Discuss the relationships between the Bisection Method fixed

Discuss the relationships between the Bisection Method, fixed-point methods, and Newton\'s Method. How fast do the methods converge to the solution to the problem? Which method is best and under what circumstances? What are the differences and similarities between the methods?

Solution

All the three algorithms helps in finding the root of the equation

a) Bisection Method works on the principle that if f(a)f(b) <0 and b>a, then there means there exists a root between a and b

In fixed point iteration,we find a method such that f(p)=p

In newton raphson method, we start with an initial approximation and then reduced the solution region by checking the accuracy of the solution

b) The methods bisection will converge in logN steps, since it will divide each iteration by half, fixed point iterations will be faster alternative and for newton raphson it depends on the choice of the initial approximation, if initial approximation is too vague then it will take a lot of steps to converge

c) If the initial approximation is appropiate, then the newton raphson will be the fastest method to converge