NAG FL Interface
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 copy

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 copy

void  fcn (const Integer *n, const double x[], double fvec[], double fjac[], Integer iuser[], double ruser[], Integer *iflag)

C++ Header Interface copy

#include <nag.h> extern "C" { void  fcn (const Integer &n, const double x[], double fvec[], double fjac[], Integer iuser[], double ruser[], Integer &iflag) }

1: $\mathbf{n}$ – Integer Input

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

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

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) array Input/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 $f_i\left(x\right)$.

4: $\mathbf{fjac}(\mathbf{n},\mathbf{n})$ – Real (Kind=nag_wp) array Input/Output

On entry: if $\mathbf{iflag}=0$, $\mathbf{fjac}(\mathit{i},\mathit{j})$ 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, $\mathbf{fjac}(\mathit{i},\mathit{j})$ 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 array User Workspace

6: $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) array User 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: $\mathbf{iflag}$ – Integer Input/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: $\mathbf{n}$ – Integer Input

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

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

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

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

5: $\mathbf{fjac}(\mathbf{n},\mathbf{n})$ – Real (Kind=nag_wp) array Output

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: $\mathbf{maxfev}$ – Integer Input

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

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

8: $\mathbf{mode}$ – Integer Input

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) array Input/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 $\mathbf{factor}\times {\Vert \mathbf{diag}\times \mathbf{x}\Vert }_2$ if this is nonzero; otherwise the bound is factor.)

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

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

11: $\mathbf{nprint}$ – Integer Input

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: $\mathbf{nfev}$ – Integer Output

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

13: $\mathbf{njev}$ – Integer Output

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

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

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) array Output

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

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

17: $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) array User 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: $\mathbf{ifail}$ – Integer Input/Output

On entry: ifail must be set to $0$, $\mathrm{-1}$ 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 $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $\mathrm{-1}$ 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

$\mathbf{ifail}=2$

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 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