Reusing computations Jordi Cortadella Department of Computer Science

Reusing computations Jordi Cortadella Department of Computer Science

Reusing computations • Introduction to Programming © Dept. CS, UPC 2

Calculating • Introduction to Programming © Dept. CS, UPC 3

Calculating • Introduction to Programming © Dept. CS, UPC 4

Calculating : naïve approach def pi(nterms): """Returns an estimation of pi using nterms of the series (nterms > 0) """ sum = 1 # Approximation of /2 # Inv: sum is an approximation of /2 with k terms, # term is the k-th term of the series. for k in range(1, nterms): term = factorial(k)/double_factorial(2 k + 1) sum += term return 2 sum Introduction to Programming © Dept. CS, UPC 5

Calculating : reusing computations • Introduction to Programming © Dept. CS, UPC 6

Calculating more efficiently def pi(nterms): """Returns an estimation of pi using nterms of the series (nterms > 0) """ sum = 1 # Approximation of /2 term = 1 # Current term of the sum # Inv: sum is an approximation of /2 with k terms, # term is the k-th term of the series. for k in range(1, nterms): term *= k/(2*k + 1) sum += term return 2 sum Introduction to Programming © Dept. CS, UPC 7

Calculating • = 3. 14159265358979323846264338327950288. . . • The series approximation: Introduction to Programming nterms 1 5 10 Pi(nterms) 2. 000000 3. 098413 3. 140578 15 20 25 3. 141566 3. 141592 3. 141593 © Dept. CS, UPC 8

Taylor and Maclaurin series • Introduction to Programming © Dept. CS, UPC 9

Calculating sin x • Maclaurin series: • It is a periodic function (period is 2 ) • Convergence improves as x gets closer to zero Introduction to Programming © Dept. CS, UPC 10

Calculating sin x • Reducing the computation to the (-2 , 2 ) interval: • Incremental computation of terms: Introduction to Programming © Dept. CS, UPC 11

Calculating sin x import math def sin_approx(x): """Returns an approximation of sin x. """ k = int(x/(2 math. pi)) x = x – 2 k math. pi # reduce to the (-2 , 2 ) interval term = x x 2 = x x i = 0 sinx = term while abs(term) >= 1 e-8: term = -term x 2/((2 i+2) (2 i+3)) sinx += term i += 1 return sinx Introduction to Programming © Dept. CS, UPC 12

Calculating sin x import math def sin_approx(x): """Returns an approximation of sin x. """ k = int(x/(2 math. pi)) x = x – 2 k math. pi # reduce to the (-2 , 2 ) interval term = x x 2 = x x d = 0 sinx = term while abs(term) >= 1 e-8: d += 2 # d = 2*i+2 term *= -x 2/(d (d+1)) sinx += term return sinx Introduction to Programming © Dept. CS, UPC 13

Combinations • Introduction to Programming © Dept. CS, UPC 14

Combinations def combinations(n, k): """Returns combinations(n k). Pre: n k 0. """ c = 1 for i in range(k, 0, -1): c = c*n/i n -= 1 return c Problem: real division c n/i may not be integer Introduction to Programming © Dept. CS, UPC 15

Combinations: recursive solution def combinations(n, k): """Returns combinations(n k). Pre: n k 0. """ if k == 0: return 1 return n*combinations(n– 1, k– 1)//k Always an integer division! Introduction to Programming © Dept. CS, UPC 16

Conclusions • Try to avoid the brute force to perform complex computations. • Reuse previous computations whenever possible. • Use your knowledge about the domain of the problem to minimize the number of operations. Introduction to Programming © Dept. CS, UPC 17