1 ;; The first three lines of this file were inserted by DrScheme. They record metadata
2 ;; about the language level of this file in a form that our tools can easily process.
3 #reader(lib "htdp-intermediate-lambda-reader.ss" "lang")((modname |27.4|) (read-case-sensitive #t) (teachpacks ((lib "draw.ss" "teachpack" "htdp") (lib "arrow.ss" "teachpack" "htdp") (lib "gui.ss" "teachpack" "htdp"))) (htdp-settings #(#t constructor repeating-decimal #f #t none #f ((lib "draw.ss" "teachpack" "htdp") (lib "arrow.ss" "teachpack" "htdp") (lib "gui.ss" "teachpack" "htdp")))))
4 (define TOLERANCE 0.0001)
5 
6 ;newton : (number -> number) number -> number
7 ;Given f, ig (initial guess), find the root of f using Newton's Method.
8 ;TERMINATION ARGUMENT: There is no guarantee that newton will terminate.
9 
10 (define (newton f ig)
11   (cond
12     [(<= (abs (f ig)) TOLERANCE) ig]
13     [else (newton f (find-root-tangent f ig))]))
14 
15 ;find-root-tangent : (number -> number) number -> number
16 ;Finds the root of the tangent line to f at point x.
17 
18 (define (find-root-tangent f x)
19   (local ((define fprime (d/dx f TOLERANCE)))
20     (- x (/ (f x)
21             (fprime x)))))
22 
23 ;d/dx : (number -> number) number -> (number -> number)
24 ;Returns f', the slope of f, given epsilon e.
25 
26 (define (d/dx f e)
27   (local ((define (slope x) (/ (- (f (+ x e))
28                                   (f (- x e)))
29                                (* 2 e))))
30     slope))
31 
32 ;find-root : (number -> number) number number -> number
33 ;Given f, left and right, approximate a root by using a binary search.  find-root returns the left interval once the interval size is less than tolerance.  find-root uses generative recursion to find the root--it splits the interval in half and finds the root for one half of the interval.
34 ;Assume that (<= (f left) 0 (f right)) or (<= (f right) 0 (f left)).
35 ;Termination argument: Each interval is half the size of the previous interval.  Hence, the nth recursive call has an interval of initial * (1/2)^n, where initial is the size of the initial interval.  As n-> infinity, the interval approaches zero, which implies that this function must terminate for any value of tolerance greater than zero.
36 
37 (define (find-root f left right)
38   (local ((define mid (/ (+ left right) 2))
39           (define f-mid (f mid)))
40     (cond
41       [(<= (- right left) TOLERANCE) left]
42       [(or (<= (f left) 0 f-mid)
43            (>= (f left) 0 f-mid)) (find-root f left mid)]
44       [else (find-root f mid right)])))
45 
46 (time (find-root (lambda (x) (- (* (expt x 3)
47                                    (exp x))
48                                 (cos x)))
49                  0 1000))
50 (time (newton (lambda (x) (- (* (expt x 3)
51                                 (exp x))
52                              (cos x)))
53               500))