def test_evalf_near_integers(): # Binet's formula f = lambda n: ((1 + sqrt(5))**n) / (2**n * sqrt(5)) assert NS(f(5000) - fibonacci(5000), 10, maxn=1500) == '5.156009964e-1046' # Some near-integer identities from # http://mathworld.wolfram.com/AlmostInteger.html assert NS('sin(2017*2**(1/5))', 15) == '-1.00000000000000' assert NS('sin(2017*2**(1/5))', 20) == '-0.99999999999999997857' assert NS('1+sin(2017*2**(1/5))', 15) == '2.14322287389390e-17' assert NS('45 - 613*E/37 + 35/991', 15) == '6.03764498766326e-11'
def test_evalf_integer_parts(): a = floor(log(8) / log(2) - exp(-1000), evaluate=False) b = floor(log(8) / log(2), evaluate=False) assert a.evalf() == 3 assert b.evalf() == 3 # equals, as a fallback, can still fail but it might succeed as here assert ceiling(10 * (sin(1)**2 + cos(1)**2)) == 10 assert int(floor(factorial(50)/E, evaluate=False).evalf(70)) == \ int(11188719610782480504630258070757734324011354208865721592720336800) assert int(ceiling(factorial(50)/E, evaluate=False).evalf(70)) == \ int(11188719610782480504630258070757734324011354208865721592720336801) assert int(floor(GoldenRatio**999 / sqrt(5) + S.Half).evalf(1000)) == fibonacci(999) assert int(floor(GoldenRatio**1000 / sqrt(5) + S.Half).evalf(1000)) == fibonacci(1000) assert ceiling(x).evalf(subs={x: 3}) == 3 assert ceiling(x).evalf(subs={x: 3 * I}) == 3.0 * I assert ceiling(x).evalf(subs={x: 2 + 3 * I}) == 2.0 + 3.0 * I assert ceiling(x).evalf(subs={x: 3.}) == 3 assert ceiling(x).evalf(subs={x: 3. * I}) == 3.0 * I assert ceiling(x).evalf(subs={x: 2. + 3 * I}) == 2.0 + 3.0 * I assert float((floor(1.5, evaluate=False) + 1 / 9).evalf()) == 1 + 1 / 9 assert float((floor(0.5, evaluate=False) + 20).evalf()) == 20 # issue 19991 n = 1169809367327212570704813632106852886389036911 r = 744723773141314414542111064094745678855643068 assert floor(n / (pi / 2)) == r assert floor(80782 * sqrt(2)) == 114242 # issue 20076 assert 260515 - floor(260515 / pi + 1 / 2) * pi == atan(tan(260515))
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_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_approximants(): x, t = symbols("x,t") g = [lucas(k) for k in range(16)] assert [e for e in approximants(g)] == ([ 2, -4 / (x - 2), (5 * x - 2) / (3 * x - 1), (x - 2) / (x**2 + x - 1) ]) g = [lucas(k) + fibonacci(k + 2) for k in range(16)] assert [e for e in approximants(g)] == ([ 3, -3 / (x - 1), (3 * x - 3) / (2 * x - 1), -3 / (x**2 + x - 1) ]) g = [lucas(k)**2 for k in range(16)] assert [e for e in approximants(g)] == ([ 4, -16 / (x - 4), (35 * x - 4) / (9 * x - 1), (37 * x - 28) / (13 * x**2 + 11 * x - 7), (50 * x**2 + 63 * x - 52) / (37 * x**2 + 19 * x - 13), (-x**2 - 7 * x + 4) / (x**3 - 2 * x**2 - 2 * x + 1) ]) p = [sum(binomial(k, i) * x**i for i in range(k + 1)) for k in range(16)] y = approximants(p, t, simplify=True) assert next(y) == 1 assert next(y) == -1 / (t * (x + 1) - 1)
def test_find_linear_recurrence(): assert sequence((0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55), \ (n, 0, 10)).find_linear_recurrence(11) == [1, 1] assert sequence((1, 2, 4, 7, 28, 128, 582, 2745, 13021, 61699, 292521, \ 1387138), (n, 0, 11)).find_linear_recurrence(12) == [5, -2, 6, -11] assert sequence(x*n**3+y*n, (n, 0, oo)).find_linear_recurrence(10) \ == [4, -6, 4, -1] assert sequence(x**n, (n,0,20)).find_linear_recurrence(21) == [x] assert sequence((1,2,3)).find_linear_recurrence(10, 5) == [0, 0, 1] assert sequence(((1 + sqrt(5))/2)**n + \ (-(1 + sqrt(5))/2)**(-n)).find_linear_recurrence(10) == [1, 1] assert sequence(x*((1 + sqrt(5))/2)**n + y*(-(1 + sqrt(5))/2)**(-n), \ (n,0,oo)).find_linear_recurrence(10) == [1, 1] assert sequence((1,2,3,4,6),(n, 0, 4)).find_linear_recurrence(5) == [] assert sequence((2,3,4,5,6,79),(n, 0, 5)).find_linear_recurrence(6,gfvar=x) \ == ([], None) assert sequence((2,3,4,5,8,30),(n, 0, 5)).find_linear_recurrence(6,gfvar=x) \ == ([Rational(19, 2), -20, Rational(27, 2)], (-31*x**2 + 32*x - 4)/(27*x**3 - 40*x**2 + 19*x -2)) assert sequence(fibonacci(n)).find_linear_recurrence(30,gfvar=x) \ == ([1, 1], -x/(x**2 + x - 1)) assert sequence(tribonacci(n)).find_linear_recurrence(30,gfvar=x) \ == ([1, 1, 1], -x/(x**3 + x**2 + x - 1))
