NAG Library Routine Document

c05qsf (sparsys_func_easy)

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 $x$.

The specification of fcn is:

Fortran Interface

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

C Header Interface

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

1:     $\mathbf{n}$ – IntegerInput

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

2:     $\mathbf{lindf}$ – IntegerInput

On entry: lindf specifies the number of indices $i$ for which values of $f_i\left(x\right)$ must be computed.

3:     $\mathbf{indf}\left(\mathbf{lindf}\right)$ – Integer arrayInput

On entry: indf specifies the indices $i$ for which values of $f_i\left(x\right)$ must be computed. The indices are specified in strictly ascending order.

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

On entry: the components of the point $x$ at which the functions must be evaluated. $\mathbf{x}\left(i\right)$ contains the coordinate $x_i$.

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

On exit: $\mathbf{fvec}\left(i\right)$ must contain the function values $f_i\left(x\right)$, for all indices $i$ in indf.

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

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

fcn is called with the arguments iuser and ruser as supplied to c05qsf. You should use the arrays iuser and ruser to supply information to fcn.

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

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

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 c05qsf 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 c05qsf. If your code inadvertently does return any NaNs or infinities, c05qsf 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. $\mathbf{x}\left(i\right)$ must contain the coordinate $x_i$.

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. $\mathbf{fvec}\left(i\right)$ contains the function values $f_i$.

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:     $\mathbf{init}$ – LogicalInput

On entry: init must be set to .TRUE. to indicate that this is the first time c05qsf is called for this specific problem. c05qsf then computes the dense Jacobian and detects and stores its sparsity pattern (in rcomm and icomm) before proceeding with the iterations. This is noticeably time consuming when n is large. If not enough storage has been provided for rcomm or icomm, c05qsf will fail. On exit with $\mathbf{ifail}=\mathbf{0}$, $\mathbf{2}$, $\mathbf{3}$ or $\mathbf{4}$, $\mathbf{icomm}\left(1\right)$ contains $\mathit{nnz}$, the number of nonzero entries found in the Jacobian. On subsequent calls, init can be set to .FALSE. if the problem has a Jacobian of the same sparsity pattern. In that case, the computation time required for the detection of the sparsity pattern will be smaller.

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

rcomm must not be altered between successive calls to c05qsf.

8:     $\mathbf{lrcomm}$ – IntegerInput

On entry: the dimension of the array rcomm as declared in the (sub)program from which c05qsf is called.

Constraint: $\mathbf{lrcomm}\ge 12+\mathit{nnz}$ where $\mathit{nnz}$ is the number of nonzero entries in the Jacobian, as computed by c05qsf.

9:     $\mathbf{icomm}\left(\mathbf{licomm}\right)$ – Integer arrayCommunication Array

If $\mathbf{ifail}=\mathbf{0}$, $\mathbf{2}$, $\mathbf{3}$ or $\mathbf{4}$ on exit, $\mathbf{icomm}\left(1\right)$ contains $\mathit{nnz}$ where $\mathit{nnz}$ is the number of nonzero entries in the Jacobian.

icomm must not be altered between successive calls to c05qsf.

10:   $\mathbf{licomm}$ – IntegerInput

On entry: the dimension of the array icomm as declared in the (sub)program from which c05qsf is called.

Constraint: $\mathbf{licomm}\ge 8\times \mathbf{n}+19+\mathit{nnz}$ where $\mathit{nnz}$ is the number of nonzero entries in the Jacobian, as computed by c05qsf.

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

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

iuser and ruser are not used by c05qsf, but are passed directly to fcn and may be used to pass information to this routine.

13:   $\mathbf{ifail}$ – 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 $200\times \left(\mathbf{n}+1\right)$ calls to fcn. Consider setting $\mathbf{init}=\mathrm{.FALSE.}$ and 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 c05qsf from a different starting point may avoid the region of difficulty. The condition number of the Jacobian is $〈\mathit{\text{value}}〉$.

$\mathbf{ifail}=5$

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

$\mathbf{ifail}=6$

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

$\mathbf{ifail}=7$

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

$\mathbf{ifail}=9$

An internal error has occurred. Code $=〈\mathit{\text{value}}〉$.

$\mathbf{ifail}=11$

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 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 (c05qsfe.f90)

10.2

Program Data

10.3

Program Results

Program Results (c05qsfe.r)