Newton Raphson Method

Sllid includes following content:
1)Definition
2) Features
3) Advantages And Disadvantages
4) Method Algo
5) Solved example

1. Newton Raphson Method

2. NEWTON RAPHSON METHOD Newton- Raphson method, is a method for finding successively better approximations to the roots(or zeroes) of a real-valued function. X:f(x)=0 . The Newton- Raphson method in one variable is implemented as follows: The method starts with a function f defined over the real no. x ,the function’s derivative f, and an initial guess x for a root of the function satisfies the assumptions made in the derivation of the formula and the initial guess is close, then a better approximation x1 is x1=x0-f(x0)/ f’(x0)

3. Features of Newton- Raphson method Efficient for small molecules, converges quickly Calculation and inversion is computationally difficult for large molecules Approximation of quadratic surface poor, particularly far from minimum Order of convergence is 2

4. Advantages of Newton- Raphson Method One of the fastest convergences to the root. Converges on the root quadratic. Near a root, the number of significant digits approximately doubles with each step. This leads to the ability of the Newton- Raphson Method to “polish” a root from another convergences technique. Easy to convert to multiple dimensions. Can be used to “polish” a root found by other methods.

5. Disadvantages of Newton- Raphson Method Must find the derivative Poor global convergence properties Dependent on initial guess May be too far from local root May encounter a zero derivative May loop indefinitely

6. Newton- Raphson Method Steps involved in the method Differentiate f(x) to find f’(x) Substitute f(x) and f’(x) into formula Choose a suitable starting value for x0

7. Example.. Find the root of the equation using Newton- Raphson method x3-2x-5=0. f(x)=x3-2x-5 f’(x)=3x2-2 xn+1=xn-x3n-2xn-5/3x2n-2 Choose x0=2,we obtain f(x0)=-1and f’(x0)=10,now Putting n=0 ,we get x1=2.1 Now finding x2 by following the same procedure , as a result we get x2=2.0945 This completes the two iterations of Newton- Raphson method