Given a function f(x) on floating number x and two numbers ‘a’ and ‘b’ such that f(a)*f(b) < 0 and f(x) is continuous in [a, b]. Here f(x) represents algebraic or transcendental equation. Find root of function in interval [a, b] (Or find a value of x such that f(x) is 0).

Input: A function of x, for example x3 – x2 + 2.     
       And two values: a = -200 and b = 300 such that
       f(a)*f(b) < 0, i.e., f(a) and f(b) have
       opposite signs.
Output: The value of root is : -1.00
        OR any other value close to root.

We strongly recommend to refer below post as a prerequisite of this post.

In this post The Method Of False Position is discussed. This method is also known as Regula Falsi or The Method of Chords.

Similarities with Bisection Method:

1. Same Assumptions: This method also assumes that function is continuous in [a, b] and given two numbers ‘a’ and ‘b’ are such that f(a) * f(b) < 0.
2. Always Converges: like Bisection, it always converges, usually considerably faster than Bisection–but sometimes very much more slowly than Bisection.

Differences with Bisection Method:
It differs in the fact that we make a chord joining the two points [a, f(a)] and [b, f(b)]. We consider the point at which the chord touches the x axis and named it as c.

Steps:

1. Write equation of the line connecting the two points.

   y – f(a) =  ( (f(b)-f(a))/(b-a) )*(x-a)
   
   Now we have to find the point which touches x axis. 
   For that we put y = 0.
   
   so x = a - (f(a)/(f(b)-f(a))) * (b-a)
      x = (a*f(b) - b*f(a)) / (f(b)-f(a)) 
   
   This will be our c that is c = x.

2. If f(c) == 0, then c is the root of the solution.
3. Else f(c) != 0
   1. If value f(a)*f(c) < 0 then root lies between a and c. So we recur for a and c
   2. Else If f(b)*f(c) < 0 then root lies between b and c. So we recur b and c.
   3. Else given function doesn’t follow one of assumptions.

Since root may be a floating point number and may converge very slow in worst case, we iterate for a very large number of times such that the answer becomes closer to the root.

Following is C++ implementation.

// C++ program for implementation of Bisection Method for
// solving equations
#include<bits/stdc++.h>
using namespace std;
#define MAX_ITER 1000000
 
// An example function whose solution is determined using
// Bisection Method. The function is x^3 - x^2  + 2
double func(double x)
{
    return x*x*x - x*x + 2;
}
 
// Prints root of func(x) in interval [a, b]
void regulaFalsi(double a, double b)
{
    if (func(a) * func(b) >= 0)
    {
        cout << "You have not assumed right a and b\n";
        return;
    }
 
    double c = a;  // Initialize result
 
    for (int i=0; i < MAX_ITER; i++)
    {
        // Find the point that touches x axis
        c = (a*func(b) - b*func(a))/ (func(b) - func(a));
 
        // Check if the above found point is root
        if (func(c)==0)
            break;
 
        // Decide the side to repeat the steps
        else if (func(c)*func(a) < 0)
            b = c;
        else
            a = c;
    }
    cout << "The value of root is : " << c;
}
 
// Driver program to test above function
int main()
{
    // Initial values assumed
    double a =-200, b = 300;
    regulaFalsi(a, b);
    return 0;
}

Output:

The value of root is : -1

This method always converges, usually considerably faster than Bisection. But worst case can be very slow.