# Source code for sympy.ntheory.egyptian_fraction

from __future__ import print_function, division

import sympy.polys
from sympy import Integer
from fractions import gcd

[docs]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 a positive rational number. 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(Rational(3, 7), "Graham Jewett") [7, 8, 9, 56, 57, 72, 3192] >>> egyptian_fraction(Rational(3, 7), "Takenouchi") [4, 7, 28] >>> egyptian_fraction(Rational(3, 7), "Golomb") [3, 15, 35] >>> egyptian_fraction(Rational(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] http://en.wikipedia.org/wiki/Egyptian_fraction .. [2] https://en.wikipedia.org/wiki/Greedy_algorithm_for_Egyptian_fractions .. [3] http://www.ics.uci.edu/~eppstein/numth/egypt/conflict.html .. [4] http://ami.ektf.hu/uploads/papers/finalpdf/AMI_42_from129to134.pdf """ if r <= 0: raise ValueError("Value must be positive") prefix, rem = egypt_harmonic(r) if rem == 0: return prefix x, y = rem.as_numer_denom() if algorithm == "Greedy": return prefix + egypt_greedy(x, y) elif algorithm == "Graham Jewett": return prefix + egypt_graham_jewett(x, y) elif algorithm == "Takenouchi": return prefix + egypt_takenouchi(x, y) elif algorithm == "Golomb": return prefix + egypt_golomb(x, y) else: raise ValueError("Entered invalid algorithm")
def egypt_greedy(x, y): 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): 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): 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): if x == 1: return [y] xp = sympy.polys.ZZ.invert(int(x), int(y)) rv = [Integer(xp*y)] rv.extend(egypt_golomb((x*xp - 1)//y, xp)) return sorted(rv) def egypt_harmonic(r): rv = [] d = Integer(1) acc = Integer(0) while acc + 1/d <= r: acc += 1/d rv.append(d) d += 1 return (rv, r - acc)