特殊形式（special form）：
1 (lambda (<formal-parameters>) <body>) 2 (let ((<var1> <exp1>) 3 ... 4 (<varn> <expn>)) 5 <body>)
等价形式：
1 (define (plus4 x) (+ x 4)) 2 等价于 3 (define plus4 (lambda (x) (+ x 4))) 4 5 (let ((<var1> <exp1>) 6 ... 7 (<varn> <expn>)) 8 <body>) 9 等价于 10 ((lambda (<var1> ... <varn>) 11 <body>) 12 <exp1> ... <expn>)
范例：区间折半法找方程的根
1 (define (average x y) 2 (/ (+ x y) 2)) 3 (define (close-enough? x y) 4 (< (abs (- x y)) 0.0001)) 5 6 (define (search f neg-point pos-point) 7 (let ((midpoint (average neg-point pos-point))) 8 (if (close-enough? neg-point pos-point) 9 midpoint 10 (let ((test-value (f midpoint))) 11 (cond ((positive? test-value) (search f neg-point midpoint)) 12 ((negative? test-value) (search f midpoint pos-point)) 13 (else midpoint)))))) 14 15 (define (half-interval-method f a b) 16 (let ((a-value (f a)) 17 (b-value (f b))) 18 (cond ((and (negative? a-value) (positive? b-value)) (search f a b)) 19 ((and (negative? b-value) (positive? a-value)) (search f b a)) 20 (else (error "Values are not of opposite sign." a b)))))
范例：找出函数不动点
1 (define (tolerance 0.00001) 2 (define (fixed-point f first-guess) 3 (define (close-enough? v1 v2) 4 (< (abs (- v1 v2)) tolerance)) 5 (define (try guess) 6 (let ((next (f guess))) 7 (if (close-enough? guess next) 8 next 9 (try next)))) 10 (try first-guess))
部分习题：
exercise 1.29
1 (define (inc x) 2 (+ x 1)) 3 (define (cube x) 4 (* x x x)) 5 (define (sum term a next b) 6 (if (> a b) 7 0 8 (+ (term a) 9 (sum term (next a) next b)))) 10 (define (simpson-integral f a b n) 11 (define h (/ (- b a) n)) 12 (define (y k) 13 (f (+ a (* h k)))) 14 (define (term k) 15 (cond ((or (= k 0) (= k n)) (y k)) 16 ((even? k) (* (y k) 2)) 17 (else (* (y k) 4)))) 18 (* (sum term 0 inc n) 19 (/ h 3)))
exercise 1.30
1 (define (sum term a next b) 2 (define (sum-iter a result) 3 (if (> a b) 4 result 5 (sum-iter (next a) (+ result (term a))))) 6 (sum-iter a 0))
exercise 1.31
1 (define (inc x) (+ x 1)) 2 (define (identity x) x) 3 ; recursive product 4 (define (product term a next b) 5 (if (> a b) 6 1 7 (* (term a) 8 (product term (next a) next b)))) 9 10 (define (factorial n) 11 (product identity 1 inc n)) 12 13 (define (quarter-pi n) 14 (define (term k) 15 (if (even? k) 16 (/ (+ k 2.0) (+ k 3.0)) 17 (/ (+ k 3.0) (+ k 2.0)))) 18 (product term 0 inc n)) 19 20 ; iterative product 21 (define (product term a next b) 22 (define (product-iter a result) 23 (if (> a b) 24 result 25 (product-iter (next a) (* (term a) result)))) 26 (product-iter a 1))
exercise 1.32
1 ;recursive process 2 (define (accumulate combiner null-value term a next b) 3 (if (> a b) 4 null-value 5 (combiner (term a) 6 (accumulate combiner null-value term (next a) next b)))) 7 ;iterative process 8 (define (accumulate combiner null-value term a next b) 9 (define (accumulate-iter a result) 10 (if (> a b) 11 result 12 (accumulate-iter (next a) 13 (combiner (term a) result)))) 14 (accumulate-iter a null-value)) 15 16 ;sum 17 (define (sum term a next b) 18 (accumulate + 0 term a next b)) 19 ;product 20 (define (product term a next b) 21 (accumulate * 1 term a next b))
exercise 1.33
1 (define (accumulate-filter combiner null-value term a next b filter) 2 (if (> a b) 3 null-value 4 (if (filter (term a)) 5 (combiner (term a) 6 (accumulate-filter combiner null-value term (next a) next b filter)) 7 (combiner null-value 8 (accumulate-filter combiner null-value term (next a) next b filter)))))
exercise 1.34
代换模型如下：
1 (f f) 2 (f 2) 3 (2 2)
exercise 1.35
1 (define (golden-ratio) 2 (fixed-point (lambda (x) (+ 1 (/ 1 x))) 3 1.0))
exercise 1.36
1 (define tolerance 0.0001) 2 (define (fixed-point f first-guess) 3 (define (close-enough? v1 v2) 4 (< (abs (- v1 v2)) tolerance)) 5 (define (try guess) 6 (newline) 7 (display guess) 8 (let ((next (f guess))) 9 (if (close-enough? guess next) 10 next 11 (try next)))) 12 (try first-guess))
设初始猜测值为 x1 且x1 != 1。则x2 = log(1000) / log(x1)，x3 = log(x1)，可以看到猜测值不停在 log(1000) / log(x1) 和 log(x1) 之间往复。
1 (define (f) 2 (fixed-point (lambda (x) (average (/ (log 1000) (log x)) 3 (log x))) 4 2.0))
exercise 1.37
1 ; recursive procedure 2 (define (cont-frac n d k) 3 (define (loop index) 4 (if (= index k) 5 0 6 (/ (n index) 7 (+ (d index) (loop (+ index 1)))))) 8 (loop 1)) 9 10 11 ; iterative procedure 12 (define (cont-frac n d k) 13 (define (loop-iter index result) 14 (if (= index k) 15 result 16 (loop-iter (+ index 1) 17 (/ (n index) 18 (+ (d index) result))))) 19 (loop-iter 1 0))
exercise 1.38
1 (define (d-item k) 2 (cond ((= k 0) 0) 3 ((= (remainder (+ k 1) 3) 0) 4 (expt 2 (/ (+ k 1) 3))) 5 (else 1))) 6 (define (euler k) 7 (+ (cont-frac (lambda (i) 1.0) 8 d-item 9 k) 10 2.0))