(ns sicp.ch1-2
- (:use (clojure.contrib trace math)))
+ (:use [sicp.utils :only (square)]
+ [clojure.contrib.math :only (sqrt expt)]
+ [clojure.contrib.trace :only (dotrace)]))
(defn factorial [n]
;; observation: clojure loop-recur construct is essentially the same as
;; the iterative version.
-;; exercise 1.9
-(defn ++ [a b]
- (if (= a 0)
- b
- (inc (++ (dec a) b))))
-
-;; (comment
-;; This version is a recursive process, where the previous call increments
-;; the sum by 1 and each call decrement the first operand by 1.
-
-;; user> (dotrace [++] (++ 4 5))
-;; TRACE t3745: (++ 4 5)
-;; TRACE t3746: | (++ 3 5)
-;; TRACE t3747: | | (++ 2 5)
-;; TRACE t3748: | | | (++ 1 5)
-;; TRACE t3749: | | | | (++ 0 5)
-;; TRACE t3749: | | | | => 5
-;; TRACE t3748: | | | => 6
-;; TRACE t3747: | | => 7
-;; TRACE t3746: | => 8
-;; TRACE t3745: => 9
-;; 9
-;; )
-
-(defn ++ [a b]
- (if (= a 0)
- b
- (++ (dec a) (inc b))))
-
-;; (comment
-
-;; user> (dotrace [++] (++ 4 5))
-;; TRACE t3766: (++ 4 5)
-;; TRACE t3767: | (++ 3 6)
-;; TRACE t3768: | | (++ 2 7)
-;; TRACE t3769: | | | (++ 1 8)
-;; TRACE t3770: | | | | (++ 0 9)
-;; TRACE t3770: | | | | => 9
-;; TRACE t3769: | | | => 9
-;; TRACE t3768: | | => 9
-;; TRACE t3767: | => 9
-;; TRACE t3766: => 9
-;; 9
-;; )
-
-;; exercise 1.10
-;; ackerman functions
-(defn A [x y]
- (cond (= y 0) 0
- (= x 0) (* 2 y)
- (= y 1) 2
- :else (A (- x 1)
- (A x (- y 1)))))
-
-(comment
-user> (A 1 10)
-1024
-user> (A 2 4)
-65536
-user> (A 3 3)
-65536
-)
-
-(defn f [n] (A 0 n)) ; f(n) = 2n
-(defn g [n] (A 1 n)) ; g(n) = 2^n
-;; (comment
-;; g (n) = A (1,n)
-;; = A (0, A (1, n-1)) = f (A(1,n-1))
-;; = f (f (1,n-2))
-;; .....
-;; = f (f (f ... f (1,(n- (n-1)))))
-;; = f (f (f ... 2))
-;; = 2 * (2^(n-1))
-;; = 2^n
-;; )
-
-(defn h [n] (A 2 n)) ; h(n) = 2^(n^2)
-
;; section 1.2.2: Tree Recursion
(defn fib [n]
(cond (= n 0) 0
(defn count-change [amount]
(cc amount 5))
-;; exercise 1.11: A function f is defined by the rule that f(n) = n if n < 3
-;; and f(n) = f(n - 1) + 2f(n - 2) + 3f(n - 3) if n> 3.
-;; Write a procedure that computes f by means of a recursive
-;; process. Write a procedure that computes f by means of an
-;; iterative process.
-(defn f [n]
- (if (< n 3)
- n
- (+ (f (- n 1))
- (* 2 (f (- n 2)))
- (* 3 (f (- n 3))))))
-
-(comment
-user> (map f (range 10))
-(0 1 2 4 11 25 59 142 335 796)
-)
-
-;; ex 1.11: iterative version
-(defn f-iter [count prev0 prev1 prev2]
- (if (= count 3)
- (+ prev0
- (* 2 prev1)
- (* 3 prev2))
- (f-iter (dec count)
- (+ prev0
- (* 2 prev1)
- (* 3 prev2))
- prev0
- prev1)))
-
-(defn f [n]
- (if (< n 3)
- n
- (f-iter n 2 1 0)))
-
-;; ex 1.11: iterative version with let
-(defn f-iter [count prev0 prev1 prev2]
- (let [res (+ prev0 (* 2 prev1) (* 3 prev2))]
- (if (= count 3)
- res
- (f-iter (dec count)
- res
- 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.
