Question

6) Use MATLAB and Newton-Raphson method to find the roots of the function, f(x) = x-exp (0.5x) and define the function as well as its derivative like so, fa@(x)x^2-exp(.5%), f primea@(x) 2*x-.5*x"exp(.5%) For each iteration, keep the x values and use 3 initial values between -10 & 10 to find more than one root. Plot each function for x with respect to the iteration #.

Answer 1

clear
x0=9;%this is the initial guess, change it accordingly
ini=x0;
f=@(x)x.^2-exp(.5*x);
f_prime=@(x)2*x-.5*x*exp(.5*x);
i=0;
while abs(f(x0))>.0000001%finding root upto 6 decimal points
    x0=x0-f(x0)/f_prime(x0);
    i=i+1;
    x(i)=x0;
end
fprintf('For initial guess:%.2f,Root found=%f\n',ini,x0);
figure
plot(1:i,f(x))
title(['Initial guess: ' num2str(ini)])

Command Window >>NewtonRaphson3v For initial guess:-2.00, Root found--0.815553 >>NewtonRaphson3v For initial guess:2.00, Root found-1. 429612 >>NewtonRaphson3v For initial guess:9.00, Root found-9.613169 Figure 1 Figure 2 Figure 3 File Edit View Insert Tools Desktop Window Help File Edit View Insert Tools Desktop Window Help File Edit View Insert Tools Desktop Window Help Initial guess: -2 Initial guess: 2 Initial guess: 9 0.4 0.4 0.2 0.3 0.2 0.2 0.1 0.4 .8 -0.6 10 0.1 -0.8 50 100 150 250 300 10 20 10