MATLAB

3. (Newton's method and the secant method) Consider the implementation of In(y) based on finding the root of f(x) = em y for given y > 0. (a) Implement the Newton's method to find the root of f(x). Test it with y= 1 and Xo = 1. Compute rn defined in the same way as equation (1) in problem 2. Uses this to show that Newton's method converges quadratically. (Notice that since en is too small, the computer may not be able to compute rn after several steps and may output rn as Inf or NaN. This is not a problem here, just focus on the values of rn for which the computer doesn't give you errors and see what the last valid rn is.) (b) Test Newton's method with y = 1000 and Xo = 1. How does the iteratives xk's behave for k not big? Write down the iterative relation between an+1 and xn and uses this relation to explain why Newton's method converges slowly. (c) Implement the secant method and test it with y = 1 and Xo = 2, X-1 = 1. Why does the iterative process have errors or give values like Inf or NaN after several steps? If we ignore these abnormal Ek's, does the method converge successfully