NAG FL Interface
c05rbf (sys_​deriv_​easy)

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

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

C++ Header Interface

#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 $\mathbf{iflag}=2$, 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) array Input/Output

On entry: if $\mathbf{iflag}=1$, fjac 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 $\mathit{j}=1,2,\dots ,n$, and 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 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 c05rbf. You should use the arrays iuser and ruser to supply information to fcn.

7: $\mathbf{iflag}$ – Integer Input/Output

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

$\mathbf{iflag}=1$

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.

fcn must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which c05rbf 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 c05rbf. If your code inadvertently does return any NaNs or infinities, c05rbf 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: $\mathbf{fjac}\left(\mathbf{n},\mathbf{n}\right)$ – 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: $\mathbf{iuser}\left(*\right)$ – Integer array User Workspace

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

iuser and ruser are not used by c05rbf, but are passed directly to fcn and may be used to pass information to this routine.

9: $\mathbf{ifail}$ – Integer Input/Output

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

If halting is not appropriate, the value $-1$ 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 $100\times \left(\mathbf{n}+1\right)$ calls to fcn. Consider restarting the calculation from the 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. 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 c05rbf from a different starting point may avoid the region of difficulty.

$\mathbf{ifail}=5$

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

$\mathbf{ifail}=-99$

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