Introduction to Iterative methods: There are number of iterative methods like Jacobi method, Gauss–Seidel method that has been tried and used successfully in various problem situations. All these methods typically generate a sequence of estimates of the solution which is expected to converge to the true solution. Newton-Raphson method is also one of the iterative methods which are used to find the roots of given expression.

### Newton-Raphson Method with MATLAB code:

If point x0 is close to the root a, then a tangent line to the graph of f(x) at x0 is a good approximation the f(x) near a. So the root of the tangent line, where the line cuts the X-axis; x1 is the better approximation to a than x0 is.

Slope of the tangent =

Therefore

Repeating process, we obtain a better approximation,

Continue in this way. If xn is the current estimate, then the next estimate xn+1 is given by

if x0 is sufficiently close to a, xn?a as n?8.

## Limitations of Newton-Raphson Method

- If initial guess is too far away from the required root, the process may converge to some other root.
- Division by zero may occur if f’(xi) is zero or very close to zero.
- A particular value in the iteration sequence may repeat, resulting in an infinite loop.

## Newton-Raphson MATLAB program:

% Newton Raphson Method clear all close all clc % Change here for different functions f=@(x) cos(x)-3*x+1 %this is the derivative of the above function df=@(x) -sin(x)-3 % Change lower limit 'a' and upper limit 'b' a=0; b=1; x=a; for i=1:1:100 x1=x-(f(x)/df(x)); x=x1; end sol=x; fprintf('Approximate Root is %.15f',sol) a=0;b=1; x=a; er(5)=0; for i=1:1:5 x1=x-(f(x)/df(x)); x=x1; er(i)=x1-sol; end plot(er) xlabel('Number of iterations') ylabel('Error') title('Error Vs. Number of iterations')

ANSWER :

f =

@(x)cos(x)-3*x+1

df =

@(x)-sin(x)-3

Approximate Root is 0.6071016481031231

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