The Newton-Raphson method is an iterative technique used to find approximate roots of a real-valued function efficiently, particularly excelling in convergence for small problems. The method requires knowledge of the function's derivative and an initial guess but can struggle with poor global convergence properties and is sensitive to starting points. Despite some disadvantages, it offers rapid convergence and can significantly improve estimates found through other methods.

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