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March 10, 2019 14:46
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import math | |
import bigfloat | |
from bigfloat import BigFloat, precision | |
MAX_MU = 50 | |
def choose(n,r): | |
f = math.factorial | |
return f(n) // f(r) // f(n-r) | |
def bigchoose(n,r): | |
f = bigfloat.factorial | |
return f(n) / f(r) / f(n-r) | |
def f1(n, mu): | |
s1 = 0 | |
for i in range(1, n + 1): | |
p = 2**(-mu) | |
for j in range(i + 1, n + 1): | |
s2 = 0 | |
for mup in range(1, mu): | |
s2 += 2**(-mup) | |
p *= s2 | |
s1 += p | |
# print(s1) | |
return s1 | |
def f2(n, mu): | |
s1 = 0 | |
for i in range(1, n + 1): | |
p = 2**(-mu) | |
for j in range(i + 1, n + 1): | |
p *= (1 - 2**(1 - mu)) | |
s1 += p | |
# print(s1) | |
return s1 | |
def f3(n, mu): | |
s1 = 0 | |
for i in range(1, n + 1): | |
s1 += (1 - 2**(1 - mu))**(n - i) | |
return 2**(-mu) * s1 | |
def f35(n, mu): | |
s1 = 0 | |
for i in range(1, n + 1): | |
s1 += (1 - 2**(1 - mu))**(-i) | |
return 2**(-mu) * (1 - 2**(1 - mu))**n * s1 | |
def f36(n, mu): | |
s1 = 0 | |
for i in range(0, n): | |
s1 += (1 - 2**(1 - mu))**(-(i + 1)) | |
return 2**(-mu) * (1 - 2**(1 - mu))**n * s1 | |
def f37(n, mu): | |
s1 = 0 | |
for i in range(0, n): | |
s1 += (1 - 2**(1 - mu))**(-i) | |
return 2**(-mu) * (1 - 2**(1 - mu))**n / (1 - 2**(1 - mu)) * s1 | |
def f4(n, mu): | |
n1 = (1 - 2**(1 - mu))**n | |
n2 = 1 - (1 - 2**(1-mu))**(-n) | |
d1 = 1 - 2**(1 - mu) | |
d2 = 1 - (1 - 2**(1 - mu))**(-1) | |
numerator = n1 * n2 | |
denominator = d1 * d2 | |
return 2**(-mu) * numerator / denominator | |
def f5(n, mu): | |
return (1.0/2)*(1 - (1 - 2**(1 - mu))**n) | |
def f6(n): | |
s1 = 0 | |
s3 = 0 | |
# s35 = 0 | |
# s36 = 0 | |
# s37 = 0 | |
# s4 = 0 | |
s5 = 0 | |
# terms = '' | |
for mu in range(1, MAX_MU): | |
Bnmu1 = f1(n, mu) | |
Bnmu3 = f3(n, mu) | |
# Bnmu35 = f35(n - 1, mu) | |
# Bnmu36 = f36(n - 1, mu) | |
# Bnmu37 = f37(n - 1, mu) | |
# Bnmu4 = f4(n - 1, mu) | |
Bnmu5 = f5(n, mu) | |
# terms += ', ' + str(Bnmu5) | |
s1 += Bnmu1 | |
s3 += Bnmu3 | |
# s35 += Bnmu35 | |
# s36 += Bnmu36 | |
# s37 += Bnmu37 | |
# s4 += Bnmu4 | |
s5 += Bnmu5 | |
# print('n = ', n, 's1 = ', s1) | |
# print('n = ', n, 's3 = ', s3) | |
# print('terms = ', terms) | |
# print('n = ', n, 's35 = ', s35) | |
# print('n = ', n, 's36 = ', s36) | |
# print('n = ', n, 's37 = ', s37) | |
# print('n = ', n, 's4 = ', s4) | |
# print('n = ', n, 's5 = ', s5) | |
return s5 | |
def f63(n): | |
s1 = 0 | |
for mu in range(1, MAX_MU): | |
