| from sympy.concrete.guess import ( | |
| find_simple_recurrence_vector, | |
| find_simple_recurrence, | |
| rationalize, | |
| guess_generating_function_rational, | |
| guess_generating_function, | |
| guess | |
| ) | |
| from sympy.concrete.products import Product | |
| from sympy.core.function import Function | |
| from sympy.core.numbers import Rational | |
| from sympy.core.singleton import S | |
| from sympy.core.symbol import (Symbol, symbols) | |
| from sympy.core.sympify import sympify | |
| from sympy.functions.combinatorial.factorials import (RisingFactorial, factorial) | |
| from sympy.functions.combinatorial.numbers import fibonacci | |
| from sympy.functions.elementary.exponential import exp | |
| def test_find_simple_recurrence_vector(): | |
| assert find_simple_recurrence_vector( | |
| [fibonacci(k) for k in range(12)]) == [1, -1, -1] | |
| def test_find_simple_recurrence(): | |
| a = Function('a') | |
| n = Symbol('n') | |
| assert find_simple_recurrence([fibonacci(k) for k in range(12)]) == ( | |
| -a(n) - a(n + 1) + a(n + 2)) | |
| f = Function('a') | |
| i = Symbol('n') | |
| a = [1, 1, 1] | |
| for k in range(15): a.append(5*a[-1]-3*a[-2]+8*a[-3]) | |
| assert find_simple_recurrence(a, A=f, N=i) == ( | |
| -8*f(i) + 3*f(i + 1) - 5*f(i + 2) + f(i + 3)) | |
| assert find_simple_recurrence([0, 2, 15, 74, 12, 3, 0, | |
| 1, 2, 85, 4, 5, 63]) == 0 | |
| def test_rationalize(): | |
| from mpmath import cos, pi, mpf | |
| assert rationalize(cos(pi/3)) == S.Half | |
| assert rationalize(mpf("0.333333333333333")) == Rational(1, 3) | |
| assert rationalize(mpf("-0.333333333333333")) == Rational(-1, 3) | |
| assert rationalize(pi, maxcoeff = 250) == Rational(355, 113) | |
| def test_guess_generating_function_rational(): | |
| x = Symbol('x') | |
| assert guess_generating_function_rational([fibonacci(k) | |
| for k in range(5, 15)]) == ((3*x + 5)/(-x**2 - x + 1)) | |
| def test_guess_generating_function(): | |
| x = Symbol('x') | |
| assert guess_generating_function([fibonacci(k) | |
| for k in range(5, 15)])['ogf'] == ((3*x + 5)/(-x**2 - x + 1)) | |
| assert guess_generating_function( | |
| [1, 2, 5, 14, 41, 124, 383, 1200, 3799, 12122, 38919])['ogf'] == ( | |
| (1/(x**4 + 2*x**2 - 4*x + 1))**S.Half) | |
| assert guess_generating_function(sympify( | |
| "[3/2, 11/2, 0, -121/2, -363/2, 121, 4719/2, 11495/2, -8712, -178717/2]") | |
| )['ogf'] == (x + Rational(3, 2))/(11*x**2 - 3*x + 1) | |
| assert guess_generating_function([factorial(k) for k in range(12)], | |
| types=['egf'])['egf'] == 1/(-x + 1) | |
| assert guess_generating_function([k+1 for k in range(12)], | |
| types=['egf']) == {'egf': (x + 1)*exp(x), 'lgdegf': (x + 2)/(x + 1)} | |
| def test_guess(): | |
| i0, i1 = symbols('i0 i1') | |
| assert guess([1, 2, 6, 24, 120], evaluate=False) == [Product(i1 + 1, (i1, 1, i0 - 1))] | |
| assert guess([1, 2, 6, 24, 120]) == [RisingFactorial(2, i0 - 1)] | |
| assert guess([1, 2, 7, 42, 429, 7436, 218348, 10850216], niter=4) == [ | |
| 2**(i0 - 1)*(Rational(27, 16))**(i0**2/2 - 3*i0/2 + | |
| 1)*Product(RisingFactorial(Rational(5, 3), i1 - 1)*RisingFactorial(Rational(7, 3), i1 | |
| - 1)/(RisingFactorial(Rational(3, 2), i1 - 1)*RisingFactorial(Rational(5, 2), i1 - | |
| 1)), (i1, 1, i0 - 1))] | |
| assert guess([1, 0, 2]) == [] | |
| x, y = symbols('x y') | |
| assert guess([1, 2, 6, 24, 120], variables=[x, y]) == [RisingFactorial(2, x - 1)] | |