SICP/ex-1_35-39.scm

(define (average a b) (/ (+ a b) 2.0))

(define (search f neg-point pos-point)
  (let ((midpoint (average neg-point pos-point)))
    (if (close-enough? neg-point pos-point)
        midpoint
        (let ((test-value (f midpoint)))
          (cond ((positive? test-value)
                 (search f neg-point midpoint))
                ((negative? test-value)
                 (search f midpoint pos-point))
                (else midpoint))))))

(define (close-enough? x y)
  (< (abs (- x y)) 0.001))

(define (half-interval-method f a b)
  (let ((a-value (f a))
        (b-value (f b)))
    (cond ((and (negative? a-value) (positive? b-value))
           (search f a b))
          ((and (negative? b-value) (positive? a-value))
           (search f b a))
          (else
           (error "Values are not of opposite sign" a b)))))

(display (half-interval-method sin 2.0 4.0)) (newline)

(define tolerance 0.00001)
(define (fixed-point f first-guess)
  (define (close-enough? v1 v2)
    (< (abs (- v1 v2)) tolerance))
  (define (try guess)
    (let ((next (f guess)))
      (if (close-enough? guess next)
          next
          (try next))))
  (try first-guess))

(display (fixed-point cos 1.0)) (newline)

(define (sqrt x)
  (fixed-point (lambda (y) (/ x y))
               1.0))

; (display (sqrt 2) (newline)) ; doesn't converge

(define (sqrt x)
  (fixed-point (lambda (y) (average y (/ x y)))
               1.0))

(display (sqrt 2))

(newline)(newline) (display "ex-1.35") (newline)

; phi^2 = phi + 1
; phi   = 1 + 1 / phi

(display (fixed-point (lambda (phi) (+ 1 (/ 1.0 phi))) 1.0)) (newline)


(newline)(newline) (display "ex-1.36") (newline)

(define (x_without_avg_damping x) (/ (log 1000) (log x)))
(define (x_with_avg_damping x)
  (average x (/ (log 1000) (log x))))

(define (fixed-point f first-guess)
  (define (close-enough? v1 v2)
    (< (abs (- v1 v2)) tolerance))
  (define (try guess)
    (display guess) (newline)
    (let ((next (f guess)))
      (if (close-enough? guess next)
          next
          (try next))))
  (try first-guess))

(newline)
(display (fixed-point x_without_avg_damping 2.0))(newline)
(display "Finished without average damping.")
(newline)
(newline)
(display (fixed-point x_with_avg_damping 2.0))(newline)
(display "Finished with average damping.")


(newline)(newline) (display "ex-1.37 a)") (newline)

; Recursive
(define (cont-frac n d k)
  (define (frac-rec i)
    (if (> i k)
      0
      (/ (n i) (+ (d i) (frac-rec (+ i 1))))))
  (frac-rec 1))

(define (phi k)
  (/ 1 (cont-frac (lambda (i) 1.0) (lambda (i) 1.0) k)))

(display (phi 100))

(define (approx-phi tolerance)
  (define (iteration current-phi i)
    (let ((next-phi (phi i)))
      (if (< (abs (- next-phi current-phi)) tolerance)
        i
        (iteration next-phi (+ i 1)))))
  (iteration 1.1 1))

; Use approx-phi to calculate value of k to get four digit precisious.
(newline)
(display (approx-phi 0.00009)) ; 12

; Phi is 1.618033 so let's see if we can get many digits right with k = 12

(newline)
(display (phi 11))
(newline)
(display (phi 12))
(newline)
(display (phi 13))
; looks like we have found exactly the right value (:

; Recursive wrong
(define (cont-frac n d k)
  (if (= k 1) 0
    (/ (n k) (+ (d k) (cont-frac n d (- k 1))))))

(newline)(newline) (display "ex-1.37 b) - implemented iterative version") (newline)

; (phi 100000) ; show that previous version is recursive.

; Iterative
(define (cont-frac n d k)
  (define (frac-iter i acc)
    (if (= i 0)
      acc
      (frac-iter (- i 1) (/ (n i) (+ (d i) acc)))))
  (frac-iter k 0))

(phi 100000) ; show that this version is iterative


(newline) (display "ex-1.38") (newline)

; 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ; Indicies
; 1, 2, 1, 1, 4, 1, 1, 6, 1,  1,  8 ; Eulers expansion

(define (eulers-expansion k)
  (define (n i) 1)
  (define (d i)
    (if (= (remainder (+ i 1) 3) 0)
      (* 2 (/ (+ i 1) 3))
      1))
  (cont-frac n d k))

(display (+ 2 (/ (eulers-expansion 1000) 1.)))

(newline)(newline) (display "ex-1.39") (newline)

(define (tan-cf x k)
  (define (n i)
    (if (= i 1)
      x
      (* x x -1)))
  (define (d i) (- (* i 2) 1))
  (cont-frac n d k))

(display (tan 1.1)) (newline)
(display (tan-cf 1.1 15)) (newline)


(newline) (display "tests") (newline)

(define (average-damp f)
  (lambda (x) (average x (f x))))

(define (sqrt x)
  (fixed-point (average-damp (lambda (y) y (/ x y)))
               1.0))

(display (sqrt 9)) (newline)

(define dx 0.0000001)
(define (deriv g)
  (lambda (x)
    (/ (- (g (+ x dx)) (g x))
       dx)))

(define (newton-transform g)
  (lambda (x)
    (- x (/ (g x) ((deriv g) x)))))
(define (newtons-method g guess)
  (fixed-point (newton-transform g) guess))

(define (sqrt x)
  (newtons-method (lambda (y) (- (* y y) x)) 1.0))

(newline)
(display "example - Newton's Method") (newline)
(display (sqrt 3)) (newline)