Fibonacci numbers through transformation

There is a clever algorithm for computing the Fibonacci numbers in a logarithmic number of steps. Recall the transformation of the state variables a and b in the fib-iter process of section 1.2.2: a ← a + b and b ← a . Call this transformation T , and 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 + aq . Show 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-expt procedure. Put this all together to complete the following procedure, which runs in a 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
                   <??>      ; compute p'
                   <??>      ; compute q'
                   (/ count 2)))
        (else (fib-iter (+ (* b q) (* a q) (* a p))
                        (+ (* b p) (* a q))
                        (- count 1)))))

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