342 lines
7.3 KiB
Plaintext
342 lines
7.3 KiB
Plaintext
|
||
(define (abs x)
|
||
(cond ((< x 0) (- x))
|
||
(else x)))
|
||
|
||
(assert (= (abs 1) 1))
|
||
(assert (= (abs (- 2)) 2))
|
||
|
||
|
||
(define (abs x)
|
||
(if (< x 0)
|
||
(- x)
|
||
x))
|
||
|
||
(assert (= (abs 12) 12))
|
||
(assert (= (abs (- 32)) 32))
|
||
|
||
|
||
(define (>= x y)
|
||
(or (> x y)
|
||
(= x y)))
|
||
|
||
(assert (>= 2 2))
|
||
(assert (>= 3 2))
|
||
(assert (not (>= 1 2)))
|
||
(assert (not (>= 12 44)))
|
||
|
||
|
||
(define (>= x y)
|
||
(not (< x y)))
|
||
|
||
(assert (>= 2 2))
|
||
(assert (>= 3 2))
|
||
(assert (not (>= 1 2)))
|
||
(assert (not (>= 12 44)))
|
||
|
||
|
||
(define (a-plus-abs-b a b)
|
||
((if (> b 0) + -) a b))
|
||
|
||
(assert (= (a-plus-abs-b 1 2) 3))
|
||
(assert (= (a-plus-abs-b 1 -2) 3))
|
||
|
||
(define (square x) (* x x))
|
||
(define (cube x) (* x x x))
|
||
|
||
(assert (= ((lambda (x y z)
|
||
(+ x y (square z)))
|
||
1 2 3)
|
||
12))
|
||
|
||
;; ;;; --------------------
|
||
;; ;;; newtons method
|
||
;; ;;; --------------------
|
||
|
||
(define tolerance 0.001)
|
||
|
||
(define (square x)
|
||
(* x x))
|
||
|
||
(define (average x y)
|
||
(/ (+ x y) 2))
|
||
|
||
(define (improve guess x)
|
||
(average guess (/ x guess)))
|
||
|
||
(define (good-enough? guess x)
|
||
(< (abs (- (square guess) x)) tolerance))
|
||
|
||
(define (sqrt-iter guess x)
|
||
(if (good-enough? guess x)
|
||
guess
|
||
(sqrt-iter (improve guess x) x)))
|
||
|
||
(define (sqrt x)
|
||
(sqrt-iter 1.0 x))
|
||
|
||
(define (sqrt2 x)
|
||
(define (good-enough? guess x)
|
||
(< (abs (- (square guess) x)) 0.001))
|
||
|
||
(define (improve guess x)
|
||
(average guess (/ x guess)))
|
||
|
||
(define (sqrt-iter guess x)
|
||
(if (good-enough? guess x)
|
||
guess
|
||
(sqrt-iter (improve guess x) x)))
|
||
|
||
(sqrt-iter 1.0 x))
|
||
|
||
(define (sqrt3 x)
|
||
(define (good-enough? guess)
|
||
(< (abs (- (square guess) x)) 0.001))
|
||
|
||
(define (improve guess)
|
||
(average guess (/ x guess)))
|
||
|
||
(define (sqrt-iter guess)
|
||
(if (good-enough? guess)
|
||
guess
|
||
(sqrt-iter (improve guess))))
|
||
|
||
(sqrt-iter 1.0))
|
||
|
||
(assert (< (abs (- 3 (sqrt 9))) tolerance))
|
||
(assert (< (abs (- 4 (sqrt 16))) tolerance))
|
||
(assert (not (< (abs (- 4 (sqrt 15))) tolerance)))
|
||
|
||
(assert (< (abs (- 3 (sqrt2 9))) tolerance))
|
||
(assert (< (abs (- 4 (sqrt2 16))) tolerance))
|
||
(assert (not (< (abs (- 4 (sqrt2 15))) tolerance)))
|
||
|
||
(assert (< (abs (- 3 (sqrt3 9))) tolerance))
|
||
(assert (< (abs (- 4 (sqrt3 16))) tolerance))
|
||
(assert (not (< (abs (- 4 (sqrt3 15))) tolerance)))
|
||
|
||
|
||
;;; -----------------
|
||
;;; factorial
|
||
;;; -----------------
