; We are not programming in ALGOL, BASIC or PASCAL, we program LISP.
; Common Abstractions:
; 2A:(1m30s) Sum of all i, for every i smaller than B, starting at A.
(DEFINE (SUM-INT A B)
  (IF (> A B)
      0
      (+ A
         (SUM-INT (1+ A) B))))
; 2A:(3m41s) Sum the squares of i=A to B.
(DEFINE (SUM-SQ A B)
  (IF (> A B)
      0
      (+ (SQUARE A)
         (SUM-SQ (1+ A) B))))
; 2A:(7m5s) Generating pi as a series.
(DEFINE (PI-SUM A B)
  (IF (> A B)
    0
    (+ (/ 1 (* A (+ A 2)))
       (PI-SUM (+ A 4) B))))
; 2A:(9m) The general pattern:
;   (define (<name> a b)
;     (if (> a b)
;         0
;         (+ (<term> a)
;         (<name (<next> a) b))))
;
; Generalizing
(DEFINE (SUM TERM A NEXT B)
  (IF (> A B)
      0
      (+ (TERM A)
         (SUM TERM
              (NEXT A)
              NEXT
              B))))
(DEFINE (SUM-INT A B)
  (DEFINE (IDENTITY A) A)
  (SUM IDENTITY A 1+ B))
(DEFINE (SUM-SQ A B)
  (SUM SQUARE A 1+ B)
(DEFINE (PI-SUM A B)
  (SUM (LAMBDA (I) (/ 1 (* I (+ 1 2))))
       A
       (LAMBDA (I) (+ I 4))
       B))
; procedures named by symbols, they are not "special".
; BREAK 2A:(22m30s)
; Can the SQRT from Lecture 1A be expressed more clearly?
; 2A:(27m)
(DEFINE (SQRT X)
  (FIXED-POINT
      (LAMBDA (Y) (AVERAGE (/ X Y) Y))
      1))
(DEFINE (FIXED-POINT F START)
  (DEFINE TOLERANCE 0.00001)
  (DEFINE (CLOSE-ENUF? U V)
    (< (ABS (- U V)) TOLERANCE))
  (DEFINE (ITER OLD NEW)
    (IF (CLOSE-ENUF? OLD NEW)
        NEW
        (ITER NEW (F NEW))))
  ;INITIALISATION
  (ITER START (F START)))
; 2A:(32m28s) Why this? Oscillation Problem.
; next try:
(DEFINE (SQRT X) ;resulting from average-damping...
  (FIXED-POINT
    (AVERAGE-DAMP (LAMBDA (Y) (/ X Y)))
    1))
(DEFINE AVERAGE-DAMP
  (LAMBDA (F) (LAMBDA (X) (AVERAGE (F X) X))))
;or the same ...
(DEFINE (AVERAGE-DAMP F)
  (LAMBDA (X) (AVERAGE (F X) X)))
;or ...
(DEFINE (AVERAGE-DAMP F)
  (DEFINE (FOO X)
    (AVERAGE (F X) X))
  FOO)
; BREAK 2A:(43m)
; including Newton's method.
(DEFINE (SQRT X)
  (NEWTON (LAMBDA (Y) (- X (SQUARE Y)))
          1))
(DEFINE (NEWTON F GUESS)
  (DEFINE DF (DERIV F))
  (FIXED-POINT
    (LAMBDA (X) (- X (/ (F X) (DF X))))
    GUESS))
(DEFINE DERIV
  (LAMBDA (F)
    (LAMBDA (X)
      (/ (- (F (+ X DX))
            (F X))
         DX))))
(DEFINE DX .0000000001)
; 2A:(53m50s) Now we have decomposed a mathematical problem and
;             defined SQRT by wishful thinking.
; 2A:(56m) The first class citizen.
; 2A:(57m56s) A memorable question. Very important.
; (END)