def rational_seq(lo,hi,step):
"""Sequence of rational numbers"""
lo = cast_to_rational(lo)
hi = cast_to_rational(hi)
step = cast_to_rational(step)
out = [lo]
while out[-1]+step <= hi:
out.append(out[-1]+step)
return out
def harmonic_progression(a=1, d=1):
a = cast_to_rational(a)
d = cast_to_rational(d)
den = a
one = Rational(1)
while True:
yield one / den
den += d
def rational_round(Q,dlim):
"""Best approximation of Q with denominator of d or less, by semi-convergents"""
Q = cast_to_rational(Q)
if type(dlim) != int:
raise TypeError("dlim must be integer")
if dlim < 1:
raise ZeroDivisionError
# https://shreevatsa.wordpress.com/2011/01/10/not-all-best-rational-approximations-are-the-convergents-of-the-continued-fraction/
a = CFrac(Q).terms
prev = Rational(a[0])
for pos,val in enumerate(a):
# Try appending the floor of half the next convergent
semi = a[:pos]+[(val-1)//2+1]
semi = CFrac(semi)
# If it is worse than the last semiconvergent add 1
if abs(semi.as_rational() - Q) > abs(prev - Q):
semi[pos] += 1
while semi.terms[pos] <= val:
if semi.as_rational().d > dlim:
semi[pos] -= 1
return prev
prev = semi.as_rational()
semi[pos] += 1
return Q
def sign(Q):
"""Sign of a rational number"""
Q = cast_to_rational(Q)
if Q.n > 0:
return 1
elif Q.n < 0:
return -1
else:
return 0
def engel_expansion(Q):
Q = cast_to_rational(Q)
u = Q
out = []
while u != 0:
a = (1/u).__ceil__()
out.append(a)
u = u*a-1
return out
def mediant(a,b):
a = cast_to_rational(a)
b = cast_to_rational(b)
return Rational(a.n+b.n,a.d+b.d)