You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
224 lines
6.8 KiB
224 lines
6.8 KiB
from sympy.core.containers import Tuple
|
|
from sympy.core.numbers import (Integer, Rational)
|
|
from sympy.core.singleton import S
|
|
import sympy.polys
|
|
|
|
from math import gcd
|
|
|
|
|
|
def egyptian_fraction(r, algorithm="Greedy"):
|
|
"""
|
|
Return the list of denominators of an Egyptian fraction
|
|
expansion [1]_ of the said rational `r`.
|
|
|
|
Parameters
|
|
==========
|
|
|
|
r : Rational or (p, q)
|
|
a positive rational number, ``p/q``.
|
|
algorithm : { "Greedy", "Graham Jewett", "Takenouchi", "Golomb" }, optional
|
|
Denotes the algorithm to be used (the default is "Greedy").
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy import Rational
|
|
>>> from sympy.ntheory.egyptian_fraction import egyptian_fraction
|
|
>>> egyptian_fraction(Rational(3, 7))
|
|
[3, 11, 231]
|
|
>>> egyptian_fraction((3, 7), "Graham Jewett")
|
|
[7, 8, 9, 56, 57, 72, 3192]
|
|
>>> egyptian_fraction((3, 7), "Takenouchi")
|
|
[4, 7, 28]
|
|
>>> egyptian_fraction((3, 7), "Golomb")
|
|
[3, 15, 35]
|
|
>>> egyptian_fraction((11, 5), "Golomb")
|
|
[1, 2, 3, 4, 9, 234, 1118, 2580]
|
|
|
|
See Also
|
|
========
|
|
|
|
sympy.core.numbers.Rational
|
|
|
|
Notes
|
|
=====
|
|
|
|
Currently the following algorithms are supported:
|
|
|
|
1) Greedy Algorithm
|
|
|
|
Also called the Fibonacci-Sylvester algorithm [2]_.
|
|
At each step, extract the largest unit fraction less
|
|
than the target and replace the target with the remainder.
|
|
|
|
It has some distinct properties:
|
|
|
|
a) Given `p/q` in lowest terms, generates an expansion of maximum
|
|
length `p`. Even as the numerators get large, the number of
|
|
terms is seldom more than a handful.
|
|
|
|
b) Uses minimal memory.
|
|
|
|
c) The terms can blow up (standard examples of this are 5/121 and
|
|
31/311). The denominator is at most squared at each step
|
|
(doubly-exponential growth) and typically exhibits
|
|
singly-exponential growth.
|
|
|
|
2) Graham Jewett Algorithm
|
|
|
|
The algorithm suggested by the result of Graham and Jewett.
|
|
Note that this has a tendency to blow up: the length of the
|
|
resulting expansion is always ``2**(x/gcd(x, y)) - 1``. See [3]_.
|
|
|
|
3) Takenouchi Algorithm
|
|
|
|
The algorithm suggested by Takenouchi (1921).
|
|
Differs from the Graham-Jewett algorithm only in the handling
|
|
of duplicates. See [3]_.
|
|
|
|
4) Golomb's Algorithm
|
|
|
|
A method given by Golumb (1962), using modular arithmetic and
|
|
inverses. It yields the same results as a method using continued
|
|
fractions proposed by Bleicher (1972). See [4]_.
|
|
|
|
If the given rational is greater than or equal to 1, a greedy algorithm
|
|
of summing the harmonic sequence 1/1 + 1/2 + 1/3 + ... is used, taking
|
|
all the unit fractions of this sequence until adding one more would be
|
|
greater than the given number. This list of denominators is prefixed
|
|
to the result from the requested algorithm used on the remainder. For
|
|
example, if r is 8/3, using the Greedy algorithm, we get [1, 2, 3, 4,
|
|
5, 6, 7, 14, 420], where the beginning of the sequence, [1, 2, 3, 4, 5,
|
|
6, 7] is part of the harmonic sequence summing to 363/140, leaving a
|
|
remainder of 31/420, which yields [14, 420] by the Greedy algorithm.
|
|
The result of egyptian_fraction(Rational(8, 3), "Golomb") is [1, 2, 3,
|
|
4, 5, 6, 7, 14, 574, 2788, 6460, 11590, 33062, 113820], and so on.
