NAG Library Function Document

nag_zero_cont_func_brent (c05ayc)


    1  Purpose
    7  Accuracy


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.


#include <nag.h>
#include <nagc05.h>
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)


nag_zero_cont_func_brent (c05ayc) attempts to obtain an approximation to a simple zero of the function fx  given an initial interval a,b  such that fa × fb 0 .
The approximation x to the zero α is determined so that at least one of the following criteria is satisfied:
(i) x-α eps ,
(ii) fxeta .


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


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: ba .
3:     eps doubleInput
On entry: the termination tolerance on x (see Section 3).
Constraint: eps>0.0 .
4:     eta doubleInput
On entry: a value such that if fxeta , x is accepted as the zero. eta may be specified as 0.0 (see Section 7).
5:     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:     x doubleInput
On entry: the point at which the function must be evaluated.
2:     comm Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to f.
userdouble *
iuserInteger *
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 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:     x double *Output
On exit: if fail.code= NE_NOERROR or NE_TOO_SMALL, x is the final approximation to the zero. If fail.code= NE_PROBABLE_POLE, x is likely to be a pole of fx. Otherwise, x contains no useful information.
7:     comm Nag_Comm *
The NAG communication argument (see Section in How to Use the NAG Library and its Documentation).
8:     fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

Error Indicators and Warnings

Dynamic memory allocation failed.
See Section in How to Use the NAG Library and its Documentation for further information.
On entry, argument value had an illegal value.
On entry, fa and fb have the same sign with neither equalling 0.0: fa=value and fb=value.
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.
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.
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.
On entry, eps=value.
Constraint: eps>0.0.
On entry, a=value and b=value.
Constraint: ab.
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.


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 fail.code= NE_TOO_SMALL, 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 fx  for values of x near the zero.

Parallelism and Performance

nag_zero_cont_func_brent (c05ayc) is not threaded in any implementation.

Further Comments

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


This example calculates an approximation to the zero of e-x - x  within the interval 0,1  using a tolerance of eps=1.0e−5.

Program Text

Program Text (c05ayce.c)

Program Data


Program Results

Program Results (c05ayce.r)

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