Hint: Using the observation that (bn/2)2=(b2)n/2, keep along with the exponent n and the base b, an additional state variable a, and define the state transformation in such a way that the product abn is unchanged from state to state. At the beginning of the process, a is taken to be 1, and the answer is given by the value of a at the end of the process. In general, the technique of defining an invariant quantity that remains unchanged from state to state is a powerful way to think about the design of iterative algorithms.
For reference, the recursive version of fast-exp is:
(define (fast-exp b n) (cond ((= n 0) 1) ((even? n) (square (fast-exp b (/ n 2)))) (else (* b (fast-exp b (- n 1))))))
a <- a + b b <- aCall this transformation T. Observe that applying T over and over again n times, starting with 1 and 0, produces the pair Fib(n+1) and Fib(n). In other words, the Fibonacci numbers are produced by applying Tn, the nth power of the transformation T, starting with the pair (1,0).
Now consider T to be the special case of p=0 and q=1 in a family of transformations Tpq, where Tpq transforms the pair (a,b) according to:
a <- bq + aq + ap b <- bp + aqShow that if we apply such a transformation Tpq twice, the effect is the same as using a single transformation Tp'q' of the same form, and compute p' and q' in terms of p and q.
This gives us an explicit way to square these transformations, and thus we
can compute Tn using successive squaring, as in the
fast-exp procedure. Put this all together to complete the following
procedure, which runs in logarithmic number of steps:
(define (fib n) (fib-iter 1 0 0 1 n)) (define (fib-iter a b p q count) (cond ((= count 0) b) ((even? count) (fib-iter a b <??> ;; compute p' <??> ;; compute q' (/ count 2))) (else (fib-iter (+ (* b q) (* a q) (* a p)) (+ (* b p) (* a q)) p q (- count 1)))))
(define (sum term a next b) (if (< b a) 0 (+ (term a) (sum term (next a) next b))))The
sumprocedure is only the simplest of a vast number of similar abstractions that can be captured as higher-order procedures. Write a similar procedure called product that returns the product of the values of a function at points over a given range. Show how the factorial function can be defined in terms of product. If your product procedure generates a recursive process, write one that generates an iterative process. If it generates an iterative process, write one that generates a recursive process.