The document discusses the bisection method for finding roots of equations. It begins by defining the bisection method as a root finding technique that repeatedly bisects an interval and selects a subinterval containing the root. It notes that while simple and robust, the bisection method converges slowly. The document then provides the step-by-step algorithm for implementing the bisection method and works through an example of finding the root of f(x) = x^2 - 2 between 1 and 2. It concludes by presenting the bisection method code in C++.

1. Prepared by
Md. Mujahid Islam
Md. Rafiqul Islam
Khaza Fahmida Akter

2. The bisection method in mathematics is
a root finding method which repeatedly bisects
an interval and then selects a subinterval in
which a root must lie for further processing.

3. It is a very simple and robust method, but it is
also relatively slow. Because of this, it is often
used to obtain a rough approximation to a
solution which is then used as a starting point
for more rapidly converging methods .The
method is also called the binary search method
or the dichotomy method.

4. Step 1: Choose two approximations A and B
(B>A) such that
f(A)*f(B)<0
Step 2: Evaluate the midpoint C of [A,B] given
by
C=(A+B)/2

5. • Step 3: If f(C)*f(B)<0 then rename B & C as A &
B. If not rename of C as B . Then apply the
formula of Step 2.
• Step 4:Stop evolution when the different of two
successive values of C obtained from Step 2 is
numerically less than E, the prescribed accuracy
.

7. Given that, f (x) = x2 - 2. Our task is finding the
root of this equation.
Solution:
Let us start with an interval of length one: a0 = 1
and b1 = 2. Note that f (a0) = f(1) = - 1 < 0,
and f (b0) = f (2) = 2 > 0. Here are the first 20
applications of the bisection algorithm:

9. #include<iostream>
#include<cmath>
using namespace std;
#define ESP 0.0000001
#define f(x) x*x-2
int main(){
double x1,x2,x3,A,B,C;
cin>>x1>>x2;
int i=1;
do{
x3=(x1+x2)/2;
A=f(x1);B=f(x2);C=f(x3);
if(A*C<0){x2=x3;}
else if(C*B){x1=x3;}i++;}while(fabs(C)>ESP);cout<<x3;}

10. • Here we define variable type & function type.
• Input two values which are lowest & height value
of the function .Then we pass it through
function for manipulation .
• Function return result until it’s condition fulfill.
• Finally we get result .

11. • Definition of Bisection method & Basic
information of Bisection method .
• Algorithm of Bisection method .
• Calculate of root by using by section method
with help of important example .
• Representing in programming Language .

12. Any Question ?
Please Give your suggestion or advice which is
very helpful to create a better presentation .