add ch1 ex6

chapter_1/ex_6.rkt

@@ -0,0 +1,80 @@
+#lang sicp
+
+;1.1.7 Example: Square Roots by Newton's Method 
+;The most common way to compute square root is to use Newton's method of successive approximations --> whenever we have a guess y for the value of the square root of a number x, we can perform a simple manipulation to get a better guess.
+;The procedure is
+;(define (sqrt-iter guess x)
+;    (if (good-enough? guess x)
+;        guess
+;        (sqrt-iter (improve guess x) x))
+;)
+;A guess is improved by averaging it with the quotient of the radicand and the old guess 
+;(define (improve guess x)
+;    (/ (+ guess (/ x guess)) 2)
+;)
+;where 
+;(define (average x y)
+;    (/ (+ x y) 2)
+;)
+;The idea is to improve the answer until it is close enough so that its square differs from the radicand by less than a predetermined tolerance (here 0.001)
+;(define (good-enough? guess x)
+;    (< (abs (- (square guess) x)) 0.001)
+;)
+;we need a way to get started. For instance, we can always guess that the square root of any number is 1 (if we start with an initial guess of 1 in our square-root program, and x is an exact integer, all subsequent values produced in the square-root computation will be rational numbers rather than decimals. Mixed operations on rational numbers
+;and decimals always yield decimals, so starting with an initial guess of 1.0 forces all subsequent values to be decimals.)
+;(define (sqrt x)
+;    (sqrt-iter 1.0 x)
+;)
+
+;(define (square x) (* x x))
+
+;(sqrt 25) ;the result should be 5.000023178253949
+;(sqrt 9) ;the result should be 3.00009155413138
+
+;--------------------------------------------------------------------------------------------------------------------------
+;ex_6
+;Alyssa P. Hacker doesn’t see why if needs to be provided as a special form. “Why can’t I just define it as an ordinary procedure in terms of cond?” she asks. Alyssa’s friend Eva Lu Ator claims this can indeed be done, and she defines a new version of if:
+;(define (new-if predicate then-clause else-clause)
+;    (cond (predicate then-clause)
+;    (else else-clause))
+;)
+;Eva demonstrates the program for Alyssa:
+;(new-if (= 2 3) 0 5)
+;5
+;(new-if (= 1 1) 0 5)
+;0
+;Delighted, Alyssa uses new-if to rewrite the square-root
+;program:
+;(define (sqrt-iter guess x)
+;    (new-if (good-enough? guess x)
+;        guess
+;        (sqrt-iter (improve guess x) x))
+;)
+;What happens when Alyssa attempts to use this to compute square roots? Explain.
+;--------------------------------------------------------------------------------------------------------------------------
+
+(define (new-if predicate then-clause else-clause)
+  (cond (predicate then-clause)
+        (else else-clause)))
+
+(define (average x y) (/ (+ x y) 2))
+
+(define (square x) (* x x))
+
+(define (improve guess x) (average guess (/ x guess)))
+
+(define (good-enough? guess x)
+(< (abs (- (square guess) x)) 0.001))
+
+
+(define (sqrt-iter guess x)
+    (new-if (good-enough? guess x) ;(good-enough? guess x), guess and (sqrt-iter (improve guess x) x) will always be evaluated before new-if
+            guess
+            (sqrt-iter (improve guess x) x))
+)
+
+(define (sqrt x)
+    (sqrt-iter 1.0 x)
+)
+
+(sqrt 25)  ;--> infinite loop// new-if will never be executed