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; summation1.scm - higher order summation functions 
; (from Cousineau and Mauny, The Functional Approach to Programming" in CAML, p42-43)


; sums f(n) from n=a to n=b 

(define (sigma f l)
  ;(display (format "~%l: ~A" l))
  (if (> (car l) (cadr l))
    0
    (+ 
      (f (car l)) 
      (sigma 
        f 
	(list 
	  (+ 1 (car l)) 
	  (cadr l) )) ))) 

(define (sqr x) (* x x))

(sigma sqr '(1 5)) ; 55

;-----------------------------------------------------------------------------

(define summation
  (lambda (incr test op e f a)
    (if (test a)
      e
      (op 
       (f a) 
       (summation incr test op e f (incr a)) ))))

(define summation_int
  (lambda (op e f a b)
    (summation 
      (lambda (x) 
        (+ x 1))
      (lambda (x)
        (> x b))
      op e f a) ))

(define pi
  (lambda (f a b)
      (summation_int * 1 f a b)))

(define sigma2
  (lambda (f a b)
      (summation_int + 0 f a b)))

(sigma2 (lambda (n) n) 0 5)         ; 15
(sigma2 (lambda (n) (/ 1 n)) 1 2)   ; 3/2

(define fact
  (lambda (b)
    (pi 
      (lambda (x) x) 
      1
      b) ))

(fact 10) ; 3628800