C05QSF (PDF version)

C05 Chapter Contents

C05 Chapter Introduction

NAG Library Manual

NAG Library Routine Document

C05QSF

+− 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

C05QSF is an easy-to-use routine that finds a solution of a sparse system of nonlinear equations by a modification of the Powell hybrid method.

2 Specification

SUBROUTINE C05QSF (

FCN, N, X, FVEC, XTOL, INIT, RCOMM, LRCOMM, ICOMM, LICOMM, IUSER, RUSER, IFAIL)

INTEGER	N, LRCOMM, ICOMM(LICOMM), LICOMM, IUSER(*), IFAIL
REAL (KIND=nag_wp)	X(N), FVEC(N), XTOL, RCOMM(LRCOMM), RUSER(*)
LOGICAL	INIT
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 .

C05QSF 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 sparse rank-1 method of Schubert (see Schubert (1970)). At the starting point, the sparsity pattern is determined and the Jacobian is approximated by forward differences, but these are not used again until the rank-1 method fails to produce satisfactory progress. Then, the sparsity structure is used to recompute an approximation to the Jacobian by forward differences with the least number of function evaluations. The subroutine you supply must be able to compute only the requested subset of the function values. The sparse Jacobian linear system is solved at each iteration with F11MEF computing the Newton step. For more details see Powell (1970) and Broyden (1965).

4 References

Broyden C G (1965) A class of methods for solving nonlinear simultaneous equations Mathematics of Computation 19(92) 577–593

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

Schubert L K (1970) Modification of a quasi-Newton method for nonlinear equations with a sparse Jacobian Mathematics of Computation 24(109) 27–30

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, LINDF, INDF, X, FVEC, IUSER, RUSER, IFLAG)

INTEGER	N, LINDF, INDF(LINDF), IUSER(*), IFLAG
REAL (KIND=nag_wp)	X(N), FVEC(N), RUSER(*)

1: N – INTEGERInput: On entry: $n$ , the number of equations.
2: LINDF – INTEGERInput: On entry: LINDF specifies the number of indices $i$ for which values of $f_{i} (x)$ must be computed.
3: INDF(LINDF) – INTEGER arrayInput: On entry: INDF specifies the indices $i$ for which values of $f_{i} (x)$ must be computed. The indices are specified in strictly ascending order.
4: X(N) – REAL (KIND=nag_wp) arrayInput: On entry: the components of the point $x$ at which the functions must be evaluated. $X (i)$ contains the coordinate $x_{i}$ .
5: FVEC(N) – REAL (KIND=nag_wp) arrayOutput: On exit: $FVEC (i)$ must contain the function values $f_{i} (x)$ , for all indices $i$ in INDF.
6: IUSER( $*$ ) – INTEGER arrayUser Workspace
7: RUSER( $*$ ) – REAL (KIND=nag_wp) arrayUser Workspace: FCN is called with the parameters IUSER and RUSER as supplied to C05QSF. You are free to use the arrays IUSER and RUSER to supply information to FCN as an alternative to using COMMON global variables.
8: 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.

FCN must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which C05QSF 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.

X (i)

must contain the coordinate

x_{i}

.

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.

FVEC (i)

contains the function values

f_{i}

.

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: INIT – LOGICALInput

On entry: INIT must be set to .TRUE. to indicate that this is the first time C05QSF is called for this specific problem. C05QSF then computes the dense Jacobian and detects and stores its sparsity pattern (in RCOMM and ICOMM) before proceeding with the iterations. This is noticeably time consuming when N is large. If not enough storage has been provided for RCOMM or ICOMM, C05QSF will fail. On exit with

IFAIL = 0

,

2

,

3

or

4

,

ICOMM (1)

contains

nnz

, the number of nonzero entries found in the Jacobian. On subsequent calls, INIT can be set to .FALSE. if the problem has a Jacobian of the same sparsity pattern. In that case, the computation time required for the detection of the sparsity pattern will be smaller.

7: RCOMM(LRCOMM) – REAL (KIND=nag_wp) arrayCommunication Array

RCOMM must not be altered between successive calls to C05QSF.

8: LRCOMM – INTEGERInput

On entry: the dimension of the array RCOMM as declared in the (sub)program from which C05QSF is called.

Constraint:

LRCOMM \geq 12 + nnz

where

nnz

is the number of nonzero entries in the Jacobian, as computed by C05QSF.

9: ICOMM(LICOMM) – INTEGER arrayCommunication Array

If

IFAIL = 0

,

2

,

3

or

4

on exit,

ICOMM (1)

contains

nnz

where

nnz

is the number of nonzero entries in the Jacobian.

ICOMM must not be altered between successive calls to C05QSF.

10: LICOMM – INTEGERInput

On entry: the dimension of the array ICOMM as declared in the (sub)program from which C05QSF is called.

Constraint:

LICOMM \geq 8 \times N + 19 + nnz

where

nnz

is the number of nonzero entries in the Jacobian, as computed by C05QSF.

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

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

IUSER and RUSER are not used by C05QSF, but are passed directly to FCN and may be used to pass information to this routine as an alternative to using COMMON global variables.

13: 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 $200 \times (N + 1)$ evaluations of FCN. Consider restarting the calculation from the final point held in X. In this case, before reentering C05QSF, set $INIT = .FALSE.$ .

$IFAIL = 3$: 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 C05QSF from a different starting point may avoid the region of difficulty. In this case, before reentering C05QSF with a different starting point, set $INIT = .FALSE.$ .

$IFAIL = 5$: You have set IFLAG negative in FCN.

$IFAIL = 6$

On entry,

LRCOMM < 12 + nnz

.

$IFAIL = 7$

On entry,

LICOMM < 8 \times N + 19 + nnz

.

$IFAIL = 9$: An internal error occurred. Please contact NAG.

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

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

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

7 Accuracy

If

\hat{x}

is the true solution, C05QSF tries to ensure that

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

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 C05QSF 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 are reasonably well behaved. If this condition is not satisfied, then C05QSF may incorrectly indicate convergence. The validity of the answer can be checked, for example, by rerunning C05QSF with a lower value for XTOL.

8 Further Comments

Local workspace arrays of fixed lengths are allocated internally by C05QSF. The total size of these arrays amounts to

8 \times n + 2 \times q

real elements and

10 \times n + 2 \times q + 5

integer elements where the integer

q

is bounded by

8 \times nnz

and

n^{2}

and depends on the sparsity pattern of the Jacobian.

The time required by C05QSF 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 C05QSF to process each evaluation of the functions depends on the number of nonzero entries in the Jacobian. The timing of C05QSF is strongly influenced by the time spent evaluating the functions.

When INIT is .TRUE., the dense Jacobian is first evaluated and that will take time proportional to

n^{2}

.

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}

It then perturbs the equations by a small amount and solves the new system.

9.1 Program Text

Program Text (c05qsfe.f90)

9.2 Program Data

None.

9.3 Program Results

Program Results (c05qsfe.r)

C05QSF (PDF version)

C05 Chapter Contents

C05 Chapter Introduction

NAG Library Manual

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