Question

B. Implement the Newton-Raphson (NR) method for solving nonlinear equations in one dimension. The program should be started from a script M-file. Prompt the user to enter an initial guess for the root. -Use an error tolerance of 107, -Allow at most 1000 iterations. .The code should be fully commented and clear 2. a) Use your NR code to find the positive root of the equation given below using the following points as initial guesses: xo = 4, 0 and-1 Plot the root and the approximate error as a function of the iteration number. Report the estimated root for each case, describe the differences in the behavior depending on the initial guess and discuss the results. o 8000x - 500x3+ 4000 0 b) Verify your NR code using the graphical approach.

Answer 1

`Hey,

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clc
clear all
close all

f = @(x) 8000*x-500*x^3+4000;
err=[];
R=[];
g=@(x) 8000-1500*x^2;
x0=input('Enter initial guess: ');
e=1e-7;
maxit=1000;
x1=x0;
x0=0;
N=0;
err=[];
while 1

if abs(f(x1))>e
x0=x1;
x1=x0-(f(x0)/g(x0));
N=N+1;
R(N)=x1;
err(N)=abs(f(x1));

else

break;

end
if(N==maxit)
break;
end

end
root=x1;
iter=N;
disp('So, final approximation is')
(root)
plot(err);
title('Plot of absolute error')
figure;
plot(R);
title('Plot of root')
figure;
fplot(f,[-2+x1,x1+2]);
hold on;
plot(x1,f(x1),'*r');
title('Plot to verify root');

MATLAB R2018a EDITOR Figure Search Documentation Log In File Edit View Insert Tools Desktop Window Help Figure File Edit View Insert Tools Desktop Window Help Window Help Plot of absolute error 400 Plot of root to verify root 4.25 350 300 4.245 4.24 150 4.235 100 50 4.23 2.5 3.5 4.225 2.5 3.5 2.5 3.5 4.5 5.5 @(x)800%-500"r ^3.. eb)8000-1500xA2 maxit 1000 [4.2500,4.2300,4.2298,. root ENG 4:22 PM O Type here to search

Kindly revert for any queries

Thanks.