Bisection

Interface

Bisection

The simplest root-finding algorithm is the bisection method: we start with two points a and b which bracket a root, and at every iteration we pick either the subinterval or , where is the midpoint between a and b. The algorithm always selects a subinterval which contains a root. It is guaranteed to converge to a root, however its progress is rather slow (the rate of convergence is linear).

To give you a better idea on the way this method works, the following graph shows different iterations in the approximation process. Here is the associated list of pairs chosen at consecutive steps

This algorithm finds the roots of the user-defined function f starting with an initial interval [x0, x1] and iterating until the accuracy eps is achieved.

References:

• Jean-Pierre Moreau's Home Page, http://perso.wanadoo.fr/jean-pierre.moreau/
• F.R. Ruckdeschel, "BASIC Scientific Subroutines", Vol. II, BYTE/McGRAWW-HILL, 1981
• Wikipedia, http://en.wikipedia.org/wiki/Root-finding_algorithm

Example 1

#include <codecogs/maths/rootfinding/bisection.h>
 
#include <iostream>
#include <iomanip>
 
// user-defined function
double f(double x) {
    return (x - 2) * (x + 1) * (x + 10);
}
 
int main() 
{
    double x = Maths::RootFinding::bisection(f, -2, 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 = -1
The associated ordinate value is Y = 0

Parameters

f	the user-defined function
x0	Default value = -1E+7
x1	Default value = 1E+7
eps	Default value = 1E-10

Authors

Lucian Bentea (August 2005)

Source Code

Source code is available when you agree to a GP Licence or buy a Commercial Licence.

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