SICP 1.19 习题答案

这道题需要一些数学知识，题目中已经提示：

引用

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

应用两次Tpq相当于一次Tp'q'，所以可以通过这样的方式进行推导：
Tpq = bq+aq+ap, bp+aq
将a<-bq+aq+ap,b<-bp+aq代入上面对式子中，得
Tp'q'=(bp+aq)q+(bq+aq+ap)q+(bq+aq+ap)p, (bp+aq)p+(bq+aq+ap)q
展开后提出a和b，可得
q'= 2pq+qq
p'= qq+pp

代入后试验结果确实如此。

(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
                   (+ (* p p) (* q q))
                   (+ (* 2 (* p q)) (* q q))
                   (/ count 2)))
        (else (fib-iter (+ (* b q) (* a q) (* a p))
                        (+ (* b p) (* a q))
                        p
                        q
                        (- count 1)))))