Convergence to a Fixed Point of a Function and Matlab Program for the Newton-Raphson Method.
1. Let g: R→R+ be such a function that g∈ C^1(R) and for all x ∈ R, -1 <g'(x) < 0.
Show that the sequence Xn+1 : = g(Xn) converges to the unique fixed point of the function g, regardless of chioce Xo ∈ R.
[ Note : Observe that the domain of function g is not a compact interval.]
2. Write a matlab program (Newton – Raphson ) for finding the root of the function f(x) = x^5 – 2x^3 + x + 2. Be as much accurate as you as you can. To proceed, store this function in an m-file, say f.m, and its derivative in df.m. Label the whole procedure newt.m.