nag_zero_nonlin_eqns_deriv_easy (c05rbc) (PDF version)

c05 Chapter Contents

c05 Chapter Introduction

NAG Library Manual

NAG Library Function Document

nag_zero_nonlin_eqns_deriv_easy (c05rbc)

+− 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_deriv_easy (c05rbc) is an easy-to-use function 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

#include <nag.h>

#include <nagc05.h>

void

nag_zero_nonlin_eqns_deriv_easy (

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

Integer n, double x[], double fvec[], double fjac[], 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_deriv_easy (c05rbc) is based on the MINPACK routine HYBRJ1 (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 Arguments

1: fcn – function, supplied by the userExternal Function

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:

void	fcn (Integer n, const double x[], double fvec[], double fjac[], 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 or the Jacobian must be evaluated.

3: fvec[n] – doubleInput/Output

On entry: if

iflag = 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[ $n \times n$ ] – doubleInput/Output

Note: the

(i, j)

th element of the matrix is stored in

fjac [(j - 1) \times n + i - 1]

.

On entry: if

iflag = 1

, fjac 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

, and must not be changed.

On exit: if

iflag = 2

on entry,

fjac [(j - 1) \times n + i - 1]

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: 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_deriv_easy (c05rbc) you may allocate memory and initialize these pointers with various quantities for use by fcn when called from nag_zero_nonlin_eqns_deriv_easy (c05rbc) (see Section 3.2.1.1 in the Essential Introduction).

6: iflag – Integer *Input/Output

On entry:

iflag = 1

or

2

.

$iflag = 1$: 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.

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: fjac[ $n \times n$ ] – doubleOutput

Note: the

(i, j)

th element of the matrix is stored in

fjac [(j - 1) \times n + i - 1]

.

On exit: the orthogonal matrix

Q

produced by the

Q R

factorization of the final approximate Jacobian, stored by columns.

6: 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

.

7: comm – Nag_Comm *Communication Structure

The NAG communication argument (see Section 3.2.1.1 in the Essential Introduction).

8: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
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.
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_deriv_easy (c05rbc) from a different starting point may avoid the region of difficulty.
NE_REAL: On entry, $xtol = ⟨value⟩$ .
Constraint: $xtol \geq 0.0$ .
NE_TOO_MANY_FEVALS: There have been at least $100 \times (n + 1)$ 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_deriv_easy (c05rbc) 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_deriv_easy (c05rbc) 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 and the Jacobian are coded consistently and that the functions are reasonably well behaved. If these conditions are not satisfied, then nag_zero_nonlin_eqns_deriv_easy (c05rbc) may incorrectly indicate convergence. The coding of the Jacobian can be checked using nag_check_derivs (c05zdc). If the Jacobian is coded correctly, then the validity of the answer can be checked by rerunning nag_zero_nonlin_eqns_deriv_easy (c05rbc) with a lower value for xtol.

8 Parallelism and Performance

nag_zero_nonlin_eqns_deriv_easy (c05rbc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_zero_nonlin_eqns_deriv_easy (c05rbc) 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 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_deriv_easy (c05rbc). The total size of these arrays amounts to

n \times (n + 13) / 2

double elements.

The time required by nag_zero_nonlin_eqns_deriv_easy (c05rbc) 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_deriv_easy (c05rbc) 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 nag_zero_nonlin_eqns_deriv_easy (c05rbc) 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 (c05rbce.c)

10.2 Program Data

None.

10.3 Program Results

Program Results (c05rbce.r)

nag_zero_nonlin_eqns_deriv_easy (c05rbc) (PDF version)

c05 Chapter Contents

c05 Chapter Introduction

NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014