def test_to_Sequence(): x = symbols('x') R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx') n = symbols('n', integer=True) _, Sn = RecurrenceOperators(ZZ.old_poly_ring(n), 'Sn') p = HolonomicFunction(x**2 * Dx**4 + x + Dx, x).to_sequence() q = [(HolonomicSequence(1 + (n + 2) * Sn**2 + (n**4 + 6 * n**3 + 11 * n**2 + 6 * n) * Sn**3), 0, 1)] assert p == q p = HolonomicFunction(x**2 * Dx**4 + x**3 + Dx**2, x).to_sequence() q = [ (HolonomicSequence(1 + (n**4 + 14 * n**3 + 72 * n**2 + 163 * n + 140) * Sn**5), 0, 0) ] assert p == q p = HolonomicFunction(x**3 * Dx**4 + 1 + Dx**2, x).to_sequence() q = [(HolonomicSequence(1 + (n**4 - 2 * n**3 - n**2 + 2 * n) * Sn + (n**2 + 3 * n + 2) * Sn**2), 0, 0)] assert p == q p = HolonomicFunction(3 * x**3 * Dx**4 + 2 * x * Dx + x * Dx**3, x).to_sequence() q = [(HolonomicSequence(2 * n + (3 * n**4 - 6 * n**3 - 3 * n**2 + 6 * n) * Sn + (n**3 + 3 * n**2 + 2 * n) * Sn**2), 0, 1)] assert p == q
def test_HolonomicFunction_multiplication(): x = symbols("x") R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), "Dx") p = HolonomicFunction(Dx + x + x * Dx ** 2, x) q = HolonomicFunction(x * Dx + Dx * x + Dx ** 2, x) r = HolonomicFunction( (8 * x ** 6 + 4 * x ** 4 + 6 * x ** 2 + 3) + (24 * x ** 5 - 4 * x ** 3 + 24 * x) * Dx + (8 * x ** 6 + 20 * x ** 4 + 12 * x ** 2 + 2) * Dx ** 2 + (8 * x ** 5 + 4 * x ** 3 + 4 * x) * Dx ** 3 + (2 * x ** 4 + x ** 2) * Dx ** 4, x, ) assert p * q == r p = HolonomicFunction(Dx ** 2 + 1, x) q = HolonomicFunction(Dx - 1, x) r = HolonomicFunction((2) + (-2) * Dx + (1) * Dx ** 2, x) assert p * q == r p = HolonomicFunction(Dx ** 2 + 1 + x + Dx, x) q = HolonomicFunction((Dx * x - 1) ** 2, x) r = HolonomicFunction( (4 * x ** 7 + 11 * x ** 6 + 16 * x ** 5 + 4 * x ** 4 - 6 * x ** 3 - 7 * x ** 2 - 8 * x - 2) + (8 * x ** 6 + 26 * x ** 5 + 24 * x ** 4 - 3 * x ** 3 - 11 * x ** 2 - 6 * x - 2) * Dx + (8 * x ** 6 + 18 * x ** 5 + 15 * x ** 4 - 3 * x ** 3 - 6 * x ** 2 - 6 * x - 2) * Dx ** 2 + (8 * x ** 5 + 10 * x ** 4 + 6 * x ** 3 - 2 * x ** 2 - 4 * x) * Dx ** 3 + (4 * x ** 5 + 3 * x ** 4 - x ** 2) * Dx ** 4, x, ) assert p * q == r p = HolonomicFunction(x * Dx ** 2 - 1, x) q = HolonomicFunction(Dx * x - x, x) r = HolonomicFunction((x - 3) + (-2 * x + 2) * Dx + (x) * Dx ** 2, x) assert p * q == r
def test_HolonomicFunction_composition(): x = symbols("x") R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), "Dx") p = HolonomicFunction(Dx - 1, x).composition(x ** 2 + x) r = HolonomicFunction((-2 * x - 1) + Dx, x) assert p == r p = HolonomicFunction(Dx ** 2 + 1, x).composition(x ** 5 + x ** 2 + 1) r = HolonomicFunction( (125 * x ** 12 + 150 * x ** 9 + 60 * x ** 6 + 8 * x ** 3) + (-20 * x ** 3 - 2) * Dx + (5 * x ** 4 + 2 * x) * Dx ** 2, x, ) assert p == r p = HolonomicFunction(Dx ** 2 * x + x, x).composition(2 * x ** 3 + x ** 2 + 1) r = HolonomicFunction( (216 * x ** 9 + 324 * x ** 8 + 180 * x ** 7 + 152 * x ** 6 + 112 * x ** 5 + 36 * x ** 4 + 4 * x ** 3) + (24 * x ** 4 + 16 * x ** 3 + 3 * x ** 2 - 6 * x - 1) * Dx + (6 * x ** 5 + 5 * x ** 4 + x ** 3 + 3 * x ** 2 + x) * Dx ** 2, x, ) assert p == r p = HolonomicFunction(Dx ** 2 + 1, x).composition(1 - x ** 2) r = HolonomicFunction((4 * x ** 3) - Dx + x * Dx ** 2, x) assert p == r p = HolonomicFunction(Dx ** 2 + 1, x).composition(x - 2 / (x ** 2 + 1)) r = HolonomicFunction( ( x ** 12 + 6 * x ** 10 + 12 * x ** 9 + 15 * x ** 8 + 48 * x ** 7 + 68 * x ** 6 + 72 * x ** 5 + 111 * x ** 4 + 112 * x ** 3 + 54 * x ** 2 + 12 * x + 1 ) + (12 * x ** 8 + 32 * x ** 6 + 24 * x ** 4 - 4) * Dx + ( x ** 12 + 6 * x ** 10 + 4 * x ** 9 + 15 * x ** 8 + 16 * x ** 7 + 20 * x ** 6 + 24 * x ** 5 + 15 * x ** 4 + 16 * x ** 3 + 6 * x ** 2 + 4 * x + 1 ) * Dx ** 2, x, ) assert p == r
