Question 1.5 -
Write programs to solve for the positive root of x 2 -5 using the Newton-Raphson and secant methods. Investigate the behaviour of the latter with changes in the initial guesses for the root.
Question 1.6 -
The function f(x)=tanh x has a root at x=0. Write a program to show that the Newton-Raphson method does not converge for an initial guess of x very much larger than 1. Can you understand what is going on by considering a graph of tanhx? From the explicit form of (1.14) for this problem, derive the critical value of the initial guess above which convergence will not occur? Try to solve th eproblem using the secant method. What happens for various initial guesses if you try to find the x =0 root of tanx using either method.
The extra material for Question 2 includes questions and the description of the Henon-Heiles model here