For Wednesday, September 29

Consider the Fibonacci sequence defined by recurrence equations (3.22) on page 59 of our textbook.

1. Write, in Scheme, a straightforward recursive procedure fibthat takes any natural number as argument and returns the number in the specified (zero-based) position in the Fibonacci sequence. Test your code. Express the number of recursive calls required to compute (fib n) as a function of n.
2. Write a memoized recursive version of the fib procedure, using a vector to hold previously computed results. Again, express the number of recursive calls required to compute (fib n) as a function of n.
3. Write an iterative (i.e., tail-recursive) version of the fib procedure, still using a vector to hold previously computed results, but filling up the vector in a plausible order.
4. All of the preceding solutions use Θ(n) memory locations to compute (fib n). Write an iterative version of fib that works in Θ(1) space.