def test_series(): x = symbols('x') R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx') p = HolonomicFunction(Dx**2 + 2*x*Dx, x, 0, [0, 1]).series(n=10) q = x - x**3/3 + x**5/10 - x**7/42 + x**9/216 + O(x**10) assert p == q p = HolonomicFunction(Dx - 1, x).composition(x**2, 0, [1]) # e^(x**2) q = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0]) # cos(x) r = (p * q).series(n=10) # expansion of cos(x) * exp(x**2) s = 1 + x**2/2 + x**4/24 - 31*x**6/720 - 179*x**8/8064 + O(x**10) assert r == s t = HolonomicFunction((1 + x)*Dx**2 + Dx, x, 0, [0, 1]) # log(1 + x) r = (p * t + q).series(n=10) s = 1 + x - x**2 + 4*x**3/3 - 17*x**4/24 + 31*x**5/30 - 481*x**6/720 +\ 71*x**7/105 - 20159*x**8/40320 + 379*x**9/840 + O(x**10) assert r == s p = HolonomicFunction((6+6*x-3*x**2) - (10*x-3*x**2-3*x**3)*Dx + \ (4-6*x**3+2*x**4)*Dx**2, x, 0, [0, 1]).series(n=7) q = x + x**3/6 - 3*x**4/16 + x**5/20 - 23*x**6/960 + O(x**7) assert p == q p = HolonomicFunction((6+6*x-3*x**2) - (10*x-3*x**2-3*x**3)*Dx + \ (4-6*x**3+2*x**4)*Dx**2, x, 0, [1, 0]).series(n=7) q = 1 - 3*x**2/4 - x**3/4 - 5*x**4/32 - 3*x**5/40 - 17*x**6/384 + O(x**7) assert p == q p = expr_to_holonomic(erf(x) + x).series(n=10) C_3 = symbols('C_3') q = (erf(x) + x).series(n=10) assert p.subs(C_3, -2/(3*sqrt(pi))) == q assert expr_to_holonomic(sqrt(x**3 + x)).series(n=10) == sqrt(x**3 + x).series(n=10) assert expr_to_holonomic((2*x - 3*x**2)**(S(1)/3)).series() == ((2*x - 3*x**2)**(S(1)/3)).series() assert expr_to_holonomic(sqrt(x**2-x)).series() == (sqrt(x**2-x)).series() assert expr_to_holonomic(cos(x)**2/x**2, y0={-2: [1, 0, -1]}).series(n=10) == (cos(x)**2/x**2).series(n=10) assert expr_to_holonomic(cos(x)**2/x**2, x0=1).series(n=10) == (cos(x)**2/x**2).series(n=10, x0=1) assert expr_to_holonomic(cos(x-1)**2/(x-1)**2, x0=1, y0={-2: [1, 0, -1]}).series(n=10) \ == (cos(x-1)**2/(x-1)**2).series(x0=1, n=10)
def test_to_expr(): x = symbols('x') R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx') p = HolonomicFunction(Dx - 1, x, 0, 1).to_expr() q = exp(x) assert p == q p = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0]).to_expr() q = cos(x) assert p == q p = HolonomicFunction(Dx**2 - 1, x, 0, [1, 0]).to_expr() q = cosh(x) assert p == q p = HolonomicFunction(2 + (4*x - 1)*Dx + \ (x**2 - x)*Dx**2, x, 0, [1, 2]).to_expr() q = 1/(x**2 - 2*x + 1) assert p == q p = expr_to_holonomic(sin(x)**2/x).integrate((x, 0, x)).to_expr() q = (sin(x)**2/x).integrate((x, 0, x)) assert p == q C_1, C_2, C_3 = symbols('C_1, C_2, C_3') p = expr_to_holonomic(log(1+x**2)).to_expr() q = C_2*log(x**2 + 1) assert p == q p = expr_to_holonomic(log(1+x**2)).diff().to_expr() q = C_1*x/(x**2 + 1) assert p == q p = expr_to_holonomic(erf(x) + x).to_expr() q = 3*C_3*x - 3*sqrt(pi)*C_3*erf(x)/2 + x + 2*x/sqrt(pi) assert p == q p = expr_to_holonomic(sqrt(x), x0=1).to_expr() assert p == sqrt(x) p = expr_to_holonomic(sqrt(1 + x**2)).to_expr() assert p == sqrt(1+x**2) p = expr_to_holonomic((2*x**2 + 1)**(S(2)/3)).to_expr() assert p == (2*x**2 + 1)**(S(2)/3)
def test_to_expr(): x = symbols('x') R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx') p = HolonomicFunction(Dx - 1, x, 0, [1]).to_expr() q = exp(x) assert p == q p = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0]).to_expr() q = cos(x) assert p == q p = HolonomicFunction(Dx**2 - 1, x, 0, [1, 0]).to_expr() q = cosh(x) assert p == q p = HolonomicFunction(2 + (4*x - 1)*Dx + \ (x**2 - x)*Dx**2, x, 0, [1, 2]).to_expr().expand() q = 1 / (x**2 - 2 * x + 1) assert p == q p = expr_to_holonomic(sin(x)**2 / x).integrate((x, 0, x)).to_expr() q = (sin(x)**2 / x).integrate((x, 0, x)) assert p == q C_0, C_1, C_2, C_3 = symbols('C_0, C_1, C_2, C_3') p = expr_to_holonomic(log(1 + x**2)).to_expr() q = C_2 * log(x**2 + 1) assert p == q p = expr_to_holonomic(log(1 + x**2)).diff().to_expr() q = C_0 * x / (x**2 + 1) assert p == q p = expr_to_holonomic(erf(x) + x).to_expr() q = 3 * C_3 * x - 3 * sqrt(pi) * C_3 * erf(x) / 2 + x + 2 * x / sqrt(pi) assert p == q p = expr_to_holonomic(sqrt(x), x0=1).to_expr() assert p == sqrt(x) assert expr_to_holonomic(sqrt(x)).to_expr() == sqrt(x) p = expr_to_holonomic(sqrt(1 + x**2)).to_expr() assert p == sqrt(1 + x**2) p = expr_to_holonomic((2 * x**2 + 1)**Rational(2, 3)).to_expr() assert p == (2 * x**2 + 1)**Rational(2, 3) p = expr_to_holonomic(sqrt(-x**2 + 2 * x)).to_expr() assert p == sqrt(x) * sqrt(-x + 2) p = expr_to_holonomic((-2 * x**3 + 7 * x)**Rational(2, 3)).to_expr() q = x**Rational(2, 3) * (-2 * x**2 + 7)**Rational(2, 3) assert p == q p = from_hyper(hyper((-2, -3), (S.Half, ), x)) s = hyperexpand(hyper((-2, -3), (S.Half, ), x)) D_0 = Symbol('D_0') C_0 = Symbol('C_0') assert (p.to_expr().subs({C_0: 1, D_0: 0}) - s).simplify() == 0 p.y0 = {0: [1], S.Half: [0]} assert p.to_expr() == s assert expr_to_holonomic(x**5).to_expr() == x**5 assert expr_to_holonomic(2*x**3-3*x**2).to_expr().expand() == \ 2*x**3-3*x**2 a = symbols("a") p = (expr_to_holonomic(1.4 * x) * expr_to_holonomic(a * x, x)).to_expr() q = 1.4 * a * x**2 assert p == q p = (expr_to_holonomic(1.4 * x) + expr_to_holonomic(a * x, x)).to_expr() q = x * (a + 1.4) assert p == q p = (expr_to_holonomic(1.4 * x) + expr_to_holonomic(x)).to_expr() assert p == 2.4 * x
