NAG Library Routine Document

C05PCF/C05PCA

1  Purpose

2  Specification

2.1  Specification for C05PCF

2.2  Specification for C05PCA

3  Description

4  References

5  Parameters

1:     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 for C05PCF is:

SUBROUTINE FCN ( N, X, FVEC, FJAC, LDFJAC, IFLAG) INTEGER  N, LDFJAC, IFLAG REAL (KIND=nag_wp)  X(N), FVEC(N), FJAC(LDFJAC,N)

The specification of FCN for C05PCA is:

SUBROUTINE FCN ( N, X, FVEC, FJAC, LDFJAC, IFLAG, IUSER, RUSER) INTEGER  N, LDFJAC, IFLAG, IUSER(*) REAL (KIND=nag_wp)  X(N), FVEC(N), FJAC(LDFJAC,N), RUSER(*)

1:     N – INTEGERInput

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

2:     X(N) – REAL (KIND=nag_wp) arrayInput

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

3:     FVEC(N) – 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:     FJAC(LDFJAC,N) – 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:     LDFJAC – INTEGERInput

On entry: the first dimension of the array FJAC as declared in the (sub)program from which C05PCF/C05PCA is called.

6:     IFLAG – INTEGERInput/Output

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

$\mathbf{IFLAG}=0$

X and FVEC 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), then IFLAG should be set to a negative integer. This value will be returned through IFAIL.

Note: the following are additional parameters for specific use with C05PCA. Users of C05PCF therefore need not read the remainder of this description.

7:     IUSER($*$) – INTEGER arrayUser Workspace

8:     RUSER($*$) – REAL (KIND=nag_wp) arrayUser Workspace

FCN is called with the parameters IUSER and RUSER as supplied to C05PCF/C05PCA. You are free to use the arrays IUSER and RUSER to supply information to FCN as an alternative to using COMMON global variables.

FCN must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which C05PCF/C05PCA is called. Parameters denoted as Input must not be changed by this procedure.

2:     N – INTEGERInput

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

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

3:     X(N) – 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:     FVEC(N) – REAL (KIND=nag_wp) arrayOutput

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

5:     FJAC(LDFJAC,N) – REAL (KIND=nag_wp) arrayOutput

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

6:     LDFJAC – INTEGERInput

On entry: the first dimension of the array FJAC as declared in the (sub)program from which C05PCF/C05PCA is called.

Constraint: $\mathbf{LDFJAC}\ge \mathbf{N}$.

7:     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: $\mathbf{XTOL}\ge 0.0$.

8:     MAXFEV – INTEGERInput

On entry: the maximum number of calls to FCN with $\mathbf{IFLAG}\ne 0$. C05PCF/C05PCA 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: $\mathbf{MAXFEV}>0$.

9:     DIAG(N) – 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:   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, the variables will be scaled internally.

11:   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: $\mathbf{FACTOR}>0.0$.

12:   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 C05PCF/C05PCA.

13:   NFEV – INTEGEROutput

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

14:   NJEV – INTEGEROutput

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

15:   R($\mathbf{N}\times \left(\mathbf{N}+1\right)/2$) – 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.

16:   LR – INTEGERDummy

This parameter is no longer accessed by C05PCF/C05PCA.

17:   QTF(N) – REAL (KIND=nag_wp) arrayOutput

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

18:   W($1$,$1$) – REAL (KIND=nag_wp) arrayInput

This parameter is no longer accessed by C05PCF/C05PCA. Workspace is provided internally by dynamic allocation instead.

19:   IFAIL – INTEGERInput/Output

Note: for C05PCA, IFAIL does not occur in this position in the parameter list. See the additional parameters described below.

On entry: IFAIL must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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 parameter, 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).

Note: the following are additional parameters for specific use with C05PCA. Users of C05PCF therefore need not read the remainder of this description.

19:   IUSER($*$) – INTEGER arrayUser Workspace

20:   RUSER($*$) – REAL (KIND=nag_wp) arrayUser Workspace

IUSER and RUSER are not used by C05PCF/C05PCA, but are passed directly to FCN and may be used to pass information to this routine as an alternative to using COMMON global variables.

21:   IFAIL – INTEGERInput/Output

Note: see the parameter description for IFAIL above.

6  Error Indicators and Warnings

$\mathbf{IFAIL}<0$

This indicates an exit from C05PCF/C05PCA because you have set IFLAG negative in FCN. The value of IFAIL will be the same as your setting of IFLAG.

$\mathbf{IFAIL}=1$

On entry,	$\mathbf{N}\le 0$,
or	$\mathbf{XTOL}<0.0$,
or	$\mathbf{MAXFEV}\le 0$,
or	$\mathbf{FACTOR}\le 0.0$,
or	$\mathbf{LDFJAC}<\mathbf{N}$,
or	$\mathbf{MODE}=2$ and $\mathbf{DIAG}\left(i\right)\le 0.0$ for some $i$, $i=1,2,\dots ,\mathbf{N}$.

$\mathbf{IFAIL}=2$

There have been MAXFEV evaluations of FCN to evaluate the functions. Consider restarting the calculation from the final point held in X.

$\mathbf{IFAIL}=3$

No further improvement in the approximate solution X is possible; XTOL is too small.

$\mathbf{IFAIL}=4$

The iteration is not making good progress, as measured by the improvement from the last five Jacobian evaluations.

$\mathbf{IFAIL}=5$

The iteration is not making good progress, as measured by the improvement from the last ten iterations.

$\mathbf{IFAIL}=-999$

Internal memory allocation failed.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (c05pcfe.f90)

Program Text (c05pcae.f90)

9.2  Program Data

9.3  Program Results

Program Results (c05pcfe.r)

Program Results (c05pcae.r)