NAG Library Routine Document

C05NBF

C05NBF is based on the MINPACK routine HYBRD1 (see Moré et al. (1980)). It chooses the correction at each step as a convex combination of the Newton and scaled gradient directions. The Jacobian is updated by the rank-1 method of Broyden. At the starting point, the Jacobian is approximated by forward differences, but these are not used again until the rank-1 method fails to produce satisfactory progress. For more details see Powell (1970).

4 References

Moré J J, Garbow B S and Hillstrom K E (1980) User guide for MINPACK-1 Technical Report ANL-80-74 Argonne National Laboratory

Powell M J D (1970) A hybrid method for nonlinear algebraic equations Numerical Methods for Nonlinear Algebraic Equations (ed P Rabinowitz) Gordon and Breach

5 Parameters

1: 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:

SUBROUTINE FCN (

N, X, FVEC, IFLAG)

INTEGER	N, IFLAG
REAL (KIND=nag_wp)	X(N), FVEC(N)

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 must be evaluated.
3: FVEC(N) – REAL (KIND=nag_wp) arrayOutput: On exit: the function values $f_{i} (x)$ (unless IFLAG is set to a negative value by FCN).
4: IFLAG – INTEGERInput/Output: On entry: $IFLAG > 0$ .

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

FCN must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which C05NBF is called. Parameters denoted as Input must not be changed by this procedure.

2: N – INTEGERInput

On entry:

n

, the number of equations.

Constraint:

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: XTOL – REAL (KIND=nag_wp)Input

On entry: the accuracy in X to which the solution is required.

Suggested value:

\sqrt{ε}

, where

ε

is the machine precision returned by X02AJF.

Constraint:

XTOL \geq 0.0

6: WA( $1$ ) – REAL (KIND=nag_wp) arrayInput

7: LWA – INTEGERInput

These parameters are no longer accessed by C05NBF. Workspace is provided internally by dynamic allocation instead.

8: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ 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 ​ 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 $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL < 0$: You have set IFLAG negative in FCN. The value of IFAIL will be the same as your setting of IFLAG.

$IFAIL = 1$

On entry,	$N \leq 0$ ,
or	$XTOL < 0.0$ .

$IFAIL = 2$: There have been at least $200 \times (N + 1)$ evaluations of FCN. Consider restarting the calculation from the final point held in X.

$IFAIL = 3$: No further improvement in the approximate solution X is possible; XTOL is too small.

$IFAIL = 4$: 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 C05NBF from a different starting point may avoid the region of difficulty.

$IFAIL = - 999$: Internal memory allocation failed.

7 Accuracy

\hat{x}

is the true solution, C05NBF tries to ensure that

‖x - \hat{x}‖ \leq XTOL \times ‖\hat{x}‖ .

If this condition is satisfied with

XTOL = 10^{- k}

, then the larger components of

x

have

k

significant decimal digits. There is a danger that the smaller components of

x

may have large relative errors, but the fast rate of convergence of C05NBF usually obviates this possibility.

If XTOL is less than machine precision and the above test is satisfied with the machine precision in place of XTOL, then the routine exits with

IFAIL = 3

Note: this convergence test is based purely on relative error, and may not indicate convergence if the solution is very close to the origin.

The test assumes that the functions are reasonably well behaved. If this condition is not satisfied, then C05NBF may incorrectly indicate convergence. The validity of the answer can be checked, for example, by rerunning C05NBF with a lower value for XTOL.

8 Further Comments

Local workspace arrays of fixed lengths are allocated internally by C05NBF. The total size of these arrays amounts to

N \times (3 \times N + 13) / 2

real elements.

The time required by C05NBF to solve a given problem depends on

n

, the behaviour of the functions, the accuracy requested and the starting point. The number of arithmetic operations executed by C05NBF to process each call of FCN is about