def test_multiplication_initial_condition(): x = symbols('x') R, Dx = DiffOperatorAlgebra(ZZ.old_poly_ring(x), 'Dx') p = HolonomicFunction(Dx**2 + x*Dx - 1, x, 0, [3, 1]) q = HolonomicFunction(Dx**2 + 1, x, 0, [1, 1]) r = HolonomicFunction((x**4 + 14*x**2 + 60) + 4*x*Dx + (x**4 + 9*x**2 + 20)*Dx**2 + \ (2*x**3 + 18*x)*Dx**3 + (x**2 + 10)*Dx**4, x, 0, [3, 4, 2, 3]) assert p * q == r p = HolonomicFunction(Dx**2 + x, x, 0, [1, 0]) q = HolonomicFunction(Dx**3 - x**2, x, 0, [3, 3, 3]) r = HolonomicFunction((27*x**8 - 37*x**7 - 10*x**6 - 492*x**5 - 552*x**4 + 160*x**3 + \ 1212*x**2 + 216*x + 360) + (162*x**7 - 384*x**6 - 294*x**5 - 84*x**4 + 24*x**3 + \ 756*x**2 + 120*x - 1080)*Dx + (81*x**6 - 246*x**5 + 228*x**4 + 36*x**3 + \ 660*x**2 - 720*x)*Dx**2 + (-54*x**6 + 128*x**5 - 18*x**4 - 240*x**2 + 600)*Dx**3 + \ (81*x**5 - 192*x**4 - 84*x**3 + 162*x**2 - 60*x - 180)*Dx**4 + (-108*x**3 + \ 192*x**2 + 72*x)*Dx**5 + (27*x**4 - 64*x**3 - 36*x**2 + 60)*Dx**6, x, 0, [3, 3, 3, -3, -12, -24]) assert p * q == r p = HolonomicFunction(Dx - 1, x, 0, [2]) q = HolonomicFunction(Dx**2 + 1, x, 0, [0, 1]) r = HolonomicFunction(2 -2*Dx + Dx**2, x, 0, [0, 2]) assert p * q == r q = HolonomicFunction(x*Dx**2+1+2*Dx,x,0,[0,1]) r = HolonomicFunction((x - 1) + (-2*x + 2)*Dx + x*Dx**2, x, 0, [0, 2]) assert p * q == r p = HolonomicFunction(Dx**2 - 1, x, 0, [1, 3]) q = HolonomicFunction(Dx**3 + 1, x, 0, [1, 2, 1]) r = HolonomicFunction(6*Dx + 3*Dx**2 + 2*Dx**3 - 3*Dx**4 + Dx**6, x, 0, [1, 5, 14, 17, 17, 2]) assert p * q == r
def test_to_expr(): x = symbols('x') R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx') p = HolonomicFunction(Dx - 1, x, 0, [1]).to_expr() q = exp(x) assert p == q p = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0]).to_expr() q = cos(x) assert p == q p = HolonomicFunction(Dx**2 - 1, x, 0, [1, 0]).to_expr() q = cosh(x) assert p == q p = HolonomicFunction(2 + (4*x - 1)*Dx + \ (x**2 - x)*Dx**2, x, 0, [1, 2]).to_expr().expand() q = 1/(x**2 - 2*x + 1) assert p == q p = expr_to_holonomic(sin(x)**2/x).integrate((x, 0, x)).to_expr() q = (sin(x)**2/x).integrate((x, 0, x)) assert p == q C_0, C_1, C_2, C_3 = symbols('C_0, C_1, C_2, C_3') p = expr_to_holonomic(log(1+x**2)).to_expr() q = C_2*log(x**2 + 1) assert p == q p = expr_to_holonomic(log(1+x**2)).diff().to_expr() q = C_0*x/(x**2 + 1) assert p == q p = expr_to_holonomic(erf(x) + x).to_expr() q = 3*C_3*x - 3*sqrt(pi)*C_3*erf(x)/2 + x + 2*x/sqrt(pi) assert p == q p = expr_to_holonomic(sqrt(x), x0=1).to_expr() assert p == sqrt(x) assert expr_to_holonomic(sqrt(x)).to_expr() == sqrt(x) p = expr_to_holonomic(sqrt(1 + x**2)).to_expr() assert p == sqrt(1+x**2) p = expr_to_holonomic((2*x**2 + 1)**(S(2)/3)).to_expr() assert p == (2*x**2 + 1)**(S(2)/3) p = expr_to_holonomic(sqrt(-x**2+2*x)).to_expr() assert p == sqrt(x)*sqrt(-x + 2) p = expr_to_holonomic((-2*x**3+7*x)**(S(2)/3)).to_expr() q = x**(S(2)/3)*(-2*x**2 + 7)**(S(2)/3) assert p == q p = from_hyper(hyper((-2, -3), (S(1)/2, ), x)) s = hyperexpand(hyper((-2, -3), (S(1)/2, ), x)) D_0 = Symbol('D_0') C_0 = Symbol('C_0') assert (p.to_expr().subs({C_0:1, D_0:0}) - s).simplify() == 0 p.y0 = {0: [1], S(1)/2: [0]} assert p.to_expr() == s assert expr_to_holonomic(x**5).to_expr() == x**5 assert expr_to_holonomic(2*x**3-3*x**2).to_expr().expand() == \ 2*x**3-3*x**2 a = symbols("a") p = (expr_to_holonomic(1.4*x)*expr_to_holonomic(a*x, x)).to_expr() q = 1.4*a*x**2 assert p == q p = (expr_to_holonomic(1.4*x)+expr_to_holonomic(a*x, x)).to_expr() q = x*(a + 1.4) assert p == q p = (expr_to_holonomic(1.4*x)+expr_to_holonomic(x)).to_expr() assert p == 2.4*x