-
-;; Note: Finding the order of space in a recursive process is sort of, equiv
-;; to finding the number of deferred operations. Which is in-turn the
-;; same as the depth of the evaluation tree.
;; 1.2.4: exponentiation
;; computing b^n
(* b (myexpt b (dec n)))))
;; fast version
-(defn square [x]
- (* x x))
-
(defn fast-expt [b n]
(cond (= n 0) 1
(even? n) (square (fast-expt b (/ n 2)))
:else (* b (fast-expt b (dec n)))))
-;; exercise 1.16:
-(defn myexpt [b n]
- (expt-iter b n 1))
-
-(defn expt-iter [b n a]
- (cond (= n 0) a
- (even? n) (expt-iter (square b) (/ n 2) a)
- :else (expt-iter b (- n 1) (* a b))))
-
-;; exercise 1.17:
-(defn mult [a b]
- (if (= b 0)
- 0
- (+ a (mult a (- b 1)))))
-
-;; double
-(defn twice [x]
- (* 2 x))
-
-(defn half [x]
- (/ x 2))
-
-;; product = 2 * (a * (b/2)) for even b
-;; = a + (a * (b - 1)) for odd b
-(defn fast-mult [a b]
- (cond (= b 0) 0
- (= b 1) a
- (even? b) (twice (fast-mult a (half b)))
- :else (+ a (fast-mult a (- b 1)))))
-
-;; exercise 1.18: iterative multiply thru addition
-;; the idea is to keep a state variable.
-(defn fast-mult-iter [a b k]
- (cond (= b 0) k
- (even? b) (fast-mult-iter (twice a) (half b) k)
- :else (fast-mult-iter a (- b 1) (+ k a))))
-
-(defn fast-mult-new [a b]
- (fast-mult-iter a b 0))
-
-;; (comment
-;; user> (dotrace [fast-mult-new fast-mult-iter] (fast-mult-new 2 3))
-;; TRACE t2915: (fast-mult-new 2 3)
-;; TRACE t2916: | (fast-mult-iter 2 3 0)
-;; TRACE t2917: | | (fast-mult-iter 2 2 2)
-;; TRACE t2918: | | | (fast-mult-iter 4 1 2)
-;; TRACE t2919: | | | | (fast-mult-iter 4 0 6)
-;; TRACE t2919: | | | | => 6
-;; TRACE t2918: | | | => 6
-;; TRACE t2917: | | => 6
-;; TRACE t2916: | => 6
-;; TRACE t2915: => 6
-;; 6
-;; )
-;; exercise 1.19: fast fibonacci
-;; see the pdf of the notebook scan for the derivation of p' and q'
-(defn ffib-iter [a b p q count]
- (cond (= count 0) b
- (even? count)
- (ffib-iter a
- b
- (+ (* p p) (* q q))
- (+ (* 2 p q) (* q q))
- (/ count 2))
- :else (ffib-iter (+ (* b q) (* a q) (* a p))
- (+ (* b p) (* a q))
- p
- q
- (- count 1))))
-
-(defn ffib [n]
- (ffib-iter 1 0 0 1 n))
;;; Section 1.2.5: GCD
(defn mygcd [a b]
(if (= b 0)
a
- (mygcd b (rem a b))))
\ No newline at end of file
+ (mygcd b (rem a b))))
+
+;;; section 1.2.6 Primality testing.
+(defn divides? [a b]
+ (= (rem b a) 0))
+
+(defn find-divisor [n test-divisor]
+ (cond (> (square test-divisor) n) n
+ (divides? test-divisor n) test-divisor
+ :else (find-divisor n (inc test-divisor))))
+
+(defn smallest-divisor [n]
+ (find-divisor n 2))
+
+(defn prime? [n]
+ (= (smallest-divisor n) n))
+
+
+;; fermat's little theorem
+(defn expmod [base exp m]
+ (cond (= exp 0) 1
+ (even? exp) (rem (square (expmod base (/ exp 2) m))
+ m)
+ :else (rem (* base (expmod base (dec exp) m))
+ m)))
+
+(defn try-it [a n]
+ (= a (expmod a n n)))
+
+(defn fermat-test [n]
+ (try-it (+ 1 (rand-int (- n 1))) n))
+
+(defn fast-prime? [n times]
+ (cond (= times 0) true
+ (fermat-test n) (fast-prime? n (dec times))
+ :else false))
+
+
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