nag_zero_nonlin_eqns_easy (c05qbc) (PDF version)

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nag_zero_nonlin_eqns_easy (c05qbc)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

nag_zero_nonlin_eqns_easy (c05qbc) is an easy-to-use function that finds a solution of a system of nonlinear equations by a modification of the Powell hybrid method.

2

Specification

#include <nag.h>

#include <nagc05.h>

void

nag_zero_nonlin_eqns_easy (

void	(fcn)(Integer n, const double x[], double fvec[], Nag_Comm comm, Integer *iflag),

Integer n, double x[], double fvec[], double xtol, Nag_Comm *comm, NagError *fail)

3

Description

The system of equations is defined as:

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

nag_zero_nonlin_eqns_easy (c05qbc) 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

Arguments

1: $fcn$ – function, supplied by the userExternal Function

fcn must return the values of the functions

f_{i}

at a point

x

.

The specification of fcn is:

void	fcn (Integer n, const double x[], double fvec[], Nag_Comm comm, Integer iflag)

1: $n$ – IntegerInput

On entry:

n

, the number of equations.

2: $x [n]$ – const doubleInput

On entry: the components of the point

x

at which the functions must be evaluated.

3: $fvec [n]$ – doubleOutput

On exit: the function values

f_{i} (x)

(unless iflag is set to a negative value by fcn).

4: $comm$ – Nag_Comm *

Pointer to structure of type Nag_Comm; the following members are relevant to fcn.

user – double *
iuser – Integer *
p – Pointer: The type Pointer will be void *. Before calling nag_zero_nonlin_eqns_easy (c05qbc) you may allocate memory and initialize these pointers with various quantities for use by fcn when called from nag_zero_nonlin_eqns_easy (c05qbc) (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).

5: $iflag$ – Integer *Input/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), iflag should be set to a negative integer.

Note: fcn should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by nag_zero_nonlin_eqns_easy (c05qbc). If your code inadvertently does return any NaNs or infinities, nag_zero_nonlin_eqns_easy (c05qbc) is likely to produce unexpected results.

2: $n$ – IntegerInput

On entry:

n

, the number of equations.

Constraint:

n > 0

.

3: $x [n]$ – doubleInput/Output

On entry: an initial guess at the solution vector.

On exit: the final estimate of the solution vector.

4: $fvec [n]$ – doubleOutput

On exit: the function values at the final point returned in x.

5: $xtol$ – doubleInput

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

Suggested value:

\sqrt{ε}

, where

ε

is the machine precision returned by nag_machine_precision (X02AJC).

Constraint:

xtol \geq 0.0

.

6: $comm$ – Nag_Comm *

The NAG communication argument (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).

7: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n > 0$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_IMPROVEMENT: 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 nag_zero_nonlin_eqns_easy (c05qbc) from a different starting point may avoid the region of difficulty.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL: On entry, $xtol = 〈value〉$ .
Constraint: $xtol \geq 0.0$ .
NE_TOO_MANY_FEVALS: There have been at least $200 \times (n + 1)$ calls to fcn. Consider restarting the calculation from the point held in x.
NE_TOO_SMALL: No further improvement in the solution is possible. xtol is too small: $xtol = 〈value〉$ .
NE_USER_STOP: iflag was set negative in fcn. $iflag = 〈value〉$ .

7

Accuracy

If

\hat{x}

is the true solution, nag_zero_nonlin_eqns_easy (c05qbc) 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 nag_zero_nonlin_eqns_easy (c05qbc) 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 function exits with

fail . code =

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 nag_zero_nonlin_eqns_easy (c05qbc) may incorrectly indicate convergence. The validity of the answer can be checked, for example, by rerunning nag_zero_nonlin_eqns_easy (c05qbc) with a lower value for xtol.

8

Parallelism and Performance

nag_zero_nonlin_eqns_easy (c05qbc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_zero_nonlin_eqns_easy (c05qbc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9

Further Comments

Local workspace arrays of fixed lengths are allocated internally by nag_zero_nonlin_eqns_easy (c05qbc). The total size of these arrays amounts to

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

double elements.

The time required by nag_zero_nonlin_eqns_easy (c05qbc) 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 nag_zero_nonlin_eqns_easy (c05qbc) to process each evaluation of the functions is approximately

11.5 \times n^{2}

. The timing of nag_zero_nonlin_eqns_easy (c05qbc) 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.

10

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}

10.1

Program Text

Program Text (c05qbce.c)

10.2

Program Data

None.

10.3

Program Results

Program Results (c05qbce.r)

nag_zero_nonlin_eqns_easy (c05qbc) (PDF version)

c05 Chapter Contents

c05 Chapter Introduction

NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017