def test_to_expr(): x = symbols('x') R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx') p = HolonomicFunction(Dx - 1, x, 0, 1).to_expr() q = exp(x) assert p == q p = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0]).to_expr() q = cos(x) assert p == q p = HolonomicFunction(Dx**2 - 1, x, 0, [1, 0]).to_expr() q = cosh(x) assert p == q p = HolonomicFunction(2 + (4*x - 1)*Dx + \ (x**2 - x)*Dx**2, x, 0, [1, 2]).to_expr() q = 1 / (x**2 - 2 * x + 1) assert p == q p = expr_to_holonomic(sin(x)**2 / x).integrate((x, 0, x)).to_expr() q = (sin(x)**2 / x).integrate((x, 0, x)) assert p == q C_1, C_2, C_3 = symbols('C_1, C_2, C_3') p = expr_to_holonomic(log(1 + x**2)).to_expr() q = C_2 * log(x**2 + 1) assert p == q p = expr_to_holonomic(log(1 + x**2)).diff().to_expr() q = C_1 * x / (x**2 + 1) assert p == q p = expr_to_holonomic(erf(x) + x).to_expr() q = 3 * C_3 * x - 3 * sqrt(pi) * C_3 * erf(x) / 2 + x + 2 * x / sqrt(pi) assert p == q p = expr_to_holonomic(sqrt(x), x0=1).to_expr() assert p == sqrt(x) p = expr_to_holonomic(sqrt(1 + x**2)).to_expr() assert p == sqrt(1 + x**2) p = expr_to_holonomic((2 * x**2 + 1)**(S(2) / 3)).to_expr() assert p == (2 * x**2 + 1)**(S(2) / 3)
def test_series(): x = symbols('x') R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx') p = HolonomicFunction(Dx**2 + 2 * x * Dx, x, 0, [0, 1]).series(n=10) q = x - x**3 / 3 + x**5 / 10 - x**7 / 42 + x**9 / 216 + O(x**10) assert p == q p = HolonomicFunction(Dx - 1, x).composition(x**2, 0, 1) # e^(x**2) q = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0]) # cos(x) r = (p * q).series(n=10) # expansion of cos(x) * exp(x**2) s = 1 + x**2 / 2 + x**4 / 24 - 31 * x**6 / 720 - 179 * x**8 / 8064 + O(x** 10) assert r == s t = HolonomicFunction((1 + x) * Dx**2 + Dx, x, 0, [0, 1]) # log(1 + x) r = (p * t + q).series(n=10) s = 1 + x - x**2 + 4*x**3/3 - 17*x**4/24 + 31*x**5/30 - 481*x**6/720 +\ 71*x**7/105 - 20159*x**8/40320 + 379*x**9/840 + O(x**10) assert r == s p = HolonomicFunction((6+6*x-3*x**2) - (10*x-3*x**2-3*x**3)*Dx + \ (4-6*x**3+2*x**4)*Dx**2, x, 0, [0, 1]).series(n=7) q = x + x**3 / 6 - 3 * x**4 / 16 + x**5 / 20 - 23 * x**6 / 960 + O(x**7) assert p == q p = HolonomicFunction((6+6*x-3*x**2) - (10*x-3*x**2-3*x**3)*Dx + \ (4-6*x**3+2*x**4)*Dx**2, x, 0, [1, 0]).series(n=7) q = 1 - 3 * x**2 / 4 - x**3 / 4 - 5 * x**4 / 32 - 3 * x**5 / 40 - 17 * x**6 / 384 + O( x**7) assert p == q
def test_HolonomicFunction_addition(): x = symbols('x') R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx') p = HolonomicFunction(Dx**2 * x, x) q = HolonomicFunction((2) * Dx + (x) * Dx**2, x) assert p == q p = HolonomicFunction(x * Dx + 1, x) q = HolonomicFunction(Dx + 1, x) r = HolonomicFunction((x - 2) + (x**2 - 2) * Dx + (x**2 - x) * Dx**2, x) assert p + q == r p = HolonomicFunction(x * Dx + Dx**2 * (x**2 + 2), x) q = HolonomicFunction(Dx - 3, x) r = HolonomicFunction((-54 * x**2 - 126 * x - 150) + (-135 * x**3 - 252 * x**2 - 270 * x + 140) * Dx +\ (-27 * x**4 - 24 * x**2 + 14 * x - 150) * Dx**2 + \ (9 * x**4 + 15 * x**3 + 38 * x**2 + 30 * x +40) * Dx**3, x) assert p + q == r p = HolonomicFunction(Dx**5 - 1, x) q = HolonomicFunction(x**3 + Dx, x) r = HolonomicFunction((-x**18 + 45*x**14 - 525*x**10 + 1575*x**6 - x**3 - 630*x**2) + \ (-x**15 + 30*x**11 - 195*x**7 + 210*x**3 - 1)*Dx + (x**18 - 45*x**14 + 525*x**10 - \ 1575*x**6 + x**3 + 630*x**2)*Dx**5 + (x**15 - 30*x**11 + 195*x**7 - 210*x**3 + \ 1)*Dx**6, x) assert p + q == r p = x**2 + 3 * x + 8 q = x**3 - 7 * x + 5 p = p * Dx - p.diff() q = q * Dx - q.diff() r = HolonomicFunction(p, x) + HolonomicFunction(q, x) s = HolonomicFunction((6*x**2 + 18*x + 14) + (-4*x**3 - 18*x**2 - 62*x + 10)*Dx +\ (x**4 + 6*x**3 + 31*x**2 - 10*x - 71)*Dx**2, x) assert r == s
def test_HolonomicFunction_addition(): x = symbols('x') R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx') p = HolonomicFunction(Dx**2 * x, x) q = HolonomicFunction((2) * Dx + (x) * Dx**2, x) assert p == q p = HolonomicFunction(x * Dx + 1, x) q = HolonomicFunction(Dx + 1, x) r = HolonomicFunction((x - 2) + (x**2 - 2) * Dx + (x**2 - x) * Dx**2, x) assert p + q == r p = HolonomicFunction(x * Dx + Dx**2 * (x**2 + 2), x) q = HolonomicFunction(Dx - 3, x) r = HolonomicFunction((-54 * x**2 - 126 * x - 150) + (-135 * x**3 - 252 * x**2 - 270 * x + 140) * Dx +\ (-27 * x**4 - 24 * x**2 + 14 * x - 150) * Dx**2 + \ (9 * x**4 + 15 * x**3 + 38 * x**2 + 30 * x +40) * Dx**3, x) assert p + q == r p = HolonomicFunction(Dx**5 - 1, x) q = HolonomicFunction(x**3 + Dx, x) r = HolonomicFunction((-x**18 + 45*x**14 - 525*x**10 + 1575*x**6 - x**3 - 630*x**2) + \ (-x**15 + 30*x**11 - 195*x**7 + 210*x**3 - 1)*Dx + (x**18 - 45*x**14 + 525*x**10 - \ 1575*x**6 + x**3 + 630*x**2)*Dx**5 + (x**15 - 30*x**11 + 195*x**7 - 210*x**3 + \ 1)*Dx**6, x) assert p+q == r p = x**2 + 3*x + 8 q = x**3 - 7*x + 5 p = p*Dx - p.diff() q = q*Dx - q.diff() r = HolonomicFunction(p, x) + HolonomicFunction(q, x) s = HolonomicFunction((6*x**2 + 18*x + 14) + (-4*x**3 - 18*x**2 - 62*x + 10)*Dx +\ (x**4 + 6*x**3 + 31*x**2 - 10*x - 71)*Dx**2, x) assert r == s
def test_integrate(): x = symbols('x') R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx') p = expr_to_holonomic(sin(x)**2/x, x0=1).integrate((x, 2, 3)) q = '0.166270406994788' assert sstr(p) == q p = expr_to_holonomic(sin(x)).integrate((x, 0, x)).to_expr() q = 1 - cos(x) assert p == q p = expr_to_holonomic(sin(x)).integrate((x, 0, 3)) q = 1 - cos(3) assert p == q p = expr_to_holonomic(sin(x)/x, x0=1).integrate((x, 1, 2)) q = '0.659329913368450' assert sstr(p) == q p = expr_to_holonomic(sin(x)**2/x, x0=1).integrate((x, 1, 0)) q = '-0.423690480850035' assert sstr(p) == q p = expr_to_holonomic(sin(x)/x) assert p.integrate(x).to_expr() == Si(x) assert p.integrate((x, 0, 2)) == Si(2) p = expr_to_holonomic(sin(x)**2/x) q = p.to_expr() assert p.integrate(x).to_expr() == q.integrate((x, 0, x)) assert p.integrate((x, 0, 1)) == q.integrate((x, 0, 1)) assert expr_to_holonomic(1/x).integrate(x).to_expr() == log(x) p = expr_to_holonomic((x + 1)**3*exp(-x), x0=-1, lenics=4).integrate(x).to_expr() q = (-x**3 - 6*x**2 - 15*x + 6*exp(x + 1) - 16)*exp(-x) assert p == q
def test_integrate(): x = symbols('x') R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx') p = expr_to_holonomic(sin(x)**2 / x, x0=1).integrate((x, 2, 3)) q = '0.166270406994788' assert sstr(p) == q p = expr_to_holonomic(sin(x)).integrate((x, 0, x)).to_expr() q = 1 - cos(x) assert p == q p = expr_to_holonomic(sin(x)).integrate((x, 0, 3)) q = 1 - cos(3) assert p == q p = expr_to_holonomic(sin(x) / x, x0=1).integrate((x, 1, 2)) q = '0.659329913368450' assert sstr(p) == q p = expr_to_holonomic(sin(x)**2 / x, x0=1).integrate((x, 1, 0)) q = '-0.423690480850035' assert sstr(p) == q p = expr_to_holonomic(sin(x) / x) assert p.integrate(x).to_expr() == Si(x) assert p.integrate((x, 0, 2)) == Si(2) p = expr_to_holonomic(sin(x)**2 / x) q = p.to_expr() assert p.integrate(x).to_expr() == q.integrate((x, 0, x)) assert p.integrate((x, 0, 1)) == q.integrate((x, 0, 1)) assert expr_to_holonomic(1 / x).integrate(x).to_expr() == log(x) p = expr_to_holonomic((x + 1)**3 * exp(-x), x0=-1, lenics=4).integrate(x).to_expr() q = (-x**3 - 6 * x**2 - 15 * x + 6 * exp(x + 1) - 16) * exp(-x) assert p == q
def test_to_Sequence(): x = symbols('x') R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx') n = symbols('n', integer=True) _, Sn = RecurrenceOperators(ZZ.old_poly_ring(n), 'Sn') p = HolonomicFunction(x**2*Dx**4 + x + Dx, x).to_sequence() q = [(HolonomicSequence(1 + (n + 2)*Sn**2 + (n**4 + 6*n**3 + 11*n**2 + 6*n)*Sn**3), 0, 1)] assert p == q p = HolonomicFunction(x**2*Dx**4 + x**3 + Dx**2, x).to_sequence() q = [(HolonomicSequence(1 + (n**4 + 14*n**3 + 72*n**2 + 163*n + 140)*Sn**5), 0, 0)] assert p == q p = HolonomicFunction(x**3*Dx**4 + 1 + Dx**2, x).to_sequence() q = [(HolonomicSequence(1 + (n**4 - 2*n**3 - n**2 + 2*n)*Sn + (n**2 + 3*n + 2)*Sn**2), 0, 0)] assert p == q p = HolonomicFunction(3*x**3*Dx**4 + 2*x*Dx + x*Dx**3, x).to_sequence() q = [(HolonomicSequence(2*n + (3*n**4 - 6*n**3 - 3*n**2 + 6*n)*Sn + (n**3 + 3*n**2 + 2*n)*Sn**2), 0, 1)] assert p == q
def test_to_Sequence(): x = symbols("x") R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), "Dx") n = symbols("n", integer=True) _, Sn = RecurrenceOperators(ZZ.old_poly_ring(n), "Sn") p = HolonomicFunction(x ** 2 * Dx ** 4 + x + Dx, x).to_sequence() q = (HolonomicSequence(1 + (n + 2) * Sn ** 2 + (n ** 4 + 6 * n ** 3 + 11 * n ** 2 + 6 * n) * Sn ** 3), 1) assert p == q p = HolonomicFunction(x ** 2 * Dx ** 4 + x ** 3 + Dx ** 2, x).to_sequence() q = (HolonomicSequence(1 + (n ** 4 + 14 * n ** 3 + 72 * n ** 2 + 163 * n + 140) * Sn ** 5, n), 0) assert p == q p = HolonomicFunction(x ** 3 * Dx ** 4 + 1 + Dx ** 2, x).to_sequence() q = (HolonomicSequence(1 + (n ** 4 - 2 * n ** 3 - n ** 2 + 2 * n) * Sn + (n ** 2 + 3 * n + 2) * Sn ** 2, n), 3) assert p == q p = HolonomicFunction(3 * x ** 3 * Dx ** 4 + 2 * x * Dx + x * Dx ** 3, x).to_sequence() q = ( HolonomicSequence( 2 * n + (3 * n ** 4 - 6 * n ** 3 - 3 * n ** 2 + 6 * n) * Sn + (n ** 3 + 3 * n ** 2 + 2 * n) * Sn ** 2, n ), 3, ) assert p == q
