Question

prove that newton-raphson method has second order convergence.

Answer 1

Convergence of Newton-Raphson method:

Suppose  is a root of  and  is an estimate of  s.t. . Then by Taylor series expansion we have,



for some  between  and . 
By Newton-Raphson method, we know that



i.e.



Using(2*) in (1*) we get



Say 

where  denote the error in the solution at n and (n+1) iterations.





 Newton Raphson Method is said to have quadratic convergence. 

Note: 
Alternatively, one can also prove the quadratic convergence of Newton-Raphson method based on the fixed - point theory. It is worth stating few comments on this approach as it is a more general approach covering most of the iteration schemes discussed earlier. 

A Brief discussion on Fixed Point Iteration: 
Suppose that we are given a function
on an interval  for which we need to find a root. Derive , from it, an equation of the form:



Any solution to (ii) is called a fixed point and it is a solution of (i). The function g(x) is called as "Iteration function". 

Example: 
Given , one may re-write it as:



or , 

or , 

where g(x) denotes possible choice iteration function.

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