s1 += 1 - (1 - 2**(1 - mu))**n | |
return 1.0/2 * s1 | |
def f64(n): | |
s1 = 0 | |
for mu in range(1, MAX_MU): | |
s2 = 0 | |
for i in range(0, n + 1): | |
s2 += choose(n, i) * 1**(n - i) * (-2**(1 - mu))**i | |
s1 += 1 - s2 | |
return 1.0/2 * s1 | |
def f65(n): | |
s1 = 0 | |
for mu in range(1, MAX_MU): | |
s2 = 0 | |
for i in range(1, n + 1): | |
s2 += choose(n, i) * 1**(n - i) * (-2)**((1 - mu) * i) | |
s1 += -s2 | |
return 1.0/2 * s1 | |
def f66(n): | |
s1 = BigFloat(0) | |
for mu in range(1, MAX_MU): | |
s2 = BigFloat(0) | |
for i in range(0, n + 1): | |
s2 += bigchoose(n, i) * (-(2**(1 - mu))) ** i | |
s1 += 1 - s2 | |
return 1.0/2 * s1 | |
def f67(n): | |
odd_bound = int(math.floor((n - 1) / 2)) | |
even_bound = int(math.floor(n / 2)) | |
s1 = BigFloat(0) | |
for mu in range(1, MAX_MU): | |
s2 = BigFloat(0) | |
for k in range(0, even_bound + 1): | |
s2 += bigchoose(n, 2*k) * 2**((1 - mu) * 2*k) | |
s3 = BigFloat(0) | |
for k in range(0, odd_bound + 1): | |
s3 += bigchoose(n, 2*k + 1) * 2**((1 - mu) * (2*k + 1)) | |
s1 += 1 - s2 + s3 | |
return 1.0/2 * s1 | |
def f68(n): | |
odd_bound = int(math.floor((n - 1) / 2)) | |
even_bound = int(math.floor(n / 2)) | |
s1 = BigFloat(0) | |
for mu in range(1, MAX_MU): | |
s2 = BigFloat(0) | |
for k in range(1, even_bound + 1): | |
s2 += bigchoose(n, 2*k) * 2**((1 - mu) * 2*k) | |
s3 = BigFloat(0) | |
for k in range(0, odd_bound + 1): | |
s3 += bigchoose(n, 2*k + 1) * 2**((1 - mu) * (2*k + 1)) | |
s1 += -s2 + s3 | |
return 1.0/2 * s1 | |
def f69(n): | |
odd_bound = int(math.floor((n - 1) / 2)) | |
even_bound = int(math.floor(n / 2)) | |
s1 = BigFloat(0) | |
for mu in range(1, MAX_MU): | |
s2 = BigFloat(0) | |
for k in range(0, odd_bound + 1): | |
s2 += bigchoose(n, 2*k + 1) * 2**((1 - mu) * (2*k + 1)) | |
s3 = BigFloat(0) | |
for k in range(1, even_bound + 1): | |
s3 += bigchoose(n, 2*k) * 2**((1 - mu) * 2*k) | |
s1 += s2 - s3 | |
return 1.0/2 * s1 | |
def f7(n): | |
odd_bound = int(math.floor((n - 1) / 2)) | |
even_bound = int(math.floor(n / 2)) | |
s1 = 0 | |
for k in range(0, odd_bound + 1): | |
for mu in range(1, MAX_MU): | |
s1 += 2**((1 - mu)*(2*k+1)) | |
s2 = 0 | |
for k in range(1, even_bound + 1): | |
for mu in range(1, MAX_MU): | |
s2 += 2**((1 - mu)*2*k) | |
return 1.0/2 * (choose(n, 2*k + 1) * s1 - choose(n, 2*k) * s2) | |
with precision(200): | |
for n in range(1, 100): | |
print('n = ', n, 'f67 = ', f67(n), ', f68 = ', f68(n)) | |
# f66(n) | |
# f67(n) |
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