C05RCF (PDF version)

C05 Chapter Contents

C05 Chapter Introduction

NAG Library Manual

NAG Library Routine Document

C05RCF

+− 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

C05RCF is a comprehensive routine that finds a solution of a system of nonlinear equations by a modification of the Powell hybrid method. You must provide the Jacobian.

2 Specification

SUBROUTINE C05RCF (

FCN, N, X, FVEC, FJAC, XTOL, MAXFEV, MODE, DIAG, FACTOR, NPRINT, NFEV, NJEV, R, QTF, IUSER, RUSER, IFAIL)

INTEGER	N, MAXFEV, MODE, NPRINT, NFEV, NJEV, IUSER(*), IFAIL
REAL (KIND=nag_wp)	X(N), FVEC(N), FJAC(N,N), XTOL, DIAG(N), FACTOR, R(N(N+1)/2), QTF(N), RUSER()
EXTERNAL	FCN

3 Description

The system of equations is defined as:

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

C05RCF is based on the MINPACK routine HYBRJ (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 requested, but it is not asked for 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

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:

SUBROUTINE FCN (

N, X, FVEC, FJAC, IUSER, RUSER, IFLAG)

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

IFLAG = 0

or

2

, FVEC contains the function values

f_{i} (x)

and must not be changed.

On exit: if

IFLAG = 1

on entry, FVEC must contain the function values

f_{i} (x)

(unless IFLAG is set to a negative value by FCN).

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

On entry: if

IFLAG = 0

,

FJAC (i, j)

contains the value of

\frac{\partial f_{i}}{\partial x_{j}}

at the point

x

, for

i = 1, 2, \dots, n

and

j = 1, 2, \dots, n

. When

IFLAG = 0

or

1

, FJAC must not be changed.

On exit: if

IFLAG = 2

on entry,

FJAC (i, j)

must contain the value of

\frac{\partial f_{i}}{\partial x_{j}}

at the point

x

, for

i = 1, 2, \dots, n

and

j = 1, 2, \dots, n

, (unless IFLAG is set to a negative value by FCN).

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

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

FCN is called with the parameters IUSER and RUSER as supplied to C05RCF. You are free to use the arrays IUSER and RUSER to supply information to FCN as an alternative to using COMMON global variables.

7: IFLAG – INTEGERInput/Output

On entry:

IFLAG = 0

,

1

or

2

.

$IFLAG = 0$: X, FVEC and FJAC are available for printing (see NPRINT).
$IFLAG = 1$: FVEC is to be updated.
$IFLAG = 2$: 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 value.

FCN must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which C05RCF 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: FJAC(N,N) – REAL (KIND=nag_wp) arrayOutput

On exit: the orthogonal matrix

Q

produced by the

Q R

factorization of the final approximate Jacobian.

6: 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

.

7: MAXFEV – INTEGERInput

On entry: the maximum number of calls to FCN with

IFLAG \neq 0

. C05RCF will exit with

IFAIL = 2

, if, at the end of an iteration, the number of calls to FCN exceeds MAXFEV.

Suggested value:

MAXFEV = 100 \times (N + 1)

.

Constraint:

MAXFEV > 0

.

8: MODE – INTEGERInput

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

If

MODE = 2

the scaling must have been specified in DIAG.

Otherwise, if

MODE = 1

, the variables will be scaled internally.

Constraint:

MODE = 1

or

2

.

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

On entry: if

MODE = 2

, DIAG must contain multiplicative scale factors for the variables.

If

MODE = 1

, DIAG need not be set.

Constraint: if

MODE = 2

,

DIAG (i) > 0.0

, for

i = 1, 2, \dots, n

.

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

MODE = 1

).

10: 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

FACTOR \times {‖DIAG \times X‖}_{2}

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

Suggested value:

FACTOR = 100.0

.

Constraint:

FACTOR > 0.0

.

11: 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.

$NPRINT \leq 0$: No calls are made.
$NPRINT > 0$: FCN is called at the beginning of the first iteration, every NPRINT iterations thereafter and immediately before the return from C05RCF.

12: NFEV – INTEGEROutput

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

13: NJEV – INTEGEROutput

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

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

On exit: the upper triangular matrix

R

produced by the

Q R

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

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

On exit: the vector

Q^{T} f

.

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

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

IUSER and RUSER are not used by C05RCF, but are passed directly to FCN and may be used to pass information to this routine as an alternative to using COMMON global variables.

18: 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

or

- 1

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

Errors or warnings detected by the routine:

$IFAIL = 2$: There have been at least MAXFEV calls to FCN to evaluate the functions. 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, 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.

$IFAIL = 6$: You have set IFLAG negative in FCN.

$IFAIL = 11$: On entry, $N \leq 0$ .

$IFAIL = 12$: On entry, $XTOL < 0.0$ .

$IFAIL = 13$: On entry, $MODE \neq 1$ or $2$ .

$IFAIL = 14$: On entry, $FACTOR \leq 0.0$ .

$IFAIL = 15$: On entry, $MODE = 2$ and $DIAG (i) \leq 0.0$ for some $i$ , for $i = 1, 2, \dots, N$ .

$IFAIL = 18$: On entry, $MAXFEV \leq 0$ .

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

A value of

IFAIL = 4

or

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 C05RCF from a different starting point may avoid the region of difficulty.

7 Accuracy

If

\hat{x}

is the true solution and

D

denotes the diagonal matrix whose entries are defined by the array DIAG, then C05RCF 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 C05RCF 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 convergence test assumes that the functions and the Jacobian are coded consistently and that the functions are reasonably well behaved. If these conditions are not satisfied, then C05RCF may incorrectly indicate convergence. The coding of the Jacobian can be checked using C05ZDF. If the Jacobian is coded correctly, then the validity of the answer can be checked by rerunning C05RCF with a lower value for XTOL.

8 Further Comments

Local workspace arrays of fixed lengths are allocated internally by C05RCF. The total size of these arrays amounts to

4 \times n

real elements.

The time required by C05RCF 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 C05RCF is approximately

11.5 \times n^{2}

to process each evaluation of the functions and approximately

1.3 \times n^{3}

to process each evaluation of the Jacobian. The timing of C05RCF is strongly influenced by the time spent evaluating the functions.

Ideally the problem should be scaled so that, at the solution, the function values are of comparable magnitude.

9 Example

This example determines the values

x_{1}, \dots, x_{9}

which satisfy the tridiagonal equations:

\begin{array}{rcl} (3 - 2 x_{1}) x_{1} - 2 x_{2} & = & - 1, \\ - x_{i - 1} + (3 - 2 x_{i}) x_{i} - 2 x_{i + 1} & = & - 1, i = 2, 3, \dots, 8 \\ - x_{8} + (3 - 2 x_{9}) x_{9} & = & - 1 . \end{array}

9.1 Program Text

Program Text (c05rcfe.f90)

9.2 Program Data

None.

9.3 Program Results

Program Results (c05rcfe.r)

C05RCF (PDF version)

C05 Chapter Contents

C05 Chapter Introduction

NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012