|
||
|
||
(define (factorial n)
|
||
(if (= n 1)
|
||
1
|
||
(* n (factorial (- n 1)))))
|
||
|
||
(define (factorial2 n)
|
||
(fact-iter 1 1 n))
|
||
|
||
(define (fact-iter product counter max-count)
|
||
(if (> counter max-count)
|
||
product
|
||
(fact-iter (* counter product) (+ counter 1) max-count)))
|
||
|
||
(define (factorial3 n)
|
||
(define (iter product counter)
|
||
(if (> counter n)
|
||
product
|
||
(iter (* counter product) (+ counter 1))))
|
||
|
||
(iter 1 1))
|
||
|
||
(assert (= (factorial 6) 720))
|
||
(assert (= (factorial2 6) 720))
|
||
(assert (= (factorial3 6) 720))
|
||
|
||
;;; ----------------
|
||
;;; ackermann
|
||
;;; ----------------
|
||
|
||
(define (A m n)
|
||
(cond ((= m 0) (+ n 1))
|
||
((= n 0) (A (- m 1) 1))
|
||
(else (A (- m 1) (A m (- n 1))))))
|
||
|
||
(assert (= (A 0 0) 1))
|
||
(assert (= (A 1 2) 4))
|
||
(assert (= (A 3 1) 13))
|
||
|
||
|
||
;; ;;; ---------------
|
||
;; ;;; Fibonacci
|
||
;; ;;; ---------------
|
||
|
||
(define (fib n)
|
||
(cond ((= n 0) 0)
|
||
((= n 1) 1)
|
||
(else (+ (fib (- n 1)) (fib (- n 2))))))
|
||
|
||
(define (fib2 n)
|
||
(fib-iter 1 0 n))
|
||
|
||
(define (fib-iter a b count)
|
||
(if (= count 0)
|
||
b
|
||
(fib-iter (+ a b) a (- count 1))))
|
||
|
||
(assert (= (fib 2) 1))
|
||
(assert (= (fib 3) 2))
|
||
(assert (= (fib 4) 3))
|
||
(assert (= (fib 5) 5))
|
||
(assert (= (fib 6) 8))
|
||
|
||
(assert (= (fib2 2) 1))
|
||
(assert (= (fib2 3) 2))
|
||
(assert (= (fib2 4) 3))
|
||
(assert (= (fib2 5) 5))
|
||
(assert (= (fib2 6) 8))
|
||
|
||
;;; ------------------
|
||
;;; count change
|
||
;;; ------------------
|
||
|
||
(define (count-change amount)
|
||
(define (cc amount kinds-of-coins)
|
||
(cond ((= amount 0) 1)
|
||
((or (< amount 0) (= kinds-of-coins 0)) 0)
|
||
(else (+ (cc amount (- kinds-of-coins 1))
|
||
(cc (- amount (first-denomination kinds-of-coins)) kinds-of-coins)))))
|
||
|
||
(define (first-denomination kinds-of-coins)
|
||
(cond ((= kinds-of-coins 1) 1)
|
||
((= kinds-of-coins 2) 5)
|
||
((= kinds-of-coins 3) 10)
|
||
((= kinds-of-coins 4) 25)
|
||
((= kinds-of-coins 5) 50)))
|
||
|
||
(cc amount 5))
|
||
|
||
;; (assert (= (count-change 100) 292))
|
||
|
||
;; ;;; --------------------
|
||
;; ;;; exponentiation
|
||
;; ;;; --------------------
|
||
|
||
(define (expt b n)
|
||
(if (= n 0)
|
||
1
|
||
(* b (expt b (- n 1)))))
|
||
|
||
(define (expt2 b n)
|
||
(define (expt-iter b counter product)
|
||
(if (= counter 0)
|
||
product
|
||
(expt-iter b (- counter 1) (* b product))))
|
||
|
||
(expt-iter b n 1))
|
||
|
||
(define (fast-expt b n)
|
||
(define (even? n)
|
||
(= (% n 2) 0))
|
||
|
||
(cond ((= n 0) 1)
|
||
((even? n) (square (fast-expt b (/ n 2))))
|
||
