Show that the initial guesses 0,1, and 2 lead to a fixed point in Exercise 21. What happens to other initial guesses close to those numbers?
Consider Fixed-Point Iteration applied to (a) Show that 1 − , 1, and 1 + are fixed points. (b) Show that none of the three fixed points is locally convergent. (Computer Problem 7 investigates this example further.)
Computer Problem 7
considered Fixed-Point Iteration applied to g(x) = 1 − 5x + − = x. Find initial guesses for which FPI (a) cycles endlessly through numbers in the interval (0,1) (b) the same as (a), but the interval is (1,2) (c) diverges to infinity. Cases (a) and (b) are examples of chaotic dynamics. In all three cases, FPI is unsuccessful.