NAG Library Routine Document

C05AYF

Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+− 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

C05AYF locates a simple zero of a continuous function in a given interval using Brent's method, which is a combination of nonlinear interpolation, linear extrapolation and bisection.

2 Specification

SUBROUTINE C05AYF ( A, B, EPS, ETA, F, X, IUSER, RUSER, IFAIL)

INTEGER IUSER(*), IFAIL

REAL (KIND=nag_wp) A, B, EPS, ETA, F, X, RUSER(*)

EXTERNAL F

3 Description

C05AYF attempts to obtain an approximation to a simple zero of the function

f (x)

given an initial interval

[a, b]

such that

f (a) \times f (b) \leq 0

. The same core algorithm is used by C05AZF whose specification should be consulted for details of the method 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: A – REAL (KIND=nag_wp)Input

On entry:

a

, the lower bound of the interval.

2: B – REAL (KIND=nag_wp)Input

On entry:

b

, the upper bound of the interval.

Constraint:

B \neq A

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 C05AYF. 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 C05AYF is called. Parameters denoted as Input must not be changed by this procedure.

6: X – REAL (KIND=nag_wp)Output

On exit: if

IFAIL = 0

2

, X is the final approximation to the zero. If

IFAIL = 3

, X is likely to be a pole of

f (x)

. Otherwise, X contains no useful information.

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

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

IUSER and RUSER are not used by C05AYF, but are passed directly to F and may be used to pass information to this routine as an alternative to using COMMON global variables.

9: 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, $EPS \leq 0.0$ ,
or $A = B$ ,
or $F (A) \times F (B) > 0.0$ .

$IFAIL = 2$: 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 C05AYF to continue beyond this point, but the final value of X returned is an accurate approximation to 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 $f (x)$ .

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 = 2

, 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 C05AYF depends primarily on the time spent evaluating F (see Section 5).

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 C05AZF is recommended. Use of C05AZF 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 C05AZF are more flexible than the direct communication of F required by C05AYF.