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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