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

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 x0
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
o One of the fastest convergences to the root.
o Converges on the root quadratic.
o Near a root, the number of significant digits
approximately doubles with each step.
o This leads to the ability of the Newton-Raphson
Method to “polish” a root from another
convergences technique.
o Easy to convert to multiple dimensions.
o Can be used to “polish” a root found by other
methods.

5. Disadvantages of
Newton-Raphson Method
o Must find the derivative
o Poor global convergence properties
o 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