def test_diff(): x, y = symbols('x, y') R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx') p = HolonomicFunction(x*Dx**2 + 1, x, 0, [0, 1]) assert p.diff().to_expr() == p.to_expr().diff().simplify() p = HolonomicFunction(Dx**2 - 1, x, 0, [1, 0]) assert p.diff(x, 2).to_expr() == p.to_expr() p = expr_to_holonomic(Si(x)) assert p.diff().to_expr() == sin(x)/x assert p.diff(y) == 0 C_0, C_1, C_2, C_3 = symbols('C_0, C_1, C_2, C_3') q = Si(x) assert p.diff(x).to_expr() == q.diff() assert p.diff(x, 2).to_expr().subs(C_0, -S(1)/3) == q.diff(x, 2).simplify() assert p.diff(x, 3).series().subs({C_3:-S(1)/3, C_0:0}) == q.diff(x, 3).series()
def test_to_sympy(): x = symbols("x") R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), "Dx") p = HolonomicFunction(Dx - 1, x, 0, 1).to_sympy() q = exp(x) assert p == q p = HolonomicFunction(Dx ** 2 + 1, x, 0, [1, 0]).to_sympy() q = cos(x) assert p == q p = HolonomicFunction(Dx ** 2 - 1, x, 0, [1, 0]).to_sympy() q = cosh(x) assert p == q p = HolonomicFunction(2 + (4 * x - 1) * Dx + (x ** 2 - x) * Dx ** 2, x, 0, [1, 2]).to_sympy() q = 1 / (x ** 2 - 2 * x + 1) assert p == q
def test_to_expr(): x = symbols('x') R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx') p = HolonomicFunction(Dx - 1, x, 0, 1).to_expr() q = exp(x) assert p == q p = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0]).to_expr() q = cos(x) assert p == q p = HolonomicFunction(Dx**2 - 1, x, 0, [1, 0]).to_expr() q = cosh(x) assert p == q p = HolonomicFunction(2 + (4*x - 1)*Dx + \ (x**2 - x)*Dx**2, x, 0, [1, 2]).to_expr().expand() q = 1 / (x**2 - 2 * x + 1) assert p == q p = expr_to_holonomic(sin(x)**2 / x).integrate((x, 0, x)).to_expr() q = (sin(x)**2 / x).integrate((x, 0, x)) assert p == q C_0, C_1, C_2, C_3 = symbols('C_0, C_1, C_2, C_3') p = expr_to_holonomic(log(1 + x**2)).to_expr() q = C_2 * log(x**2 + 1) assert p == q p = expr_to_holonomic(log(1 + x**2)).diff().to_expr() q = C_0 * x / (x**2 + 1) assert p == q p = expr_to_holonomic(erf(x) + x).to_expr() q = 3 * C_3 * x - 3 * sqrt(pi) * C_3 * erf(x) / 2 + x + 2 * x / sqrt(pi) assert p == q p = expr_to_holonomic(sqrt(x), x0=1).to_expr() assert p == sqrt(x) assert expr_to_holonomic(sqrt(x)).to_expr() == sqrt(x) p = expr_to_holonomic(sqrt(1 + x**2)).to_expr() assert p == sqrt(1 + x**2) p = expr_to_holonomic((2 * x**2 + 1)**(S(2) / 3)).to_expr() assert p == (2 * x**2 + 1)**(S(2) / 3) p = expr_to_holonomic(sqrt(-x**2 + 2 * x)).to_expr() assert p == sqrt(x) * sqrt(-x + 2) p = expr_to_holonomic((-2 * x**3 + 7 * x)**(S(2) / 3)).to_expr() q = x**(S(2) / 3) * (-2 * x**2 + 7)**(S(2) / 3) assert p == q p = from_hyper(hyper((-2, -3), (S(1) / 2, ), x)) s = hyperexpand(hyper((-2, -3), (S(1) / 2, ), x)) D_0 = Symbol('D_0') C_0 = Symbol('C_0') assert (p.to_expr().subs({C_0: 1, D_0: 0}) - s).simplify() == 0 p.singular_ics = [(0, [1]), (S(1) / 2, [0])] assert p.to_expr() == s
def test_integrate(): x = symbols('x') R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx') p = expr_to_holonomic(sin(x)**2 / x, x0=1).integrate((x, 2, 3)) q = '0.166270406994788' assert sstr(p) == q p = expr_to_holonomic(sin(x)).integrate((x, 0, x)).to_expr() q = 1 - cos(x) assert p == q p = expr_to_holonomic(sin(x)).integrate((x, 0, 3)) q = 1 - cos(3) assert p == q p = expr_to_holonomic(sin(x) / x, x0=1).integrate((x, 1, 2)) q = '0.659329913368450' assert sstr(p) == q p = expr_to_holonomic(sin(x)**2 / x, x0=1).integrate((x, 1, 0)) q = '-0.423690480850035' assert sstr(p) == q p = expr_to_holonomic(sin(x) / x) assert p.integrate(x).to_expr() == Si(x) assert p.integrate((x, 0, 2)) == Si(2) p = expr_to_holonomic(sin(x)**2 / x) q = p.to_expr() assert p.integrate(x).to_expr() == q.integrate((x, 0, x)) assert p.integrate((x, 0, 1)) == q.integrate((x, 