|
|
|
|
References
|
|
==========
|
|
|
|
.. [1] https://en.wikipedia.org/wiki/Egyptian_fraction
|
|
.. [2] https://en.wikipedia.org/wiki/Greedy_algorithm_for_Egyptian_fractions
|
|
.. [3] https://www.ics.uci.edu/~eppstein/numth/egypt/conflict.html
|
|
.. [4] https://web.archive.org/web/20180413004012/https://ami.ektf.hu/uploads/papers/finalpdf/AMI_42_from129to134.pdf
|
|
|
|
"""
|
|
|
|
if not isinstance(r, Rational):
|
|
if isinstance(r, (Tuple, tuple)) and len(r) == 2:
|
|
r = Rational(*r)
|
|
else:
|
|
raise ValueError("Value must be a Rational or tuple of ints")
|
|
if r <= 0:
|
|
raise ValueError("Value must be positive")
|
|
|
|
# common cases that all methods agree on
|
|
x, y = r.as_numer_denom()
|
|
if y == 1 and x == 2:
|
|
return [Integer(i) for i in [1, 2, 3, 6]]
|
|
if x == y + 1:
|
|
return [S.One, y]
|
|
|
|
prefix, rem = egypt_harmonic(r)
|
|
if rem == 0:
|
|
return prefix
|
|
# work in Python ints
|
|
x, y = rem.p, rem.q
|
|
# assert x < y and gcd(x, y) = 1
|
|
|
|
if algorithm == "Greedy":
|
|
postfix = egypt_greedy(x, y)
|
|
elif algorithm == "Graham Jewett":
|
|
postfix = egypt_graham_jewett(x, y)
|
|
elif algorithm == "Takenouchi":
|
|
postfix = egypt_takenouchi(x, y)
|
|
elif algorithm == "Golomb":
|
|
postfix = egypt_golomb(x, y)
|
|
else:
|
|
raise ValueError("Entered invalid algorithm")
|
|
return prefix + [Integer(i) for i in postfix]
|
|
|
|
|
|
def egypt_greedy(x, y):
|
|
# assumes gcd(x, y) == 1
|
|
if x == 1:
|
|
return [y]
|
|
else:
|
|
a = (-y) % x
|
|
b = y*(y//x + 1)
|
|
c = gcd(a, b)
|
|
if c > 1:
|
|
num, denom = a//c, b//c
|
|
else:
|
|
num, denom = a, b
|
|
return [y//x + 1] + egypt_greedy(num, denom)
|
|
|
|
|
|
def egypt_graham_jewett(x, y):
|
|
# assumes gcd(x, y) == 1
|
|
l = [y] * x
|
|
|
|
# l is now a list of integers whose reciprocals sum to x/y.
|
|
# we shall now proceed to manipulate the elements of l without
|
|
# changing the reciprocated sum until all elements are unique.
|
|
|
|
while len(l) != len(set(l)):
|
|
l.sort() # so the list has duplicates. find a smallest pair
|
|
for i in range(len(l) - 1):
|
|
if l[i] == l[i + 1]:
|
|
break
|
|
# we have now identified a pair of identical
|
|
# elements: l[i] and l[i + 1].
|
|
# now comes the application of the result of graham and jewett:
|
|
l[i + 1] = l[i] + 1
|
|
# and we just iterate that until the list has no duplicates.
|
|
l.append(l[i]*(l[i] + 1))
|
|
return sorted(l)
|
|
|
|
|
|
def egypt_takenouchi(x, y):
|
|
# assumes gcd(x, y) == 1
|
|
# special cases for 3/y
|
|
if x == 3:
|
|
if y % 2 == 0:
|
|
return [y//2, y]
|
|
i = (y - 1)//2
|
|
j = i + 1
|
|
k = j + i
|
|
return [j, k, j*k]
|
|
l = [y] * x
|
|
while len(l) != len(set(l)):
|
|
l.sort()
|
|
for i in range(len(l) - 1):
|
|
if l[i] == l[i + 1]:
|
|
break
|
|
k = l[i]
|
|
if k % 2 == 0:
|
|
l[i] = l[i] // 2
|
|
del l[i + 1]
|
|
else:
|
|
l[i], l[i + 1] = (k + 1)//2, k*(k + 1)//2
|
|
return sorted(l)
|
|
|
|
|
|
def egypt_golomb(x, y):
|
|
# assumes x < y and gcd(x, y) == 1
|
|
if x == 1:
|
|
return [y]
|
|
xp = sympy.polys.ZZ.invert(int(x), int(y))
|
|
rv = [xp*y]
|
|
rv.extend(egypt_golomb((x*xp - 1)//y, xp))
|
|
return sorted(rv)
|
|
|
|
|
|
def egypt_harmonic(r):
|
|
# assumes r is Rational
|
|
rv = []
|
|
d = S.One
|
|
acc = S.Zero
|
|
while acc + 1/d <= r:
|
|
acc += 1/d
|
|
rv.append(d)
|
|
d += 1
|
|
return (rv, r - acc)
|