本文共 5621 字，大约阅读时间需要 18 分钟。
练习 1.40
(define (cubic a b c) (lambda (x) (+ (cube x) (* a (square x)) (* b x) c)))1 ]=> (newtons-method (cubic 1.0 1.0 -3.0) 1.0);Value: 1.
练习 1.41
(define (double f) (lambda (x) (f (f x))))(define (inc x) (+ x 1)) 1 ]=> (((double (double double)) inc) 5) ;Value: 21
练习 1.42
(define (compose f g) (lambda (x) (f (g x)))) 1 ]=> ((compose square inc) 6);Value: 49
练习 1.43
(define (repeated f n) (cond ((= n 1) f) ((even? n) (compose f (repeated f (/ n 2)))) (else (compose f (repeated f (- n 1)))))) 1 ]=> ((repeated square 2) 5);Value: 625
练习 1.44
(define (smooth f) (lambda (x) (/ (+ (f (+ x dx)) (f x) (f (- x dx))) 3)))(define (smooth-n f n) (repeated smooth n))
练习 1.45
(define (sqrt x) (fixed-point (average-damp (lambda (y) (/ x y))) 1.0))1 ]=> (sqrt 4.0) *** 1. *** 2.5 *** 2.05 *** 2.000609756097561 *** 2.0000000929222947;Value: 2.000000000000002(define (cube-root x) (fixed-point (average-damp (lambda (y) (/ x (square y)))) 1.0))1 ]=> (cube-root 8.0) *** 1. *** 4.5 *** 2.447530864197531 *** 1.8914996576441667 *** 2.0637643832634476 *** 1.9710425766479744 *** 2.0151199754332096 *** 1.992609760395472 *** 2.0037362842809587 *** 1.998142301706526 *** 2.0009314406381735 *** 1.9995349299633447 *** 2.0002326972862416 *** 1.9998836919616 *** 2.0000581641656563 *** 1.999970920454376 *** 2.0000145404070393 *** 1.9999927299550464 *** 2.000003635062117;Value: 1.9999981824788517(define (fourth-root x) (fixed-point (average-damp (lambda (y) (/ x (cube y)))) 1.0)) 1 ]=> (fourth-root 16.0) *** 1. *** 8.5 *** 4.263026663952778 *** 2.2347742162681095 *** 1.8341723103276433 *** 2.2135774006557023 *** 1.844363126744897 ...... *** 1.983599149847315 *** 2.016809915138478 *** 1.983608081177677 *** 2.0168005353295855 *** 1.983616997866416 *** 2.0167911711462776 *** 1.983625899953568;Quit!;; 出现振荡,Ctrl+C杀掉;; 对x求n次方根时,使用repeated调用m次average-damp(即做m次平均阻尼)(define (fast-expt x n) (fast-expt-iter x n 1)) (define (fast-expt-iter x counter product) (cond ((= counter 0) product) ((even? counter) (fast-expt-iter (square x) (/ counter 2) product)) (else (fast-expt-iter x (- counter 1) (* x product))))) (define (find-root x n m) (fixed-point ((repeated average-damp m) (lambda (y) (/ x (fast-expt y (- n 1))))) 1.0)) 1 ]=> (find-root 4.0 2 1) *** 1. *** 2.5 *** 2.05 *** 2.000609756097561 *** 2.0000000929222947;Value: 2.0000000000000021 ]=> (find-root 8.0 3 1) *** 1. *** 4.5 *** 2.447530864197531 *** 1.8914996576441667 *** 2.0637643832634476 *** 1.9710425766479744 *** 2.0151199754332096 *** 1.992609760395472 *** 2.0037362842809587 *** 1.998142301706526 *** 2.0009314406381735 *** 1.9995349299633447 *** 2.0002326972862416 *** 1.9998836919616 *** 2.0000581641656563 *** 1.999970920454376 *** 2.0000145404070393 *** 1.9999927299550464 *** 2.000003635062117;Value: 