def test_tribonacci(): assert [tribonacci(n) for n in range(8)] == [0, 1, 1, 2, 4, 7, 13, 24] assert tribonacci(100) == 98079530178586034536500564 assert tribonacci(0, x) == 0 assert tribonacci(1, x) == 1 assert tribonacci(2, x) == x**2 assert tribonacci(3, x) == x**4 + x assert tribonacci(4, x) == x**6 + 2 * x**3 + 1 assert tribonacci(5, x) == x**8 + 3 * x**5 + 3 * x**2 n = Dummy('n') assert tribonacci(n).limit(n, S.Infinity) == S.Infinity w = (-1 + S.ImaginaryUnit * sqrt(3)) / 2 a = (1 + cbrt(19 + 3 * sqrt(33)) + cbrt(19 - 3 * sqrt(33))) / 3 b = (1 + w * cbrt(19 + 3 * sqrt(33)) + w**2 * cbrt(19 - 3 * sqrt(33))) / 3 c = (1 + w**2 * cbrt(19 + 3 * sqrt(33)) + w * cbrt(19 - 3 * sqrt(33))) / 3 assert tribonacci(n).rewrite(sqrt) == \ (a**(n + 1)/((a - b)*(a - c)) + b**(n + 1)/((b - a)*(b - c)) + c**(n + 1)/((c - a)*(c - b))) assert tribonacci(n).rewrite(sqrt).subs(n, 4).simplify() == tribonacci(4) assert tribonacci(n).rewrite(GoldenRatio).subs(n,10).evalf() == \ tribonacci(10)
def test_tribonacci(): assert [tribonacci(n) for n in range(8)] == [0, 1, 1, 2, 4, 7, 13, 24] assert tribonacci(100) == 98079530178586034536500564 assert tribonacci(0, x) == 0 assert tribonacci(1, x) == 1 assert tribonacci(2, x) == x**2 assert tribonacci(3, x) == x**4 + x assert tribonacci(4, x) == x**6 + 2 * x**3 + 1 assert tribonacci(5, x) == x**8 + 3 * x**5 + 3 * x**2 n = Dummy('n') assert tribonacci(n).limit(n, S.Infinity) is S.Infinity w = (-1 + S.ImaginaryUnit * sqrt(3)) / 2 a = (1 + cbrt(19 + 3 * sqrt(33)) + cbrt(19 - 3 * sqrt(33))) / 3 b = (1 + w * cbrt(19 + 3 * sqrt(33)) + w**2 * cbrt(19 - 3 * sqrt(33))) / 3 c = (1 + w**2 * cbrt(19 + 3 * sqrt(33)) + w * cbrt(19 - 3 * sqrt(33))) / 3 assert tribonacci(n).rewrite(sqrt) == \ (a**(n + 1)/((a - b)*(a - c)) + b**(n + 1)/((b - a)*(b - c)) + c**(n + 1)/((c - a)*(c - b))) assert tribonacci(n).rewrite(sqrt).subs(n, 4).simplify() == tribonacci(4) assert tribonacci(n).rewrite(GoldenRatio).subs(n,10).evalf() == \ tribonacci(10) assert tribonacci(n).rewrite(TribonacciConstant) == floor( 3 * TribonacciConstant**n * (102 * sqrt(33) + 586)**Rational(1, 3) / (-2 * (102 * sqrt(33) + 586)**Rational(1, 3) + 4 + (102 * sqrt(33) + 586)**Rational(2, 3)) + S.Half) raises(ValueError, lambda: tribonacci(-1, x))
def test_tribonacci(): assert [tribonacci(n) for n in range(8)] == [0, 1, 1, 2, 4, 7, 13, 24] assert tribonacci(100) == 98079530178586034536500564 assert tribonacci(0, x) == 0 assert tribonacci(1, x) == 1 assert tribonacci(2, x) == x**2 assert tribonacci(3, x) == x**4 + x assert tribonacci(4, x) == x**6 + 2*x**3 + 1 assert tribonacci(5, x) == x**8 + 3*x**5 + 3*x**2 n = Dummy('n') assert tribonacci(n).limit(n, S.Infinity) == S.Infinity w = (-1 + S.ImaginaryUnit * sqrt(3)) / 2 a = (1 + cbrt(19 + 3*sqrt(33)) + cbrt(19 - 3*sqrt(33))) / 3 b = (1 + w*cbrt(19 + 3*sqrt(33)) + w**2*cbrt(19 - 3*sqrt(33))) / 3 c = (1 + w**2*cbrt(19 + 3*sqrt(33)) + w*cbrt(19 - 3*sqrt(33))) / 3 assert tribonacci(n).rewrite(sqrt) == \ (a**(n + 1)/((a - b)*(a - c)) + b**(n + 1)/((b - a)*(b - c)) + c**(n + 1)/((c - a)*(c - b))) assert tribonacci(n).rewrite(sqrt).subs(n, 4).simplify() == tribonacci(4) assert tribonacci(n).rewrite(GoldenRatio).subs(n,10).evalf() == \ tribonacci(10)
