(ns sicp.chapter1.ch1_2
- (:use (clojure.contrib trace)))
+ (:use (clojure.contrib trace math)))
(defn factorial [n]
(= 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))
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]
:else (* b (fast-expt b (dec n)))))
(defn even? [x]
- (= (rem x 2) 0))
\ No newline at end of file
+ (= (rem x 2) 0))
projects / sicp.git / commitdiff