Table of Contents

## Bisection Method

**The bisection method** is an Algorithm or an** Iterative Method** for finding the roots of a **Non-Linear** equation.

The convergence in the bisection method is linear which is slow as compared to the other Iterative methods.

However, it is the **simplest method and it never fails**.

## Rule | Method

This method is actually using Intermediate Value Property repeatedly.

If a function **f(x)** is continuous in a closed interval **[a,b]** and **f(a)** and **f(b)** have opposite sign.

Then The root lies between **a **and **b** and the first approximation of the root is **x1=(a+b)/2**.

Now the root lies between **a** and **x1** or **x1** and **b **accordingly if **f(a)** and **f(x1) have an **opposite sign or **f(b)** and **f(x1)** have opposite signs respectively.

Let the root lies between **a **and **x1**, then we again bisect the interval to find the next approximation of the root i.e. **x2=(a+x1)/2,** and continue the process until the root is found to the desired accuracy.

In the above figure the** f(x1)** is positive and **f(x0)** is negative so the root lies between x1 and x0.

Then we bisect the interval and find x2 and f(x2) is also positive so the root lies between x0 and x2 , and we find x3 and so on ..

## Bisection Method Example

### Find the Root of the equation x^3 – 4x – 9 = 0, using the bisection method correct to three decimal places.

Sol. Let **f(x) = x ^{3} – 4x – 9**

**Some Observations**

Since every interval is half of its previous interval, i.e in each step the length of interval is reduced by a factor of 1/2.

So the length of nth interval will be (b-a)/x^n. let ‘e’ be the required accuracy, then *(b-a)/x^n ≤ε*. taking log on both sides we get ‘n ≥[log(b-a) – log*ε*] / log2. This gives the number of iterations required for achieving an accuracy *ε*.

**MCQ**: * The convergence in the bisection method is linear*.

## Bisection Method C++ code

```
//Biscection method c++
//wikkihut.com
#include<iostream>
#include<math.h> //used for fabs() function.
#include<iomanip>//used for setw() and setprecision()
//they are used to just manipulate the output.
using namespace std;
//function definition
//it calculates the value of xsinx-1 for different values of x.
float f(float x)
{
return x*sin(x)-1;
}
//bisects the interval and counts the number of iterations
//by incrementing the value of itr.
void bisect(float *x,float a,float b,int*itr)
{
*x=(a+b)/2;
++(*itr);
cout<<"iteration no. "<<*itr<< " X = "<<setw(3)<< setprecision(5)<< *x<<endl;
}
int main()
{
int itr=0,maxitr;
float x,a,b,aerr,x1;
cout<<"Enter the values of a and b , allowed error, maximum iterations"<<endl;
cin>>a>>b>>aerr>>maxitr;
cout<<fixed;
bisect(&x,a,b,&itr);
do
{
if(f(a)*f(x)<0)
b=x;
else
a=x;
bisect(&x1,a,b,&itr);
if(fabs(x1-x)<aerr)//fabs() calculate the absolute value of (x1-x).
{
cout<<"After "<< itr <<" iterations, root"<< " = "<<setw(6)<< setprecision(4)<<x1<<endl;
return 0;
}
x=x1;
}while(itr<maxitr);
cout<<"Solution does not converege,"<<"iterations not sufficient"<<endl;
return 0;
}
```

Code language: PHP (php)

*Suggested Read: Related Numerical Methods*