# NAG Library Function Document

## 1Purpose

nag_zero_cont_func_brent (c05ayc) 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.

## 2Specification

 #include #include
void  nag_zero_cont_func_brent (double a, double b, double eps, double eta,
 double (*f)(double x, Nag_Comm *comm),
double *x, Nag_Comm *comm, NagError *fail)

## 3Description

nag_zero_cont_func_brent (c05ayc) 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) $\left|x-\alpha \right|\le {\mathbf{eps}}$, (ii) $\left|f\left(x\right)\right|\le {\mathbf{eta}}$.

## 4References

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

## 5Arguments

1:    $\mathbf{a}$doubleInput
On entry: $a$, the lower bound of the interval.
2:    $\mathbf{b}$doubleInput
On entry: $b$, the upper bound of the interval.
Constraint: ${\mathbf{b}}\ne {\mathbf{a}}$.
3:    $\mathbf{eps}$doubleInput
On entry: the termination tolerance on $x$ (see Section 3).
Constraint: ${\mathbf{eps}}>0.0$.
4:    $\mathbf{eta}$doubleInput
On entry: a value such that if $\left|f\left(x\right)\right|\le {\mathbf{eta}}$, $x$ is accepted as the zero. eta may be specified as $0.0$ (see Section 7).
5:    $\mathbf{f}$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 x, Nag_Comm *comm)
1:    $\mathbf{x}$doubleInput
On entry: the point at which the function must be evaluated.
2:    $\mathbf{comm}$Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to f.
userdouble *
iuserInteger *
pPointer
The type Pointer will be void *. Before calling nag_zero_cont_func_brent (c05ayc) you may allocate memory and initialize these pointers with various quantities for use by f when called from nag_zero_cont_func_brent (c05ayc) (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
Note: f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by nag_zero_cont_func_brent (c05ayc). If your code inadvertently does return any NaNs or infinities, nag_zero_cont_func_brent (c05ayc) is likely to produce unexpected results.
6:    $\mathbf{x}$double *Output
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR or NE_TOO_SMALL, x is the final approximation to the zero. If ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_PROBABLE_POLE, x is likely to be a pole of $f\left(x\right)$. Otherwise, x contains no useful information.
7:    $\mathbf{comm}$Nag_Comm *
The NAG communication argument (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
8:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
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_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_PROBABLE_POLE
The function values in the interval $\left[{\mathbf{a}},{\mathbf{b}}\right]$ might contain a pole rather than a zero. Reducing eps may help in distinguishing between a pole and a zero.
NE_REAL
On entry, ${\mathbf{eps}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{eps}}>0.0$.
NE_REAL_2
On entry, ${\mathbf{a}}=〈\mathit{\text{value}}〉$ and ${\mathbf{b}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{a}}\ne {\mathbf{b}}$.
NE_TOO_SMALL
No further improvement in the solution is possible. eps is too small: ${\mathbf{eps}}=〈\mathit{\text{value}}〉$. The final value of x returned is an accurate approximation to the zero.

## 7Accuracy

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

## 8Parallelism and Performance

nag_zero_cont_func_brent (c05ayc) is not threaded in any implementation.

The time taken by nag_zero_cont_func_brent (c05ayc) depends primarily on the time spent evaluating f (see Section 5).

## 10Example

This example calculates an approximation to the zero of ${e}^{-x}-x$ within the interval $\left[0,1\right]$ using a tolerance of ${\mathbf{eps}}=\text{1.0e−5}$.

### 10.1Program Text

Program Text (c05ayce.c)

None.

### 10.3Program Results

Program Results (c05ayce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017