def test_sympy__functions__combinatorial__numbers__fibonacci(): from sympy.functions.combinatorial.numbers import fibonacci assert _test_args(fibonacci(x))
def test_issue_10382(): n = Symbol('n', integer=True) assert limit(fibonacci(n+1)/fibonacci(n), n, oo) == S.GoldenRatio
def test_sympy__functions__combinatorial__numbers__fibonacci(): from sympy.functions.combinatorial.numbers import fibonacci assert _test_args(fibonacci(x))
def fibonacci(n: Union[int, float]) -> Union[int, float]: try: import sympy.functions.combinatorial.numbers as ns except ModuleNotFoundError: raise Exception("Install sympy to use number-theoretic functions!") return ns.fibonacci(n)
def test_linrec(): assert linrec(coeffs=[1, 1], init=[1, 1], n=20) == 10946 assert linrec(coeffs=[1, 2, 3, 4, 5], init=[1, 1, 0, 2], n=10) == 1040 assert linrec(coeffs=[0, 0, 11, 13], init=[23, 27], n=25) == 59628567384 assert linrec(coeffs=[0, 0, 1, 1, 2], init=[1, 5, 3], n=15) == 165 assert linrec(coeffs=[11, 13, 15, 17], init=[1, 2, 3, 4], n=70) == \ 56889923441670659718376223533331214868804815612050381493741233489928913241 assert linrec(coeffs=[0]*55 + [1, 1, 2, 3], init=[0]*50 + [1, 2, 3], n=4000) == \ 702633573874937994980598979769135096432444135301118916539 assert linrec(coeffs=[11, 13, 15, 17], init=[1, 2, 3, 4], n=10**4) assert linrec(coeffs=[11, 13, 15, 17], init=[1, 2, 3, 4], n=10**5) assert all( linrec(coeffs=[1, 1], init=[0, 1], n=n) == fibonacci(n) for n in range(95, 115)) assert all( linrec(coeffs=[1, 1], init=[1, 1], n=n) == fibonacci(n + 1) for n in range(595, 615)) a = [ S.Half, Rational(3, 4), Rational(5, 6), 7, Rational(8, 9), Rational(3, 5) ] b = [1, 2, 8, Rational(5, 7), Rational(3, 7), Rational(2, 9), 6] x, y, z = symbols('x y z') assert linrec(coeffs=a[:5], init=b[:4], n=80) == \ Rational(1726244235456268979436592226626304376013002142588105090705187189, 1960143456748895967474334873705475211264) assert linrec(coeffs=a[:4], init=b[:4], n=50) == \ Rational(368949940033050147080268092104304441, 504857282956046106624) assert linrec(coeffs=a[3:], init=b[:3], n=35) == \ Rational(97409272177295731943657945116791049305244422833125109, 814315512679031689453125) assert linrec(coeffs=[0]*60 + [Rational(2, 3), Rational(4, 5)], init=b, n=3000) == \ Rational(26777668739896791448594650497024, 48084516708184142230517578125) raises(TypeError, lambda: linrec(coeffs=[11, 13, 15, 17], init=[1, 2, 3, 4, 5], n=1)) raises(TypeError, lambda: linrec(coeffs=a[:4], init=b[:5], n=10000)) raises(ValueError, lambda: linrec(coeffs=a[:4], init=b[:4], n=-10000)) raises(TypeError, lambda: linrec(x, b, n=10000)) raises(TypeError, lambda: linrec(a, y, n=10000)) assert linrec(coeffs=[x, y, z], init=[1, 1, 1], n=4) == \ x**2 + x*y + x*z + y + z assert linrec(coeffs=[1, 2, 1], init=[x, y, z], n=20) == \ 269542*x + 664575*y + 578949*z assert linrec(coeffs=[0, 3, 1, 2], init=[x, y], n=30) == \ 58516436*x + 56372788*y assert linrec(coeffs=[0]*50 + [1, 2, 3], init=[x, y, z], n=1000) == \ 11477135884896*x + 25999077948732*y + 41975630244216*z assert linrec(coeffs=[], init=[1, 1], n=20) == 0 assert linrec(coeffs=[x, y, z], init=[1, 2, 3], n=2) == 3
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_find_simple_recurrence_vector(): assert find_simple_recurrence_vector([fibonacci(k) for k in range(12)]) == [1, -1, -1]