0, 1)) assert expr_to_holonomic(1 / x, x0=1).integrate(x).to_expr() == log(x) p = expr_to_holonomic((x + 1)**3 * exp(-x), x0=-1).integrate(x).to_expr() q = (-x**3 - 6 * x**2 - 15 * x + 6 * exp(x + 1) - 16) * exp(-x) assert p == q p = expr_to_holonomic(cos(x)**2 / x**2, y0={ -2: [1, 0, -1] }).integrate(x).to_expr() q = -Si(2 * x) - cos(x)**2 / x assert p == q p = expr_to_holonomic(sqrt(x**2 + x)).integrate(x).to_expr() q = (x**Rational(3, 2) * (2 * x**2 + 3 * x + 1) - x * sqrt(x + 1) * asinh(sqrt(x))) / (4 * x * sqrt(x + 1)) assert p == q p = expr_to_holonomic(sqrt(x**2 + 1)).integrate(x).to_expr() q = (sqrt(x**2 + 1)).integrate(x) assert (p - q).simplify() == 0 p = expr_to_holonomic(1 / x**2, y0={-2: [1, 0, 0]}) r = expr_to_holonomic(1 / x**2, lenics=3) assert p == r q = expr_to_holonomic(cos(x)**2) assert (r * q).integrate(x).to_expr() == -Si(2 * x) - cos(x)**2 / x
def test_integrate(): x = symbols('x') R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx') p = from_sympy(sin(x)**2 / x, x0=1).integrate((x, 2, 3)) q = '0.166270406994788' assert sstr(p) == q p = from_sympy(sin(x)).integrate((x, 0, x)).to_sympy() q = 1 - cos(x) assert p == q p = from_sympy(sin(x)).integrate((x, 0, 3)) q = '1.98999246812687' assert sstr(p) == q p = from_sympy(sin(x) / x, x0=1).integrate((x, 1, 2)) q = '0.659329913368450' assert sstr(p) == q p = from_sympy(sin(x)**2 / x, x0=1).integrate((x, 1, 0)) q = '-0.423690480850035' assert sstr(p) == q
def test_integrate(): x = symbols('x') R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx') p = from_sympy(sin(x)**2/x, x0=1).integrate((x, 2, 3)) q = '0.166270406994788' assert sstr(p) == q p = from_sympy(sin(x)).integrate((x, 0, x)).to_sympy() q = 1 - cos(x) assert p == q p = from_sympy(sin(x)).integrate((x, 0, 3)) q = '1.98999246812687' assert sstr(p) == q p = from_sympy(sin(x)/x, x0=1).integrate((x, 1, 2)) q = '0.659329913368450' assert sstr(p) == q p = from_sympy(sin(x)**2/x, x0=1).integrate((x, 1, 0)) q = '-0.423690480850035' assert sstr(p) == q
def test_series(): x = symbols("x") R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), "Dx") p = HolonomicFunction(Dx ** 2 + 2 * x * Dx, x, 0, [0, 1]).series(n=10) q = x - x ** 3 / 3 + x ** 5 / 10 - x ** 7 / 42 + x ** 9 / 216 + O(x ** 10) assert p == q p = HolonomicFunction(Dx - 1, x).composition(x ** 2, 0, 1) # e^(x**2) q = HolonomicFunction(Dx ** 2 + 1, x, 0, [1, 0]) # cos(x) r = (p * q).series(n=10) # expansion of cos(x) * exp(x**2) s = 1 + x ** 2 / 2 + x ** 4 / 24 - 31 * x ** 6 / 720 - 179 * x ** 8 / 8064 + O(x ** 10) assert r == s t = HolonomicFunction((1 + x) * Dx ** 2 + Dx, x, 0, [0, 1]) # log(1 + x) r = (p * t + q).series(n=10) s = ( 1 + x - x ** 2 + 4 * x ** 3 / 3 - 17 * x ** 4 / 24 + 31 * x ** 5 / 30 - 481 * x ** 6 / 720 + 71 * x ** 7 / 105 - 20159 * x ** 8 / 40320 + 379 * x ** 9 / 840 + O(x ** 10) ) assert r == s p = HolonomicFunction( (6 + 6 * x - 3 * x ** 2) - (10 * x - 3 * x ** 2 - 3 * x ** 3) * Dx + (4 - 6 * x ** 3 + 2 * x ** 4) * Dx ** 2, x, 0, [0, 1], ).series(n=7) q = x + x ** 3 / 6 - 3 * x ** 4 / 16 + x ** 5 / 20 - 23 * x ** 6 / 960 + O(x ** 7) assert p == q p = HolonomicFunction( (6 + 6 * x - 3 * x ** 2) - (10 * x - 3 * x ** 2 - 3 * x ** 3) * Dx + (4 - 6 * x ** 3 + 2 * x ** 4) * Dx ** 2, x, 0, [1, 0], ).series(n=7) q = 1 - 3 * x ** 2 / 4 - x ** 3 / 4 - 5 * x ** 4 / 32 - 3 * x ** 5 / 40 - 17 * x ** 6 / 384 + O(x ** 7) assert p == q
def test_integrate(): x = symbols('x') R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx') p = expr_to_holonomic(sin(x)**2/x, x0=1).integrate((x, 2, 3)) q = '0.166270406994788' assert sstr(p) == q p = expr_to_holonomic(sin(x)).integrate((x, 0, x)).to_expr() q = 1 - cos(x) assert p == q p = expr_to_holonomic(sin(x)).integrate((x, 0, 3)) q = 1 - cos(3) assert p == q p = expr_to_holonomic(sin(x)/x, x0=1).integrate((x, 1, 2)) q = '0.659329913368450' assert sstr(p) == q p = expr_to_holonomic(sin(x)**2/x, x0=1).integrate((x, 1, 0)) q = '-0.423690480850035' assert sstr(p) == q p = expr_to_holonomic(sin(x)/x) assert p.integrate(x).to_expr() == Si(x) assert p.integrate((x, 0, 2)) == Si(2) p = expr_to_holonomic(sin(x)**2/x) q = p.to_expr() assert p.integrate(x).to_expr() == q.integrate((x, 0, x)) assert p.integrate((x, 0, 