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

(1)

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

#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;
}

The calculated zero is X = -1
The associated ordinate value is Y = 0

Parameters