Brent

Calculates the zeros of a function using Brent's method.

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Brent
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Interface

C++

Brent

doublebrent(	double	(`f`)(double)[function pointer]*
	double	`x1` `= -1E+7`
	double	`x2` `= 1E+7`
	double	`eps` `= 1E-10`
	int	`maxit` `= 1000`	)

Brent's method is a root finding algorithm which combines root bracketing, interval bisection and inverse quadratic interpolation. It is sometimes known as the van Wijngaarden-Deker-Brent method.

The main algorithm uses a Lagrange interpolating polynomial of degree 2. Brent (1973) claims that this method will always converge as long as the values of the function are computable within a given region containing a root. Given three points $\inline x_1$ , $\inline x_2$ and $\inline x_3$ , Brent's method fits x as a quadratic function of y, then uses the interpolation formula

$x = \sum_{k = 1}^3 x_k \prod_{1 \leq i \leq 3 \quad i \neq k} \frac {y - f(x_i)} {f(x_k) - f(x_i)}$

(2)

Subsequent root estimates are obtained by setting $\inline y = 0$ , giving

$x = x_2 + \frac {P} {Q}$

(3)

where

$P = S[T(R - T)(x_3 - x_2) - (1 - R)(x_2 - x_1)] \f] \f[ Q = (T - 1)(R - 1)(S - 1)$

(4)

with

$R = \frac {f(x_2)} {f(x_3)} \qquad S = \frac {f(x_2)} {f(x_1)} \qquad T = \frac {f(x_1)} {f(x_3)}$

(5)

This algorithm finds the roots of the user-defined function f starting with an initial interval [x1, x2] and iterating the sequence above until either the accuracy eps is achieved or the maximum number of iterations maxit is exceeded.

References:

Jean-Pierre Moreau's Home Page, http://perso.wanadoo.fr/jean-pierre.moreau/
MathWorld, http://mathworld.wolfram.com/BrentsMethod.html

Example 1

#include <codecogs/maths/rootfinding/brent.h>
 
#include <iostream>
#include <iomanip>
 
// user-defined function
double f(double x) {
    return (x + 1) * (x + 1) * (x + 1);
}
 
int main()  
{
  double x = Maths::RootFinding::brent(f, -3, 0);
 
  std::cout << "The calculated zero is X = " << std::setprecision(12) << x << std::endl;
  std::cout << "The associated ordinate value is Y = " << f(x) << std::endl;
  return 0;
}

Output:

The calculated zero is X = -0.999999878599
The associated ordinate value is Y = 1.78923630993e-021

Parameters

f	the user-defined function
x1	Default value = -1E+7
x2	Default value = 1E+7
eps	Default value = 1E-10
maxit	Default value = 1000

Authors

Lucian Bentea (August 2005)

Source Code

Source code is available when you buy a Commercial licence.

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