1)) == q.integrate((x, 0, 1)) assert expr_to_holonomic(1/x, x0=1).integrate(x).to_expr() == log(x) p = expr_to_holonomic((x + 1)**3*exp(-x), x0=-1).integrate(x).to_expr() q = (-x**3 - 6*x**2 - 15*x + 6*exp(x + 1) - 16)*exp(-x) assert p == q p = expr_to_holonomic(cos(x)**2/x**2, y0={-2: [1, 0, -1]}).integrate(x).to_expr() q = -Si(2*x) - cos(x)**2/x assert p == q p = expr_to_holonomic(sqrt(x**2+x)).integrate(x).to_expr() q = (x**(3/2)*(2*x**2 + 3*x + 1) - x*sqrt(x + 1)*asinh(sqrt(x)))/(4*x*sqrt(x + 1)) assert p == q p = expr_to_holonomic(sqrt(x**2+1)).integrate(x).to_expr() q = (sqrt(x**2+1)).integrate(x) assert (p-q).simplify() == 0 p = expr_to_holonomic(1/x**2, y0={-2:[1, 0, 0]}) r = expr_to_holonomic(1/x**2, lenics=3) assert p == r q = expr_to_holonomic(cos(x)**2) assert (r*q).integrate(x).to_expr() == -Si(2*x) - cos(x)**2/x
def test_HolonomicFunction_addition(): x = symbols('x') R, Dx = DiffOperatorAlgebra(ZZ.old_poly_ring(x), 'Dx') p = HolonomicFunction(Dx**2 * x, x) q = HolonomicFunction((2) * Dx + (x) * Dx**2, x) assert p == q p = HolonomicFunction(x * Dx + 1, x) q = HolonomicFunction(Dx + 1, x) r = HolonomicFunction((x - 2) + (x**2 - 2) * Dx + (x**2 - x) * Dx**2, x) assert p + q == r p = HolonomicFunction(x * Dx + Dx**2 * (x**2 + 2), x) q = HolonomicFunction(Dx - 3, x) r = HolonomicFunction((-54 * x**2 - 126 * x - 150) + (-135 * x**3 - 252 * x**2 - 270 * x + 140) * Dx +\ (-27 * x**4 - 24 * x**2 + 14 * x - 150) * Dx**2 + \ (9 * x**4 + 15 * x**3 + 38 * x**2 + 30 * x +40) * Dx**3, x) assert p + q == r p = HolonomicFunction(Dx**5 - 1, x) q = HolonomicFunction(x**3 + Dx, x) r = HolonomicFunction((-x**18 + 45*x**14 - 525*x**10 + 1575*x**6 - x**3 - 630*x**2) + \ (-x**15 + 30*x**11 - 195*x**7 + 210*x**3 - 1)*Dx + (x**18 - 45*x**14 + 525*x**10 - \ 1575*x**6 + x**3 + 630*x**2)*Dx**5 + (x**15 - 30*x**11 + 195*x**7 - 210*x**3 + \ 1)*Dx**6, x) assert p+q == r
def test_HolonomicFunction_multiplication(): x = symbols('x') R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx') p = HolonomicFunction(Dx + x + x * Dx**2, x) q = HolonomicFunction(x * Dx + Dx * x + Dx**2, x) r = HolonomicFunction((8*x**6 + 4*x**4 + 6*x**2 + 3) + (24*x**5 - 4*x**3 + 24*x)*Dx + \ (8*x**6 + 20*x**4 + 12*x**2 + 2)*Dx**2 + (8*x**5 + 4*x**3 + 4*x)*Dx**3 + \ (2*x**4 + x**2)*Dx**4, x) assert p * q == r p = HolonomicFunction(Dx**2 + 1, x) q = HolonomicFunction(Dx - 1, x) r = HolonomicFunction((2) + (-2) * Dx + (1) * Dx**2, x) assert p * q == r p = HolonomicFunction(Dx**2 + 1 + x + Dx, x) q = HolonomicFunction((Dx * x - 1)**2, x) r = HolonomicFunction((4*x**7 + 11*x**6 + 16*x**5 + 4*x**4 - 6*x**3 - 7*x**2 - 8*x - 2) + \ (8*x**6 + 26*x**5 + 24*x**4 - 3*x**3 - 11*x**2 - 6*x - 2)*Dx + \ (8*x**6 + 18*x**5 + 15*x**4 - 3*x**3 - 6*x**2 - 6*x - 2)*Dx**2 + (8*x**5 + \ 10*x**4 + 6*x**3 - 2*x**2 - 4*x)*Dx**3 + (4*x**5 + 3*x**4 - x**2)*Dx**4, x) assert p * q == r p = HolonomicFunction(x * Dx**2 - 1, x) q = HolonomicFunction(Dx * x - x, x) r = HolonomicFunction((x - 3) + (-2 * x + 2) * Dx + (x) * Dx**2, x) assert p * q == r
def test_HolonomicFunction_composition(): x = symbols('x') R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx') p = HolonomicFunction(Dx - 1, x).composition(x**2 + x) r = HolonomicFunction((-2 * x - 1) + Dx, x) assert p == r p = HolonomicFunction(Dx**2 + 1, x).composition(x**5 + x**2 + 1) r = HolonomicFunction((125*x**12 + 150*x**9 + 60*x**6 + 8*x**3) + (-20*x**3 - 2)*Dx + \ (5*x**4 + 2*x)*Dx**2, x) assert p == r p = HolonomicFunction(Dx**2 * x + x, x).composition(2 * x**3 + x**2 + 1) r = HolonomicFunction((216*x**9 + 324*x**8 + 180*x**7 + 152*x**6 + 112*x**5 + \ 36*x**4 + 4*x**3) + (24*x**4 + 16*x**3 + 3*x**2 - 6*x - 1)*Dx + (6*x**5 + 5*x**4 + \ x**3 + 3*x**2 + x)*Dx**2, x) assert p == r p = HolonomicFunction(Dx**2 + 1, x).composition(1 - x**2) r = HolonomicFunction((4 * x**3) - Dx + x * Dx**2, x) assert p == r p = HolonomicFunction(Dx**2 + 1, x).composition(x - 2 / (x**2 + 1)) r = HolonomicFunction((x**12 + 6*x**10 + 12*x**9 + 15*x**8 + 48*x**7 + 68*x**6 + \ 72*x**5 + 111*x**4 + 112*x**3 + 54*x**2 + 12*x + 1) + (12*x**8 + 32*x**6 + \ 24*x**4 - 4)*Dx + (x**12 + 6*x**10 + 4*x**9 + 15*x**8 + 16*x**7 + 20*x**6 + 24*x**5+ \ 15*x**4 + 16*x**3 + 6*x**2 + 4*x + 1)*Dx**2, x) assert p == r