sicp/zapiski/g1smo/1_3_zapiski_vaje.org

Zapiski #3 srečanja programerskega bralnega krožka SICP

Teme

Funkcije višjega reda

Vaje

1.29 Simpsonovo pravilo

  (define(cube x)
    (* x x x))

  (define(sum-integers a b)
    (if ( > a b)
        0
        (+ a (sum-integers (+ a 1) b))))

  (define(sum-cubes a b)
    (if (> a b)
        0
        (+ (cube a) (sum-cubes(+ a 1) b))))


  (define (sum term a next b)
    (if (> a b)
        0
        (+ (term a)
           (sum term (next a) next b))))

  (define (integral f a b dx)
    (define (add-dx x) (+ x dx))
    (* (sum f (+ a (/ dx 2.0)) add-dx b)
       dx))

  (define (simpson f a b n)
    (define h (/ (- b a) n))
    (define (add-hh a) (+ a h h))
    (* (+ (- (f a))
          (* 2 (sum f a add-hh b))
          (* 4 (sum f (+ a h) add-hh b))
          (- (f b)))
       (/ h 3.0)))


  (list (simpson cube 1 2 100)
        (simpson cube 1 2 1000))

3.7500000000000004

3.75

1.30 linearen sum

Rekurzivne procedure:

  (define (sum term a next b)
    (if (> a b)
        0
        (+ (term a)
           (sum term (next a) next b))))
         
  (define (cube n) (* n n n))
  (define (inc n) (+ n 1))

  (define (sum-cubes a b)
    (sum cube a inc b))
  
  (sum-cubes 1 10)

3025

Iterativne procedure:

  (define (sum-iter term a next b)
    (define (iter a result)
      (if (> a b)
         result
         (iter (next a) (+ result (term a)))))
    (iter a 0))
    
  (define (cube n) (* n n n))
  (define (inc n) (+ n 1))
  
  (define (sum-cubes-iter a b)
    (sum-iter cube a inc b))

  (sum-cubes-iter 1 10)

3025

1.31 Produkt višjega reda

  (define (prod-iter term a next b)
    (define (iter a result)
      (if (> a b)
         result
         (iter (next a) (* result (term a)))))
    (iter a 1))

  (define (piblizek n)
    (define (stevec m)
      (+ m 2))
    (* 4 (/ (prod-iter (lambda (a) (* (- a 1) (+ a 1)))
                       3
                       stevec
                       n)
            (prod-iter (lambda (b) (* b b))
                       3
                       stevec
                       n))))

  (+ (piblizek 10000) 0.0)

3.1417497371492673

1.32 Akumulator (reduce funkcija)

  (define (accumulate combiner null-value term a next b)
    (if (> a b)
        null-value
        (combiner (term a) (accumulate combiner null-value term (next a) next b))))

  (define (sum term a next b)
    (accumulate + 0 term a next b))

  (define (cube n) (* n n n))

  (define (sum-cubes a b) (sum cube a (lambda (n) (+ n 1)) b))

  (define (prod term a next b)
    (accumulate * 1 term a next b))

  (define (piblizek n)
    (define (stevec m)
      (+ m 2))
    (* 4 (/ (prod (lambda (a) (* (- a 1) (+ a 1)))
                  3
                  stevec
                  n)
            (prod (lambda (b) (* b b))
                  3
                  stevec
                  n))))

  (list (sum-cubes 1 10) (+ (piblizek 10000) 0.0))

3025	3.1417497371492673

1.32b Akumulator iterativno (reduce funkcija)

  (define (accumulate-iter combiner null-value term a next b)
    (define (combine a combination)
      (if (> a b)
          combination
          (combine (next a) (combiner combination (term a)))))
    (combine a null-value))

  (define (sum term a next b)
    (accumulate-iter + 0 term a next b))

  (define (cube n) (* n n n))

  (define (sum-cubes a b) (sum cube a (lambda (n) (+ n 1)) b))

  (define (prod term a next b)
    (accumulate-iter * 1 term a next b))

  (define (piblizek n)
    (define (stevec m)
      (+ m 2))
    (* 4 (/ (prod (lambda (a) (* (- a 1) (+ a 1)))
                  3
                  stevec
                  n)
            (prod (lambda (b) (* b b))
                  3
                  stevec
                  n))))

  (list (sum-cubes 1 10) (+ (piblizek 10000) 0.0))

3025	3.1417497371492673

1.33 filtriran akumulator

a) vsota kvadratov praštevil med a in b

  (define (accumulate-filter combiner null-value term a next b predikat?)
    (if (> a b)
        null-value
        (if (predikat? a)
            (combiner (term a) (accumulate-filter combiner null-value term (next a) next b predikat?))
            (accumulate-filter combiner null-value term (next a) next b predikat?))))

  (define (je-prastevilo? a)
    (define (preveri-prastevilo? x a)
      (if (> x (/ a 2))
          #t
          (and (not (integer? (/ a x)))
               (preveri-prastevilo? (+ x 1) a))))
    (preveri-prastevilo? 2 a))

  ;; (vsota kvadratov praštevil med 1 in 12 je 209)
  (= (accumulate-filter + 0 (lambda (n) (* n n)) 1 (lambda (n) (+ n 1)) 12 je-prastevilo?)
     209)

#t

a) produkt vseh naravnih števil manjših od n ki so "relativno pra-n" (torej si z n ne delijo skupnega delitelja razen n)

  (define (accumulate-filter combiner null-value term a next b predikat?)
    (if (> a b)
        null-value
        (if (predikat? a)
            (combiner (term a) (accumulate-filter combiner null-value term (next a) next b predikat?))
            (accumulate-filter combiner null-value term (next a) next b predikat?))))

  (define (id a) a)
  (define (inc a) (+ a 1))
  (define (IN a b) (and a b))

  (define (relativni-prastevili? a b)
    (accumulate-filter IN #t (lambda (x) (and (not (integer? (/ a x)))
                                              (not (integer? (/ b x)))))
                       2 inc a id))

  (define (produkt-relativnih-prastevil n)
    (accumulate-filter * 1 id 1 inc n (lambda (x) (relativni-prastevili? x n))))

  (produkt-relativnih-prastevil 9)

TODO popravi!! ^