(\mathit{i},\mathit{j}\right)$ contains the value of $\frac{\partial f_{\mathit{i}}}{\partial x_{\mathit{j}}}$ at the point $x$, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,n$. When $\mathbf{iflag}=0$ or $1$, fjac must not be changed.

On exit: if $\mathbf{iflag}=2$ on entry, $\mathbf{fjac}\left(\mathit{i},\mathit{j}\right)$ must contain the value of $\frac{\partial f_{\mathit{i}}}{\partial x_{\mathit{j}}}$ at the point $x$, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,n$, (unless iflag is set to a negative value by fcn).

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

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

fcn is called with the arguments iuser and ruser as supplied to c05rcf. You should use the arrays iuser and ruser to supply information to fcn.

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

On entry: $\mathbf{iflag}=0$, $1$ or $2$.

$\mathbf{iflag}=0$

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

$\mathbf{iflag}=1$

fvec is to be updated.

$\mathbf{iflag}=2$

fjac is to 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 value.

fcn must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which c05rcf 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 c05rcf. If your code inadvertently does return any NaNs or infinities, c05rcf 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:     $\mathbf{fjac}\left(\mathbf{n},\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayOutput

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

6:     $\mathbf{xtol}$ – 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: $\mathbf{xtol}\ge 0.0$.

7:     $\mathbf{maxfev}$ – IntegerInput

On entry: the maximum number of calls to fcn with $\mathbf{iflag}\ne 0$. c05rcf will exit with $\mathbf{ifail}=\mathbf{2}$, if, at the end of an iteration, the number of calls to fcn exceeds maxfev.

Suggested value: $\mathbf{maxfev}=100\times \left(\mathbf{n}+1\right)$.

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

8:     $\mathbf{mode}$ – 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 $2$.

9:     $\mathbf{diag}\left(\mathbf{n}\right)$ – 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 $\mathbf{mode}=1$).

10:   $\mathbf{factor}$ – 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: $\mathbf{factor}>0.0$.

11:   $\mathbf{nprint}$ – 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 c05rcf.

12:   $\mathbf{nfev}$ – IntegerOutput

On exit: the number of calls made to fcn to evaluate the functions.

13:   $\mathbf{njev}$ – IntegerOutput

On exit: the number of calls made to fcn to evaluate the Jacobian.

14:   $\mathbf{r}\left(\mathbf{n}\times \left(\mathbf{n}+1\right)/2\right)$ – 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.

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

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

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

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

iuser and ruser are not used by c05rcf, but are passed directly to fcn and may be used to pass information to this routine.

18:   $\mathbf{ifail}$ – 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. This failure exit may indicate that the system does not have a zero, or that the solution is very close to the origin (see Section 7). Otherwise, rerunning c05rcf from a different starting point may avoid the region of difficulty.

$\mathbf{ifail}=5$

The iteration is not making good progress, as measured by the improvement from the last $〈\mathit{\text{value}}〉$ iterations. This failure exit may indicate that the system does not have a zero, or that the solution is very close to the origin (see Section 7). Otherwise, rerunning c05rcf from a different starting point may avoid the region of difficulty.

$\mathbf{ifail}=6$

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.

$\mathbf{ifail}=18$

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 (c05rcfe.f90)

10.2

Program Data

10.3

Program Results

Program Results (c05rcfe.r)