From: Ramakrishnan Muthukrishnan Date: Thu, 8 Apr 2010 15:13:04 +0000 (+0530) Subject: Added solution to 1.12, 1.13 and 1.15. X-Git-Url: https://git.rkrishnan.org/listings?a=commitdiff_plain;h=9b1c0373d0261194a279c1f29a6772bde404043a;p=sicp.git Added solution to 1.12, 1.13 and 1.15. --- diff --git a/chapter1/ch1_2.clj b/chapter1/ch1_2.clj index 19a3e99..89ae7db 100644 --- a/chapter1/ch1_2.clj +++ b/chapter1/ch1_2.clj @@ -1,5 +1,5 @@ (ns sicp.chapter1.ch1_2 - (:use (clojure.contrib trace))) + (:use (clojure.contrib trace math))) (defn factorial [n] @@ -198,6 +198,7 @@ user> (A 3 3) (= kinds-of-coins 3) 10 (= kinds-of-coins 4) 25 (= kinds-of-coins 5) 50)) + (defn first-denomination [kinds-of-coins] ({1 1 2 5 3 10 4 25 5 50} kinds-of-coins)) @@ -246,6 +247,207 @@ user> (map f (range 10)) prev0 prev1)))) +;; exercise 1.12. The following pattern of numbers is called Pascal's triangle. +;; 1 +;; 1 1 +;; 1 2 1 +;; 1 3 3 1 +;; 1 4 6 4 1 +;; ................... +;; +;; The numbers at the edge of the triangle are all 1, and each +;; number inside the triangle is the sum of the two numbers above +;; it. Write a procedure that computes elements of Pascal's triangle +;; by means of a recursive process. +(defn pascal [row col] + (when (<= col row) + (if (or (= col 0) (= row col)) + 1 + (+ (pascal (dec row) col) + (pascal (dec row) (dec col)))))) + +;; exercise 1.13: +(comment +See the pdfs in the directory for the answers. +) + +;; ex 1.13 (contd) +(defn fib-approx [n] + (let [phi (/ (+ 1 (sqrt 5)) 2)] + (/ (expt phi n) (sqrt 5)))) + +(comment +user> (map fib-approx (range 10)) +(0.4472135954999579 0.7236067977499789 1.1708203932499368 1.8944271909999157 3.065247584249853 4.959674775249769 8.024922359499623 12.984597134749393 21.009519494249016 33.99411662899841) +) + +;; exercise 1.14: tree of (count-changes 11) +(comment + See the pdf for the tree representation. +) + + +;; order of size and computation +;; see PDF, but below is the trace tree. +(comment +user> (use 'clojure.contrib.trace) +nil +user> (dotrace [count-change cc] (count-change 11)) +TRACE t2609: (count-change 11) +TRACE t2610: | (cc 11 5) +TRACE t2611: | | (cc 11 4) +TRACE t2612: | | | (cc 11 3) +TRACE t2613: | | | | (cc 11 2) +TRACE t2614: | | | | | (cc 11 1) +TRACE t2615: | | | | | | (cc 11 0) +TRACE t2615: | | | | | | => 0 +TRACE t2616: | | | | | | (cc 10 1) +TRACE t2617: | | | | | | | (cc 10 0) +TRACE t2617: | | | | | | | => 0 +TRACE t2618: | | | | | | | (cc 9 1) +TRACE t2619: | | | | | | | | (cc 9 0) +TRACE t2619: | | | | | | | | => 0 +TRACE t2620: | | | | | | | | (cc 8 1) +TRACE t2621: | | | | | | | | | (cc 8 0) +TRACE t2621: | | | | | | | | | => 0 +TRACE t2622: | | | | | | | | | (cc 7 1) +TRACE t2623: | | | | | | | | | | (cc 7 0) +TRACE t2623: | | | | | | | | | | => 0 +TRACE t2624: | | | | | | | | | | (cc 6 1) +TRACE t2625: | | | | | | | | | | | (cc 6 0) +TRACE t2625: | | | | | | | | | | | => 0 +TRACE t2626: | | | | | | | | | | | (cc 5 1) +TRACE t2627: | | | | | | | | | | | | (cc 5 0) +TRACE t2627: | | | | | | | | | | | | => 0 +TRACE t2628: | | | | | | | | | | | | (cc 4 1) +TRACE t2629: | | | | | | | | | | | | | (cc 4 0) +TRACE t2629: | | | | | | | | | | | | | => 0 +TRACE t2630: | | | | | | | | | | | | | (cc 3 1) +TRACE t2631: | | | | | | | | | | | | | | (cc 3 0) +TRACE t2631: | | | | | | | | | | | | | | => 0 +TRACE t2632: | | | | | | | | | | | | | | (cc 2 1) +TRACE t2633: | | | | | | | | | | | | | | | (cc 2 0) +TRACE t2633: | | | | | | | | | | | | | | | => 0 +TRACE t2634: | | | | | | | | | | | | | | | (cc 1 1) +TRACE t2635: | | | | | | | | | | | | | | | | (cc 1 0) +TRACE t2635: | | | | | | | | | | | | | | | | => 0 +TRACE t2636: | | | | | | | | | | | | | | | | (cc 0 1) +TRACE t2636: | | | | | | | | | | | | | | | | => 1 +TRACE t2634: | | | | | | | | | | | | | | | => 1 +TRACE t2632: | | | | | | | | | | | | | | => 