1.99999818247885171 ]=> (find-root 16.0 4 2) *** 1. *** 4.75 *** 3.5998232249599065 *** 2.7856139316659103 *** 2.274263910561008 *** 2.045743730517053 *** 2.0015115314098866 *** 2.000001711389449;Value: 2.0000000000021965;; 尝试对求5次方根时做两次平均阻尼以查看收敛性, 可以得到近似正确的结果1 ]=> (find-root 32.0 5 2) *** 1. ......;Value: 2.000001512995761;; 求6次方根, 做两次平均阻尼仍然能得到正确的近似结果1 ]=> (find-root 64.0 6 2) *** 1. ......;Value: 2.0000029334662086;; 7次方根1 ]=> (find-root 128.0 7 2)......;Value: 2.0000035538623377;; 8次方根1 ]=> (find-root 256.0 8 2)...... *** 2.0036413071907813 *** 1.9964048473987912 *** 2.0036406355855547 *** 1.9964055020263292 *** 2.0036399643510747 *** 1.9964061562955964 C-c C-c *** 2.0036392934869998;Quit!;; 出现振荡,Ctrl+C杀掉;; 在求8次方根时又出现振荡,因此再多做一次平均阻尼,即如下调用,得到正确的近似结果1 ]=> (find-root 256.0 8 3)......;Value: 2.000000000003967;; 发现如下规律;; 2、3 次方根时做一次平均阻尼;; 4、5、6、7 次方根时做两次平均阻尼;; 8 次方根时做三次平均阻尼;; 可见对 X 开 N 次方根与做 M 次平均阻尼有如下关系;; M = (/ (log N) (log 2));; 对9、15次方根验证1 ]=> (find-root 512.0 9 3);Value: 1.99999971068401021 ]=> (find-root (fast-expt 2.0 15) 15 3);Value: 2.0000040951543028;; 因此修改过程为(define (find-root x n) (let ((m (round (/ (log n) (log 2))))) (fixed-point ((repeated average-damp m) (lambda (y) (/ x (fast-expt y (- n 1))))) 1.0)));; 但是继续验证, 发现还有问题 ;; 继续对3^16验证, 做四次平均阻尼仍然会出现振荡, 需要做五次平均阻尼才会得到正确的近似结果;; 继续对3^32验证, 做六次平均阻尼仍然会出现振荡, 需要做七次平均阻尼才会得到正确的近似结果;; 继续对3^64验证, 做八次平均阻尼仍然会出现振荡, 需要做十一次平均阻尼才会得到正确的近似结果;; 于是又发现变化过程;; 取 K = (/ (log N) (log 2));; 取 M = 能得到正确近似结果时需要做的平均阻尼次数;; K M ;; 1 1;; 2 2;; 3 3;; 4 5;; 5 7;; 6 11;; 于是推测;; 7 15;; 漫长的等待后, 终于收敛到了近似结果 ..... -_-|;; 继续推测;; 8 21 ?;; 更加漫长的等待, 不收敛！！！ -_-|;; 那就试试;; 8 23 !;; MBP MD102 主频i7 2.9 内存8G 跑了近10分钟才收敛到近似结果 -_-|;; 再往后收敛时间太过漫长, 没有继续测试, 应该有简单的办法可以得到求N次方根与做M次平均阻尼的关系
练习 1.46
;; 对iterative-improve过程的实现(define (iterative-improve good-enough? improve) (define (iter guess) (if (good-enough? guess) guess (iter (improve guess)))) (lambda (guess) (iter guess))) ;; 根据iterative-improve过程对sqrt重新实现(define (iterative-improve-sqrt x) (define (sqrt-good-enough? guess) (< (abs (- (square guess) x)) tolerance)) (define (sqrt-improve guess) (average guess (/ x guess))) ((iterative-improve sqrt-good-enough? sqrt-improve) 1.0)) 1 ]=> (iterative-improve-sqrt 9.0);Value: 3.000000001396984;; 根据iterative-improve过程对fixed-point重新实现(define (iterative-improve-fixed-point f) (define (fixed-point-good-enough? guess) (< (abs (- guess (f guess))) tolerance)) ((iterative-improve fixed-point-good-enough? f) 1.0)) 1 ]=> (iterative-improve-fixed-point cos);Value: .7390893414033928;; 计算结果与原始fixed-point过程计算结果精度略低, 经查1 ]=> (cos 0.7390893414033928);Value: .7390822985224023;; 即原始fixed-point过程中定义了let ((next (f guess)));; 即对于guess, 原始过程经过next处理后进行检测, 如果足够好返回的是(f guess);; 而iterative-improve过程对于guess无此处理, 返回的是guess
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