NAG FL Interface
c05ayf (contfn_brent)

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

Fortran Interface

Subroutine c05ayf (

a, b, eps, eta, f, x, iuser, ruser, ifail)

Integer, Intent (Inout)	::	iuser(*), ifail
Real (Kind=nag_wp), External	::	f
Real (Kind=nag_wp), Intent (In)	::	a, b, eps, eta
Real (Kind=nag_wp), Intent (Inout)	::	ruser(*)
Real (Kind=nag_wp), Intent (Out)	::	x

C Header Interface

#include <nag.h>

void	c05ayf_ (const double a, const double b, const double eps, const double eta, double (NAG_CALL f)(const double x, Integer iuser[], double ruser[]), double x, Integer iuser[], double ruser[], Integer ifail)

The routine may be called by the names c05ayf or nagf_roots_contfn_brent.

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 Arguments

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:

Fortran Interface

Function f (

x, iuser, ruser)

Real (Kind=nag_wp)	::	f
Integer, Intent (Inout)	::	iuser(*)
Real (Kind=nag_wp), Intent (In)	::	x
Real (Kind=nag_wp), Intent (Inout)	::	ruser(*)

C Header Interface

double

f_ (const double *x, Integer iuser[], double ruser[])

1: $x$ – Real (Kind=nag_wp) Input: On entry: the point at which the function must be evaluated.
2: $iuser (*)$ – Integer array User Workspace
3: $ruser (*)$ – Real (Kind=nag_wp) array User Workspace: f is called with the arguments iuser and ruser as supplied to c05ayf. You should use the arrays iuser and ruser to supply information to f.

f must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which c05ayf is called. Arguments denoted as Input must not be changed by this procedure.

Note: f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by c05ayf. If your code inadvertently does return any NaNs or infinities, c05ayf is likely to produce unexpected results.

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 array User Workspace

8: $ruser (*)$ – Real (Kind=nag_wp) array User 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.

9: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

- 1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

- 1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

- 1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. 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, $a = 〈value〉$ and $b = 〈value〉$ .
Constraint: $a \neq b$ .

On entry, $eps = 〈value〉$ .
Constraint: $eps > 0.0$ .

On entry, $f (a)$ and $f (b)$ have the same sign with neither equalling $0.0$ : $f (a) = 〈value〉$ and $f (b) = 〈value〉$ .

$ifail = 2$: No further improvement in the solution is possible. eps is too small: $eps = 〈value〉$ . The final value of x returned is an accurate approximation to the zero.

$ifail = 3$: The function values in the interval $[a, b]$ might contain a pole rather than a zero. Reducing eps may help in distinguishing between a pole and a zero.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

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 Parallelism and Performance

c05ayf is not threaded in any implementation.

9 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.