What is the bisection method?

Finding the roots of equations can be tiresome. Sometimes, the process to find the exact roots of equations is very complex, and some equations can’t be solved using the usual methods.

There are two main types of graphical methods that can be used to approximate the roots of the equations. These types are:

Bracketing methods
Open methods

The bracketing category is further divided into two categories, including the bisection method. The bisection method, which is also known as the interval-halving or binary chopping method, is a method in which the interval is divided into half in each increment.

In the bisection method, the sign of the function is changed in the interval. The midpoint is then calculated and the interval in which the sign of the function changed is selected. The midpoint of the new interval then has the root of the equation. This process is repeated until we get very close to the actual root.

Algorithm

Select an interval in which you think the answer might lie. Label the lower limit as $xl$ and the upper limit as $xu$ .
The estimated root can be calculated with the following formula: $xr = (xl + xu)/2$ , where $xr$ is the new root.
To check whether the actual root lies in the interval $xl-xr$ or $xr-xu$ , use the following formula: $f(xl)*f(xr)$ .
If the answer to the above identity is negative, the root is present in this interval, and $xr$ becomes the new $xu$ . $xl$ remains the same. This increment is done and the process starts again.
If the answer to the above identity is positive, the root is present in the upper interval, and $xr$ becomes the new $xl$ . $xu$ remains the same. This increment is done and the process starts again.
If the answer to the above identity is zero, the root is equal to $xr$ . The process is terminated.

Code

import math
def func(x):
  func = 0.95*(pow(x,3))-5.9*(pow(x,2))+10.9*x-6 #Example equation
  return func
def bisection():
  ea = 100 #Absolute Error
  xl = 3   #Lower Limit
  xu = 4   #Upper Limit
  xold=0
  i=1
  while ea > 0.1:  #This loop will run until the absolute error becomes less than 0.1
    xr = (xu+xl)/ 2  #xr is the estimated root 
    ea = (abs(xr - xold) / xr) * 100 #Re-computing error. This takes the error from the previous estimation.
    sign_check=func(xr) * func(xl)
    if sign_check < 0: #Checking where the root lies (Mentioned in the Algorithm above)
      xu = xr
    else:
      xl = xr
    
    xold = xr #xold is used to store the previous value of xr in order to find absolute error
    print("Iteration",i)
    print("Absolute Error", ea)
    print("Root",xr)
    i=i+1
    
def main():
    bisection()
main()

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