c05 Chapter Contents
c05 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_zero_cont_func_bd (c05adc)

## 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 #include
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 $f\left(x\right)$ given an initial interval $\left[a,b\right]$ such that $f\left(a\right)×f\left(b\right)\le 0$.
The approximation $x$ to the zero $\alpha$ is determined so that at least one of the following criteria is satisfied:
 (i) $\left|x-\alpha \right|\le {\mathbf{xtol}}$, (ii) $\left|f\left(x\right)\right|\le {\mathbf{ftol}}$.

## 4  References

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

## 5  Arguments

On entry: $a$, the lower bound of the interval.
2:     bdoubleInput
On entry: $b$, the upper bound of the interval.
Constraint: ${\mathbf{b}}\ne {\mathbf{a}}$.
3:     xdouble *Output
On exit: the approximation to the zero.
4:     ffunction, 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:     xxdoubleInput
On entry: the point at which the function must be evaluated.
5:     xtoldoubleInput
On entry: the termination tolerance on $x$ (see Section 3).
Constraint: ${\mathbf{xtol}}>0.0$.
6:     ftoldoubleInput
On entry: a value such that if $\left|f\left(x\right)\right|\le {\mathbf{ftol}}$, $x$ is accepted as the zero. ftol may be specified as $0.0$ (see Section 7).
7:     failNagError *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, ${\mathbf{a}}=〈\mathit{\text{value}}〉$ and ${\mathbf{b}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{a}}\ne {\mathbf{b}}$.
NE_FUNC_END_VAL
On entry, ${\mathbf{f}}\left({\mathbf{a}}\right)$ and ${\mathbf{f}}\left({\mathbf{b}}\right)$ have the same sign with neither equalling $0.0$: ${\mathbf{f}}\left({\mathbf{a}}\right)=〈\mathit{\text{value}}〉$ and ${\mathbf{f}}\left({\mathbf{b}}\right)=〈\mathit{\text{value}}〉$.
NE_PROBABLE_POLE
The function values in the interval $\left[{\mathbf{a}},{\mathbf{b}}\right]$ 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, ${\mathbf{xtol}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{xtol}}>0.0$.
NE_XTOL_TOO_SMALL
No further improvement in the solution is possible. xtol is too small: ${\mathbf{xtol}}=〈\mathit{\text{value}}〉$.

## 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 NE_XTOL_TOO_SMALL, although this may involve many more iterations than a lesser accuracy. You are recommended to set ${\mathbf{ftol}}=0.0$ and to use xtol to control the accuracy, unless you have considerable knowledge of the size of $f\left(x\right)$ for values of $x$ near the zero.

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 $\left[0,1\right]$ using a tolerance of ${\mathbf{xtol}}=\text{1.0e−5}$.

None.