For those who are unaware, Newton Raphson method fondly known as NR method is an iterative method for finding the solution or roots of equations arising from the differential equations. I studied Newton Raphson method in 3 of my courses namely “Applied Numerical Methods”, “Optimization and Soft Computing” and “Power System Analysis”.

To learn more about Newton Raphson method, you can refer various literature texts available online and offline.

A concise explanation of NR method is as follows:

- Consider a function f(x), say f(x) = x^2 - 5
- To find the approximate root of f(x), we will assume an initial solution of f(x). Let us consider x=1
- Then we need to find out the amount of accuracy required in the approximations. Let us consider it as ‘e’.
- The formula of NR method is given by “x(n+1) = x(n) - (f(x)/f'(x))”. Using this relation, the value of ‘x’ is calculated in each iteration and the difference between the current and previous values of ‘x’ is estimated.
- After every iteration, the difference is compared with the accuracy value.
- If the difference is less than the accuracy value, then iteration is stopped and the current value of x is considered as the approximate root of the solution.
- If the difference remains larger than the accuracy value even after a large number of iteration, then the solution is said to be non-converging.

The MATLAB implementation of Newton Raphson Method is given below:

```
%***** MATLAB PROGRAM FOR NEWTON RAPHSON METHOD****!
%*************************************************!
%
% PROGRAMMED BY: AMIT BISWAL
% URL : http://amitbiswal.blogspot.com
%
%*************************************************!
%---------NEWTON RAPHSON METHOD FORMULA-----------!
%
% x(n+1) = x(n) - (f(x)/f'(x))
%
% ************************************************!
close all
clc
clear all
syms x
%---START OF FUNCTION-----------!
f= (x^2)+ (2*x) - 10
%---END OF FUNCTION-------------!
fdash=diff(f);
x=input('Enter the initial solution, x0=');
e=input('Enter the value of error, e=');
iteration=0;
error=1;
while (error>e)
iteration=iteration+1;
fprintf('********************Iteration no. %0.0f********************',iteration)
error=abs(x);
x=x-subs(f/fdash)
error=abs(x-error)
if(iteration>100) %exit condition for non-coverging solution
break
end
end
fprintf('*************************RESULT****************************');
if(iteration>100)
fprintf('\nSOLUTION NOT CONVERGING !\n TRY WITH DIFFERENT INITIAL SOLUTION/ERROR')
else
fprintf('\nSolution x= %0.20f\nNumber of Iteration= %0.0f',x,iteration);
end
fprintf('\n***********************************************************');
```

If you have any query or issue with the code then you can comment below. Thanks for reading this post.