1 +TRACE t2630: | | | | | | | | | | | | | => 1 +TRACE t2628: | | | | | | | | | | | | => 1 +TRACE t2626: | | | | | | | | | | | => 1 +TRACE t2624: | | | | | | | | | | => 1 +TRACE t2622: | | | | | | | | | => 1 +TRACE t2620: | | | | | | | | => 1 +TRACE t2618: | | | | | | | => 1 +TRACE t2616: | | | | | | => 1 +TRACE t2614: | | | | | => 1 +TRACE t2637: | | | | | (cc 6 2) +TRACE t2638: | | | | | | (cc 6 1) +TRACE t2639: | | | | | | | (cc 6 0) +TRACE t2639: | | | | | | | => 0 +TRACE t2640: | | | | | | | (cc 5 1) +TRACE t2641: | | | | | | | | (cc 5 0) +TRACE t2641: | | | | | | | | => 0 +TRACE t2642: | | | | | | | | (cc 4 1) +TRACE t2643: | | | | | | | | | (cc 4 0) +TRACE t2643: | | | | | | | | | => 0 +TRACE t2644: | | | | | | | | | (cc 3 1) +TRACE t2645: | | | | | | | | | | (cc 3 0) +TRACE t2645: | | | | | | | | | | => 0 +TRACE t2646: | | | | | | | | | | (cc 2 1) +TRACE t2647: | | | | | | | | | | | (cc 2 0) +TRACE t2647: | | | | | | | | | | | => 0 +TRACE t2648: | | | | | | | | | | | (cc 1 1) +TRACE t2649: | | | | | | | | | | | | (cc 1 0) +TRACE t2649: | | | | | | | | | | | | => 0 +TRACE t2650: | | | | | | | | | | | | (cc 0 1) +TRACE t2650: | | | | | | | | | | | | => 1 +TRACE t2648: | | | | | | | | | | | => 1 +TRACE t2646: | | | | | | | | | | => 1 +TRACE t2644: | | | | | | | | | => 1 +TRACE t2642: | | | | | | | | => 1 +TRACE t2640: | | | | | | | => 1 +TRACE t2638: | | | | | | => 1 +TRACE t2651: | | | | | | (cc 1 2) +TRACE t2652: | | | | | | | (cc 1 1) +TRACE t2653: | | | | | | | | (cc 1 0) +TRACE t2653: | | | | | | | | => 0 +TRACE t2654: | | | | | | | | (cc 0 1) +TRACE t2654: | | | | | | | | => 1 +TRACE t2652: | | | | | | | => 1 +TRACE t2655: | | | | | | | (cc -4 2) +TRACE t2655: | | | | | | | => 0 +TRACE t2651: | | | | | | => 1 +TRACE t2637: | | | | | => 2 +TRACE t2613: | | | | => 3 +TRACE t2656: | | | | (cc 1 3) +TRACE t2657: | | | | | (cc 1 2) +TRACE t2658: | | | | | | (cc 1 1) +TRACE t2659: | | | | | | | (cc 1 0) +TRACE t2659: | | | | | | | => 0 +TRACE t2660: | | | | | | | (cc 0 1) +TRACE t2660: | | | | | | | => 1 +TRACE t2658: | | | | | | => 1 +TRACE t2661: | | | | | | (cc -4 2) +TRACE t2661: | | | | | | => 0 +TRACE t2657: | | | | | => 1 +TRACE t2662: | | | | | (cc -9 3) +TRACE t2662: | | | | | => 0 +TRACE t2656: | | | | => 1 +TRACE t2612: | | | => 4 +TRACE t2663: | | | (cc -14 4) +TRACE t2663: | | | => 0 +TRACE t2611: | | => 4 +TRACE t2664: | | (cc -39 5) +TRACE t2664: | | => 0 +TRACE t2610: | => 4 +TRACE t2609: => 4 +4 +) +;; TODO: orders of growth in space and number of steps. + + +;; exercise 1.15: sin (x) calculation +;; a. How many times is the procedure p applied when (sine 12.15) +;; is evaluated? +;; b. What is the order of growth in space and number of steps (as +;; a function of a) used by the process generated by the sine +;; procedure when (sine a) is evaluated? +(defn cube [x] (* x x x)) + +(defn p [x] (- (* 3 x) (* 4 (cube x)))) + +(defn sine [angle] + (if (not (> (abs angle) 0.1)) + angle + (p (sine (/ angle 3.0))))) + +;; solution to (a) => 5 +(comment +user> (dotrace [p] (sine 12.15)) +TRACE t2490: (p 0.049999999999999996) +TRACE t2490: => 0.1495 +TRACE t2491: (p 0.1495) +TRACE t2491: => 0.4351345505 +TRACE t2492: (p 0.4351345505) +TRACE t2492: => 0.9758465331678772 +TRACE t2493: (p 0.9758465331678772) +TRACE t2493: => -0.7895631144708228 +TRACE t2494: (p -0.7895631144708228) +TRACE t2494: => -0.39980345741334 +-0.39980345741334 +) + +;; solution to b +;; both space and number of steps grows as log3(a) -> log a to the base 3. +;; +;; proof: +;; a * (1/3)^n <= 0.1 +;; => take log to the base 3 on both the sides. + ;; 1.2.4: exponentiation ;; computing b^n (defn expt [b n] @@ -271,4 +473,4 @@ user> (map f (range 10)) :else (* b (fast-expt b (dec n))))) (defn even? [x] - (= (rem x 2) 0)) \ No newline at end of file + (= (rem x 2) 0))