(else (* b (fast-expt b (- n 1))))))
|
||
|
||
(assert (= (expt 1 2) 1))
|
||
(assert (= (expt 2 2) 4))
|
||
(assert (= (expt 2 3) 8))
|
||
|
||
(assert (= (expt2 1 2) 1))
|
||
(assert (= (expt2 2 2) 4))
|
||
(assert (= (expt2 2 3) 8))
|
||
|
||
(assert (= (fast-expt 1 2) 1))
|
||
(assert (= (fast-expt 2 2) 4))
|
||
(assert (= (fast-expt 2 3) 8))
|
||
|
||
;;; ----------
|
||
;;; gcd
|
||
;;; ----------
|
||
|
||
(define (gcd a b)
|
||
(if (= b 0)
|
||
a
|
||
(gcd b (% a b))))
|
||
|
||
(assert (= (gcd 40 6) 2))
|
||
(assert (= (gcd 13 4) 1))
|
||
|
||
|
||
;;; ----------
|
||
;;; primes
|
||
;;; ----------
|
||
|
||
(define (smallest-divisor n)
|
||
(find-divisor n 2))
|
||
|
||
(define (find-divisor n test-divisor)
|
||
(cond ((> (square test-divisor) n) n)
|
||
((divides? test-divisor n) test-divisor)
|
||
(else (find-divisor n (+ test-divisor 1)))))
|
||
|
||
(define (divides? a b)
|
||
(= (% b a) 0))
|
||
|
||
(define (prime? n)
|
||
(= n (smallest-divisor n)))
|
||
|
||
(assert (prime? 13))
|
||
(assert (prime? 11))
|
||
(assert (not (prime? 12)))
|
||
|
||
|
||
;; ----------------------
|
||
;; simple integral
|
||
;; ----------------------
|
||
|
||
(define (sum term a next b)
|
||
(if (> a b)
|
||
0
|
||
(+ (term a) (sum term (next a) next b))))
|
||
|
||
(define (integral f a b dx)
|
||
(define (add-dx x) (+ x dx))
|
||
(* (sum f (+ a (/ dx 2.0)) add-dx b) dx))
|
||
|
||
(define (pi-sum a b)
|
||
(define (pi-term x) (/ 1.0 (* x (+ x 2))))
|
||
(define (pi-next x) (+ x 4))
|
||
(sum pi-term a pi-next b))
|
||
|
||
(assert (< (abs (- (* 8 (pi-sum 1 100)) 3.121595)) 0.0001))
|
||
(assert (< (abs (- (integral cube 0 1 0.02) 0.249950)) 0.0001))
|
||
|
||
|
||
;; ------------------------------------------------------------
|
||
;; F(x,y) = x(1 + xy)^2 + y(1 − y) + (1 + xy)(1 − y)
|
||
;; ------------------------------------------------------------
|
||
|
||
(define (f x y)
|
||
(let ((a (+ 1 (* x y)))
|
||
(b (- 1 y)))
|
||
(+ (* x (square a))
|
||
(* y b)
|
||
(* a b))))
|
||
|
||
(assert (= (f 0 0) 1))
|
||
(assert (= (f 1 1) 4))
|
||
|
||
|
||
;;; ---------------
|
||
;;; find zero
|
||
;;; ---------------
|
||
|
||
(define (positive? x) (< 0 x))
|
||
(define (negative? x) (< x 0))
|
||
|
||
(define (search f neg-point pos-point)
|
||
(let ((midpoint (average neg-point pos-point)))
|
||
(if (close-enough? neg-point pos-point)
|
||
midpoint
|
||
(let ((test-value (f midpoint)))
|
||
(cond ((positive? test-value) (search f neg-point midpoint))
|
||
((negative? test-value) (search f midpoint pos-point))
|
||
(else midpoint))))))
|
||
|
||
(define (close-enough? x y) (< (abs (- x y)) 0.001))
|
||
|
||
(assert (close-enough? (search (lambda (x) (- 1 (square x))) -3 3) -1))
|