NAG Library Routine Document

C05NDF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

C05NDF is a comprehensive reverse communication routine that finds a solution of a system of nonlinear equations by a modification of the Powell hybrid method.

2 Specification

SUBROUTINE C05NDF (

IREVCM, N, X, FVEC, XTOL, ML, MU, EPSFCN, DIAG, MODE, FACTOR, FJAC, LDFJAC, R, LR, QTF, W, IFAIL)

INTEGER	IREVCM, N, ML, MU, MODE, LDFJAC, LR, IFAIL
REAL (KIND=nag_wp)	X(N), FVEC(N), XTOL, EPSFCN, DIAG(N), FACTOR, FJAC(LDFJAC,N), R(N*(N+1)/2), QTF(N), W(N,4)

3 Description

The system of equations is defined as:

f_{i} (x_{1}, x_{2}, \dots, x_{n}) = 0, i = 1, 2, \dots, n .

C05NDF is based on the MINPACK routine HYBRD (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

Note: this routine uses reverse communication. Its use involves an initial entry, intermediate exits and re-entries, and a final exit, as indicated by the parameter IREVCM. Between intermediate exits and re-entries, all parameters other than FVEC must remain unchanged.

1: IREVCM – INTEGERInput/Output

On initial entry: must have the value

0

On intermediate exit: specifies what action you must take before re-entering C05NDF with IREVCM unchanged. The value of IREVCM should be interpreted as follows:

$IREVCM = 1$: Indicates the start of a new iteration. No action is required by you, but X and FVEC are available for printing.
$IREVCM = 2$: Indicates that before re-entry to C05NDF, FVEC must contain the function values $f_{i} (x)$ .

On final exit:

IREVCM = 0

, and the algorithm has terminated.

Constraint:

IREVCM = 0

1

2

2: N – INTEGERInput

On initial entry:

n

, the number of equations.

Constraint:

N > 0

3: X(N) – REAL (KIND=nag_wp) arrayInput/Output

On initial entry: an initial guess at the solution vector.

On intermediate exit: contains the current point.

On final exit: the final estimate of the solution vector.

4: FVEC(N) – REAL (KIND=nag_wp) arrayInput/Output

On initial entry: need not be set.

On intermediate re-entry: if

IREVCM = 1

, FVEC must not be changed.

IREVCM = 2

, FVEC must be set to the values of the functions computed at the current point X.

On final exit: the function values at the final point, X.

5: XTOL – REAL (KIND=nag_wp)Input

On initial 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: ML – INTEGERInput

On initial entry: the number of subdiagonals within the band of the Jacobian matrix. (If the Jacobian is not banded, or you are unsure, set

ML = N - 1

Constraint:

ML \geq 0

7: MU – INTEGERInput

On initial entry: the number of superdiagonals within the band of the Jacobian matrix. (If the Jacobian is not banded, or you are unsure, set

MU = N - 1

Constraint:

MU \geq 0

8: EPSFCN – REAL (KIND=nag_wp)Input

On initial entry: the order of the largest relative error in the functions. It is used in determining a suitable step for a forward difference approximation to the Jacobian. If EPSFCN is less than machine precision (returned by X02AJF) then machine precision is used. Consequently a value of

0.0

will often be suitable.

Suggested value:

EPSFCN = 0.0

9: DIAG(N) – REAL (KIND=nag_wp) arrayInput/Output

On initial entry: if

MODE = 2

, DIAG must contain multiplicative scale factors for the variables.

Constraint:

DIAG (i) > 0.0

, for

i = 1, 2, \dots, n

On intermediate exit: the scale factors actually used (computed internally if

MODE \neq 2

10: MODE – INTEGERInput

On initial entry: indicates whether or not you have provided scaling factors in DIAG.

MODE = 2

the scaling must have been supplied in DIAG.

Otherwise, the variables will be scaled internally.

11: FACTOR – REAL (KIND=nag_wp)Input

On initial 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

FACTOR \times {‖DIAG \times X‖}_{2}

if this is nonzero; otherwise the bound is FACTOR.)

Suggested value:

FACTOR = 100.0

Constraint:

FACTOR > 0.0

12: FJAC(LDFJAC,N) – REAL (KIND=nag_wp) arrayInput/Output

On initial entry: need not be set.

On intermediate exit: must not be changed.

On final exit: the orthogonal matrix

Q

produced by the

Q R

factorization of the final approximate Jacobian.

13: LDFJAC – INTEGERInput

On initial entry: the first dimension of the array FJAC as declared in the (sub)program from which C05NDF is called.

Constraint:

LDFJAC \geq N

14: R( $N \times (N + 1) / 2$ ) – REAL (KIND=nag_wp) arrayInput/Output

On initial entry: need not be set.

On intermediate exit: must not be changed.

On final exit: the upper triangular matrix

R

produced by the

Q R

factorization of the final approximate Jacobian, stored row-wise.

15: LR – INTEGERDummy

This parameter is no longer accessed by C05NDF.

16: QTF(N) – REAL (KIND=nag_wp) arrayInput/Output

On initial entry: need not be set.

On intermediate exit: must not be changed.

On final exit: the vector

Q^{T} f

17: W(N, $4$ ) – REAL (KIND=nag_wp) arrayCommunication Array

18: IFAIL – INTEGERInput/Output

On initial 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, because for this routine the values of the output parameters may be useful even if

IFAIL \neq 0

on exit, the recommended value is

- 1

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

On final 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:

$IFAIL = 1$

On entry,	$N \leq 0$ ,
or	$XTOL < 0.0$ ,
or	$ML < 0$ ,
or	$MU < 0$ ,
or	$FACTOR \leq 0.0$ ,
or	$LDFJAC < N$ ,
or	$MODE = 2$ and $DIAG (i) \leq 0.0$ for some $i$ , $i = 1, 2, \dots, N$ .

$IFAIL = 2$

On entry,

IREVCM < 0

IREVCM > 2

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

$IFAIL = 4$: The iteration is not making good progress, as measured by the improvement from the last five Jacobian evaluations.

$IFAIL = 5$: The iteration is not making good progress, as measured by the improvement from the last ten iterations.

A value of

IFAIL = 4

5

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 C05NDF from a different starting point may avoid the region of difficulty.

7 Accuracy

\hat{x}

is the true solution and

D

denotes the diagonal matrix whose entries are defined by the array DIAG, then C05NDF tries to ensure that

{‖D (x - \hat{x})‖}_{2} \leq XTOL \times {‖D \hat{x}‖}_{2} .

If this condition is satisfied with

XTOL = 10^{- k}

, then the larger components of

D x

have

k

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

D x

may have large relative errors, but the fast rate of convergence of C05NDF 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 C05NDF may incorrectly indicate convergence. The validity of the answer can be checked, for example, by rerunning C05NDF with a lower value for XTOL.

8 Further Comments

The time required by C05NDF 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 C05NDF to process the evaluation of functions in the main program in each exit is about