NAG Library Function Document

nag_zero_cont_func_bd (c05adc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

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

nag_zero_cont_func_bd (c05adc) locates a zero of a continuous function in a given interval by a combination of the methods of nonlinear interpolation, linear extrapolation and bisection.

2 Specification

#include <nag.h>

#include <nagc05.h>

void

nag_zero_cont_func_bd (double a, double b, double *x,

double

(*f)(double xx),

double xtol, double ftol, NagError *fail)

3 Description

nag_zero_cont_func_bd (c05adc) 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 approximation

x

to the zero

α

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

(i)	$\|x - α\| \leq xtol$ ,
(ii)	$\|f (x)\| \leq ftol$ .

4 References

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

5 Arguments

1: a – doubleInput

On entry:

a

, the lower bound of the interval.

2: b – doubleInput

On entry:

b

, the upper bound of the interval.

Constraint:

b \neq a

3: x – double *Output

On exit: the approximation to the zero.

4: f – function, supplied by the userExternal Function

f must evaluate the function

f

whose zero is to be determined.

The specification of f is:

double

f (double xx)

1: xx – doubleInput: On entry: the point at which the function must be evaluated.

5: xtol – doubleInput

On entry: the termination tolerance on

x

(see Section 3).

Constraint:

xtol > 0.0

6: ftol – doubleInput

On entry: a value such that if

|f (x)| \leq ftol

x

is accepted as the zero. ftol may be specified as

0.0

(see Section 7).

7: fail – NagError *Input/Output

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

6 Error Indicators and Warnings

NE_2_REAL_ARG_EQ: On entry, $a = 〈value〉$ and $b = 〈value〉$ .
Constraint: $a \neq b$ .
NE_FUNC_END_VAL: On entry, $f (a)$ and $f (b)$ have the same sign with neither equalling $0.0$ : $f (a) = 〈value〉$ and $f (b) = 〈value〉$ .
NE_PROBABLE_POLE: The function values in the interval $[a, b]$ might contain a pole rather than a zero. Reducing xtol may help in distinguishing between a pole and a zero.
NE_REAL_ARG_LE: On entry, $xtol = 〈value〉$ .
Constraint: $xtol > 0.0$ .
NE_XTOL_TOO_SMALL: No further improvement in the solution is possible. xtol is too small: $xtol = 〈value〉$ .

7 Accuracy

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

fail . code =

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

ftol = 0.0

and to use xtol 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 nag_zero_cont_func_bd (c05adc) depends primarily on the time spent evaluating f (see Section 5).

9 Example

This example calculates an approximation to the zero of

e^{- x} - x

within the interval

[0, 1]

using a tolerance of

xtol = 1.0e−5

9.1 Program Text

Program Text (c05adce.c)

9.2 Program Data

None.

9.3 Program Results

Program Results (c05adce.r)

nag_zero_cont_func_bd (c05adc) (PDF version)

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NAG C Library Manual

NAG Library Function Documentnag_zero_cont_func_bd (c05adc)