NAG Library Routine Document

C05AUF

+− 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

C05AUF locates a simple zero of a continuous function from a given starting value. It uses a binary search to locate an interval containing a zero of the function, then Brent's method, which is a combination of nonlinear interpolation, linear extrapolation and bisection, to locate the zero precisely.

2 Specification

SUBROUTINE C05AUF (

X, H, EPS, ETA, F, A, B, IUSER, RUSER, IFAIL)

INTEGER	IUSER(*), IFAIL
REAL (KIND=nag_wp)	X, H, EPS, ETA, F, A, B, RUSER(*)
EXTERNAL	F

3 Description

C05AUF attempts to locate an interval

[a, b]

containing a simple zero of the function

f (x)

by a binary search starting from the initial point

x = X

and using repeated calls to C05AVF. If this search succeeds, then the zero is determined to a user-specified accuracy by a call to C05AYF. The specifications of routines C05AVF and C05AYF should be consulted for details of the methods used.

The approximation

x

to the zero

α

is determined so that at least one of the following criteria is satisfied:

(i)	$\|x - α\| \leq EPS$ ,
(ii)	$\|f (x)\| \leq ETA$ .

4 References

Brent R P (1973) Algorithms for Minimization Without Derivatives Prentice–Hall

5 Parameters

1: X – REAL (KIND=nag_wp)Input/Output

On entry: an initial approximation to the zero.

On exit: if

IFAIL = 0

4

, X is the final approximation to the zero.

IFAIL = 3

, X is likely to be a pole of

f (x)

Otherwise, X contains no useful information.

2: H – REAL (KIND=nag_wp)Input

On entry: a step length for use in the binary search for an interval containing the zero. The maximum interval searched is

[X - 256.0 \times H, X + 256.0 \times H]

Constraint:

H

must be sufficiently large that

X + H \neq X

on the computer.

3: EPS – REAL (KIND=nag_wp)Input

On entry: the termination tolerance on

x

(see Section 3).

Constraint:

EPS > 0.0

4: ETA – REAL (KIND=nag_wp)Input

On entry: a value such that if

|f (x)| \leq ETA

x

is accepted as the zero. ETA may be specified as

0.0

(see Section 7).

5: F – REAL (KIND=nag_wp) FUNCTION, supplied by the user.External Procedure

F must evaluate the function

f

whose zero is to be determined.

The specification of F is:

FUNCTION F (

X, IUSER, RUSER)

REAL (KIND=nag_wp) F

INTEGER	IUSER(*)
REAL (KIND=nag_wp)	X, RUSER(*)

1: X – REAL (KIND=nag_wp)Input: On entry: the point at which the function must be evaluated.
2: IUSER( $*$ ) – INTEGER arrayUser Workspace
3: RUSER( $*$ ) – REAL (KIND=nag_wp) arrayUser Workspace: F is called with the parameters IUSER and RUSER as supplied to C05AUF. You are free to use the arrays IUSER and RUSER to supply information to F as an alternative to using COMMON global variables.

F must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which C05AUF is called. Parameters denoted as Input must not be changed by this procedure.

6: A – REAL (KIND=nag_wp)Output

7: B – REAL (KIND=nag_wp)Output

On exit: the lower and upper bounds respectively of the interval resulting from the binary search. If the zero is determined exactly such that

f (x) = 0.0

or is determined so that

|f (x)| \leq ETA

at any stage in the calculation, then on exit

A = B = x

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

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

IUSER and RUSER are not used by C05AUF, but are passed directly to F and may be used to pass information to this routine as an alternative to using COMMON global variables.

10: 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

- 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, either $EPS \leq 0.0$ , or $X + H = X$ to machine accuracy (meaning that the search for an interval containing the zero cannot commence).

$IFAIL = 2$: An interval containing the zero could not be found. Increasing H and calling C05AUF again will increase the range searched for the zero. Decreasing H and calling C05AUF again will refine the mesh used in the search for the zero.

$IFAIL = 3$: A change in sign of $f (x)$ has been determined as occurring near the point defined by the final value of X. However, there is some evidence that this sign-change corresponds to a pole of $f (x)$ .

$IFAIL = 4$: Too much accuracy has been requested in the computation; that is, the zero has been located to relative accuracy at least $ε$ , where $ε$ is the machine precision, but the exit conditions described in Section 3 are not satisfied. It is unsafe for C05AUF to continue beyond this point, but the final value of X returned is an accurate approximation to the zero.

7 Accuracy

The levels of accuracy depend on the values of EPS and ETA. If full machine accuracy is required, they may be set very small, resulting in an exit with

IFAIL = 4

, although this may involve many more iterations than a lesser accuracy. You are recommended to set

ETA = 0.0

and to use EPS to control the accuracy, unless you have considerable knowledge of the size of

f (x)

for values of

x

near the zero.

8 Further Comments

The time taken by C05AUF depends primarily on the time spent evaluating F (see Section 5). The accuracy of the initial approximation X and the value of H will have a somewhat unpredictable effect on the timing.

If it is important to determine an interval of relative length less than

2 \times EPS

containing the zero, or if F is expensive to evaluate and the number of calls to F is to be restricted, then use of C05AVF followed by C05AZF is recommended. Use of this combination is also recommended when the structure of the problem to be solved does not permit a simple F to be written: the reverse communication facilities of these routines are more flexible than the direct communication of F required by C05AUF.

If the iteration terminates with successful exit and

A = B = X

there is no guarantee that the value returned in X corresponds to a simple zero and you should check whether it does.

One way to check this is to compute the derivative of

f

at the point X, preferably analytically, or, if this is not possible, numerically, perhaps by using a central difference estimate. If

f^{'} (X) = 0.0

, then X must correspond to a multiple zero of

f

rather than a simple zero.

9 Example

This example calculates an approximation to the zero of

x - e^{- x}

using a tolerance of

EPS = 1.0E−5

starting from

X = 1.0

and using an initial search step

H = 0.1

9.1 Program Text

Program Text (c05aufe.f90)

9.2 Program Data

None.

9.3 Program Results

Program Results (c05aufe.r)

C05AUF (PDF version)

C05 Chapter Contents

C05 Chapter Introduction

NAG Library Manual

NAG Library Routine DocumentC05AUF