Use the following pseudocode for the Newton-Raphson method to write MATLAB code to approximate the cube root (a)^1/3 of a given number a with accuracy roughly within 10^-8 using x0 = a/2. Use at most 100 iterations. Explain steps by commenting on them. Use f(x) = x3 a. Choose a = 2 + w, where w = 3 Algorithm : Newton-Raphson Iteration Input: f(x)=x³a, x₀ =a/2, tolerance 10^-8, maximum number of iterations100 Output: an approximation (a)^1/3 within 10^-8 or a message of failure set x = x₀, xold = x₀;

for i = 1 to 100 do x = x f(x)/f(x); if |x xold| < 10^-8 then. % checking required accuracy FoundSolution = true; % done break; % leave for environment end if xold = x; % update xold for the next iteration end for if FoundSolution then print x, The number of iterations =, i else print The required accuracy is not reached in 100 iterations end if For f, define the function df(x) = 3x² by using the command

df = @(x) 3 x.²;