BISECTION METHOD

The bisection method is a very simple method. It is the method to calculate the root of the function. The only disadvantage of the bisection method is that it is very slow for calculation. Because this method is very slow that is why it is used as a starting point to obtain the approximate value of the solution which is used later as a starting point. We have discussed in this article, the definition of the bisection method. The theorem related to the bisection method has been discussed in detail. We will also be talking about the algorithm workflow for any function f(x) by the bisection method. We will also come across the topic of absolute error. We will understand the definition of absolute error and also the theorem related to the more absolute error for the bisection method. 

BISECTION METHOD-

Bisection is the method to find the root. For any given function  

Theorem: if a function f(x) is continuous on an interval [a, b] and f(a). f(b) < 0, then the value c ϵ ( a, b) exists for which f(c) = 0. 

A function is said to be continuous when small changes in the input results in small changes in the result. In other words, we can say that if x changes in small proportion, f(x) also changes in small proportion. 

f(a). f(b) < 0 means that f(a) and f(b) have different signs, in which one of them is below x-axis and another above x-axis. Suppose that if you want to plot this on the graph, then f(x) at some point, will cross the x-axis. 

The bi-section method calculates the value of c for which the plot of the function f(x) crosses the x-axis. The value of c is the root of the function f(x). 

For any given function f(x), the step-by-step working for the bisection method is-

• Two values are a and b are calculated such that f(a) > 0 and f(b) < 0. 
• f(x) is calculated for the value c.
• C is the midpoint of a and b. So, c is the arithmetic mean. 
• f(c)= 0 , root of the function is c
• f(c) ≠ 0, then check the sign as f(c)

1. f(c) has the same sign as f(a). Here a is replaced with c and the value of b is the same.
2. f(c) has the same sign as f(b). Here, b is replaced with c and the value of a is the same. 

• Repeat the above method until f(c) becomes zero. 

   This bisection method algorithm is completed when the value of f(c) is less than the defined value. 

BISECTION METHOD MEANING-

The bisection method is used to find the roots of an equation. The intermediate theorem for the continuous function is the main principle behind the bisector method. This method takes into account the average of positive and negative intervals. The bisector method can also be called a binary search method, root-finding method, and dichotomy method. 

Let us talk about the absolute error. 

The bisection method never gives the exact solution of any given equation f(x)= 0. But you can calculate the absolute error.

Theorem: let f(x) be a continuous function on [a, b] in such a way that f(a) f(b) < 0. In the bisection method, after n iterations, xn be the midpoint in the nth subinterval [ an, bn]                                           xn=an+ bn2

There exists an exact value of the given function f(x) = 0 in the subinterval [ an, bn]. Hence the absolute error is given by                                                  xtrue–xn ≤b-a2n+1 

 You can rearrange the error to see the number of iterations required to guarantee absolute error than the required ϵ. 

                                                        b-a2n+1 < ϵ

                                                         b-a < 2n+1

                                            ln b-a <n+1ln (2)

                                             ln b-a ln 2 -1

Implementation of the bisection method-

Write a function f(x) which takes 4 input parameters and gives the approximation of a solution f(x)=0 by n number of iterations of the bisection method. 

Let us suppose if f (an) f bn≥0 at any point in the iteration, which is caused by a bad interval or rounding error in computations. Then you have to print ‘ Bisection method fails’ and return. 

Conclusion-

As discussed above, we have talked about the definition of the bisection method. This theorem of the bisection method applies to the continuous function. We have even talked about the step-by-step algorithm workflow of the bisection method. We know from the above article that the bisection method does not give the exact solution of any given function f(x). There is always a slight error in the approximate result. This slight error is referred to as absolute error. The bisector method can also be called a binary search method, root-finding method, and dichotomy method.