def test_MatPow(): A = MatrixSymbol('A', n, n) AA = MatPow(A, 2) assert AA.exp == 2 assert AA.base == A assert (A**n).exp == n assert A**0 == Identity(n) assert A**1 == A assert A**2 == AA assert A**-1 == Inverse(A) assert (A**-1)**-1 == A assert (A**2)**3 == A**6 assert A**S.Half == sqrt(A) assert A**(S(1) / 3) == cbrt(A) raises(ShapeError, lambda: MatrixSymbol('B', 3, 2)**2)
def test_MatPow(): A = MatrixSymbol('A', n, n) AA = MatPow(A, 2) assert AA.exp == 2 assert AA.base == A assert (A**n).exp == n assert A**0 == Identity(n) assert A**1 == A assert A**2 == AA assert A**-1 == Inverse(A) assert (A**-1)**-1 == A assert (A**2)**3 == A**6 assert A**S.Half == sqrt(A) assert A**(S(1)/3) == cbrt(A) raises(ShapeError, lambda: MatrixSymbol('B', 3, 2)**2)
def _minpoly_compose(ex, x, dom): """ Computes the minimal polynomial of an algebraic element using operations on minimal polynomials Examples ======== >>> from sympy import minimal_polynomial, sqrt, Rational >>> from sympy.abc import x, y >>> minimal_polynomial(sqrt(2) + 3*Rational(1, 3), x, compose=True) x**2 - 2*x - 1 >>> minimal_polynomial(sqrt(y) + 1/y, x, compose=True) x**2*y**2 - 2*x*y - y**3 + 1 """ if ex.is_Rational: return ex.q * x - ex.p if ex is I: _, factors = factor_list(x**2 + 1, x, domain=dom) return x**2 + 1 if len(factors) == 1 else x - I if ex is GoldenRatio: _, factors = factor_list(x**2 - x - 1, x, domain=dom) if len(factors) == 1: return x**2 - x - 1 else: return _choose_factor(factors, x, (1 + sqrt(5)) / 2, dom=dom) if ex is TribonacciConstant: _, factors = factor_list(x**3 - x**2 - x - 1, x, domain=dom) if len(factors) == 1: return x**3 - x**2 - x - 1 else: fac = (1 + cbrt(19 - 3 * sqrt(33)) + cbrt(19 + 3 * sqrt(33))) / 3 return _choose_factor(factors, x, fac, dom=dom) if hasattr(dom, 'symbols') and ex in dom.symbols: return x - ex if dom.is_QQ and _is_sum_surds(ex): # eliminate the square roots ex -= x while 1: ex1 = _separate_sq(ex) if ex1 is ex: return ex else: ex = ex1 if ex.is_Add: res = _minpoly_add(x, dom, *ex.args) elif ex.is_Mul: f = Factors(ex).factors r = sift(f.items(), lambda itx: itx[0].is_Rational and itx[1].is_Rational) if r[True] and dom == QQ: ex1 = Mul(*[bx**ex for bx, ex in r[False] + r[None]]) r1 = dict(r[True]) dens = [y.q for y in r1.values()] lcmdens = reduce(lcm, dens, 1) neg1 = S.NegativeOne expn1 = r1.pop(neg1, S.Zero) nums = [base**(y.p * lcmdens // y.q) for base, y in r1.items()] ex2 = Mul(*nums) mp1 = minimal_polynomial(ex1, x) # use the fact that in SymPy canonicalization products of integers # raised to rational powers are organized in relatively prime # bases, and that in ``base**(n/d)`` a perfect power is # simplified with the root # Powers of -1 have to be treated separately to preserve sign. mp2 = ex2.q * x**lcmdens - ex2.p * neg1**(expn1 * lcmdens) ex2 = neg1**expn1 * ex2**Rational(1, lcmdens) res = _minpoly_op_algebraic_element(Mul, ex1, ex2, x, dom, mp1=mp1, mp2=mp2) else: res = _minpoly_mul(x, dom, *ex.args) elif ex.is_Pow: res = _minpoly_pow(ex.base, ex.exp, x, dom) elif ex.__class__ is sin: res = _minpoly_sin(ex, x) elif ex.__class__ is cos: res = _minpoly_cos(ex, x) elif ex.__class__ is exp: res = _minpoly_exp(ex, x) elif ex.__class__ is CRootOf: res = _minpoly_rootof(ex, x) else: raise NotAlgebraic("%s doesn't seem to be an algebraic element" % ex) return res