NAG Library Routine Document

C05PDF/C05PDA

+− Contents

1 Purpose

+− 2 Specification

2.1 Specification for C05PDF

2.2 Specification for C05PDA

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

C05PDF/C05PDA is a comprehensive reverse communication routine that finds a solution of a system of nonlinear equations by a modification of the Powell hybrid method. You must provide the Jacobian.

C05PDA is a version of C05PDF that has additional parameters in order to make it safe for use in multithreaded applications (see Section 5).

2 Specification

2.1 Specification for C05PDF

SUBROUTINE C05PDF (

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

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

2.2 Specification for C05PDA

SUBROUTINE C05PDA (

IREVCM, N, X, FVEC, FJAC, LDFJAC, XTOL, DIAG, MODE, FACTOR, R, LR, QTF, W, LWSAV, IWSAV, RWSAV, IFAIL)

INTEGER	IREVCM, N, LDFJAC, MODE, LR, IWSAV(15), IFAIL
REAL (KIND=nag_wp)	X(N), FVEC(N), FJAC(LDFJAC,N), XTOL, DIAG(N), FACTOR, R(N*(N+1)/2), QTF(N), W(N,4), RWSAV(10)
LOGICAL	LWSAV(2)

3 Description

The system of equations is defined as:

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

C05PDF/C05PDA 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. 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 and FJAC 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 C05PDF/C05PDA 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 C05PDF/C05PDA, FVEC must contain the function values $f_{i} (x)$ .
$IREVCM = 3$: Indicates that before re-entry to C05PDF/C05PDA, $FJAC (i, j)$ must contain the value of $\frac{\partial f_{i}}{\partial x_{j}}$ at the point $x$ , for $i = 1, 2, \dots, n$ and $j = 1, 2, \dots, n$ .

On final exit:

IREVCM = 0

, and the algorithm has terminated.

Constraint:

IREVCM = 0

1

2

3

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

On initial entry: must be set to the values of the Jacobian evaluated at the initial point X.

On intermediate re-entry: if

IREVCM \neq 3

, FJAC must not be changed.

IREVCM = 3

, FJAC must be set to the value of the Jacobian computed at the current point X.

On final exit: the orthogonal matrix

Q

produced by the

Q R

factorization of the final approximate Jacobian.

6: LDFJAC – INTEGERInput

On initial entry: the first dimension of the array FJAC as declared in the (sub)program from which C05PDF/C05PDA is called.

Constraint:

LDFJAC \geq N

7: 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

8: 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

9: 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.

10: 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

11: 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.

12: LR – INTEGERDummy

This parameter is no longer accessed by C05PDF/C05PDA.

13: 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

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

15: IFAIL – INTEGERInput/Output

Note: for C05PDA, IFAIL does not occur in this position in the parameter list. See the additional parameters described below.

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).

Note: the following are additional parameters for specific use with C05PDA. Users of C05PDF therefore need not read the remainder of this description.

15: LWSAV( $2$ ) – LOGICAL arrayCommunication Array

16: IWSAV( $15$ ) – INTEGER arrayCommunication Array

17: RWSAV( $10$ ) – REAL (KIND=nag_wp) arrayCommunication Array

The arrays LWSAV, IWSAV and RWSAV must not be altered between calls to C05PDA.

18: IFAIL – INTEGERInput/Output

Note: see the parameter description for IFAIL above.

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	$FACTOR \leq 0.0$ ,
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 > 3

$IFAIL = 3$: 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 C05PDF/C05PDA 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 C05PDF/C05PDA 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 C05PDF/C05PDA 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 and the Jacobian are coded consistently and that the functions are reasonably well behaved. If these conditions are not satisfied, then C05PDF/C05PDA may incorrectly indicate convergence. The coding of the Jacobian can be checked using C05ZAF. If the Jacobian is coded correctly, then the validity of the answer can be checked by rerunning C05PDF/C05PDA with a lower value for XTOL.

8 Further Comments

The time required by C05PDF/C05PDA 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 C05PDF/C05PDA is about

11.5 \times n^{2}

to process each evaluation of the functions and about

1.3 \times n^{3}

to process each evaluation of the Jacobian. The timing of C05PDF/C05PDA is strongly influenced by the time spent in the evaluation of the functions and the Jacobian.

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