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_to_hyper(): x = symbols('x') R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx') p = HolonomicFunction(Dx - 2, x, 0, [3]).to_hyper() q = 3 * hyper([], [], 2*x) assert p == q p = hyperexpand(HolonomicFunction((1 + x) * Dx - 3, x, 0, [2]).to_hyper()).expand() q = 2*x**3 + 6*x**2 + 6*x + 2 assert p == q p = HolonomicFunction((1 + x)*Dx**2 + Dx, x, 0, [0, 1]).to_hyper() q = -x**2*hyper((2, 2, 1), (2, 3), -x)/2 + x assert p == q p = HolonomicFunction(2*x*Dx + Dx**2, x, 0, [0, 2/sqrt(pi)]).to_hyper() q = 2*x*hyper((1/2,), (3/2,), -x**2)/sqrt(pi) assert p == q p = hyperexpand(HolonomicFunction(2*x*Dx + Dx**2, x, 0, [1, -2/sqrt(pi)]).to_hyper()) q = erfc(x) assert p.rewrite(erfc) == q p = hyperexpand(HolonomicFunction((x**2 - 1) + x*Dx + x**2*Dx**2, x, 0, [0, S(1)/2]).to_hyper()) q = besselj(1, x) assert p == q p = hyperexpand(HolonomicFunction(x*Dx**2 + Dx + x, x, 0, [1, 0]).to_hyper()) q = besselj(0, x) 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() 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_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_negative_power(): x = symbols("x") _, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx') h1 = HolonomicFunction((-1) + (x) * Dx, x)**-2 h2 = HolonomicFunction((2) + (x) * Dx, x) assert h1 == h2
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_expr_in_power(): x, n = symbols("x n") Q = QQ[n].get_field() _, Dx = DifferentialOperators(Q.old_poly_ring(x), 'Dx') h1 = HolonomicFunction((-1) + (x) * Dx, x)**(n - 3) h2 = HolonomicFunction((-n + 3) + (x) * Dx, x) assert h1 == h2
def test_from_sympy(): x = symbols('x') R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx') p = from_sympy((sin(x) / x)**2) q = HolonomicFunction(24*x + (24*x**2 + 12)*Dx + (4*x**3 + 24*x)*Dx**2 + \ 10*x**2*Dx**3 + x**3*Dx**4, x, 1, [-cos(2)/2 + 1/2, -1 + cos(2) + sin(2), \ -4*sin(2) - cos(2) + 3, -12 + 14*sin(2)]) assert p == q p = from_sympy(1 / (1 + x**2)**2) q = HolonomicFunction(4 * x + (x**2 + 1) * Dx, x, 0, 1) assert p == q
def test_from_hyper(): x = symbols('x') R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx') p = hyper([1, 1], [S(3)/2], x**2/4) q = HolonomicFunction((4*x) + (5*x**2 - 8)*Dx + (x**3 - 4*x)*Dx**2, x, 1, [2*sqrt(3)*pi/9, -4*sqrt(3)*pi/27 + 4/3]) r = from_hyper(p) assert r == q p = from_hyper(hyper([1], [S(3)/2], x**2/4)) q = HolonomicFunction(-x + (-x**2/2 + 2)*Dx + x*Dx**2, x) x0 = 1 y0 = '[sqrt(pi)*exp(1/4)*erf(1/2), -sqrt(pi)*exp(1/4)*erf(1/2)/2 + 1]' assert sstr(p.y0) == y0 assert q.annihilator == p.annihilator
def test_extended_domain_in_expr_to_holonomic(): x = symbols('x') p = expr_to_holonomic(1.2 * cos(3.1 * x)) assert p.to_expr() == 1.2 * cos(3.1 * x) assert sstr(p.integrate(x).to_expr()) == '0.387096774193548*sin(3.1*x)' _, Dx = DifferentialOperators(RR.old_poly_ring(x), 'Dx') p = expr_to_holonomic(1.1329138213 * x) q = HolonomicFunction((-1.1329138213) + (1.1329138213 * x) * Dx, x, 0, {1: [1.1329138213]}) assert p == q assert p.to_expr() == 1.1329138213 * x assert sstr(p.integrate((x, 1, 2))) == sstr((1.1329138213 * x).integrate( (x, 1, 2))) y, z = symbols('y, z') p = expr_to_holonomic(sin(x * y * z), x=x) assert p.to_expr() == sin(x * y * z) assert p.integrate(x).to_expr() == (-cos(x * y * z) + 1) / (y * z) p = expr_to_holonomic(sin(x * y + z), x=x).integrate(x).to_expr() q = (cos(z) - cos(x * y + z)) / y assert p == q a = symbols('a') p = expr_to_holonomic(a * x, x) assert p.to_expr() == a * x assert p.integrate(x).to_expr() == a * x**2 / 2 D_2, C_1 = symbols("D_2, C_1") p = expr_to_holonomic(x) + expr_to_holonomic(1.2 * cos(x)) p = p.to_expr().subs(D_2, 0) assert p - x - 1.2 * cos(1.0 * x) == 0 p = expr_to_holonomic(x) * expr_to_holonomic(1.2 * cos(x)) p = p.to_expr().subs(C_1, 0) assert p - 1.2 * x * cos(1.0 * x) == 0
def test_from_meijerg(): x = symbols('x') R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx') p = from_meijerg(meijerg(([], [S(3)/2]), ([S(1)/2], [S(1)/2, 1]), x)) q = HolonomicFunction(x/2 - 1/4 + (-x**2 + x/4)*Dx + x**2*Dx**2 + x**3*Dx**3, x, 1, \ [1/sqrt(pi), 1/(2*sqrt(pi)), -1/(4*sqrt(pi))]) assert p == q p = from_meijerg(meijerg(([], []), ([0], []), x)) q = HolonomicFunction(1 + Dx, x, 0, [1]) assert p == q p = from_meijerg(meijerg(([1], []), ([S(1)/2], [0]), x)) q = HolonomicFunction((x + 1/2)*Dx + x*Dx**2, x, 1, [sqrt(pi)*erf(1), exp(-1)]) assert p == q p = from_meijerg(meijerg(([0], [1]), ([0], []), 2*x**2)) q = HolonomicFunction((3*x**2 - 1)*Dx + x**3*Dx**2, x, 1, [-exp(-S(1)/2) + 1, -exp(-S(1)/2)]) assert p == q
def test_to_Sequence_Initial_Coniditons(): x = symbols('x') R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx') n = symbols('n', integer=True) _, Sn = RecurrenceOperators(QQ.old_poly_ring(n), 'Sn') p = HolonomicFunction(Dx - 1, x, 0, 1).to_sequence() q = HolonomicSequence(-1 + (n + 1)*Sn, 1) assert p == q p = HolonomicFunction(Dx**2 + 1, x, 0, [0, 1]).to_sequence() q = HolonomicSequence(1 + (n**2 + 3*n + 2)*Sn**2, [0, 1]) assert p == q p = HolonomicFunction(Dx**2 + 1 + x**3*Dx, x, 0, [2, 3]).to_sequence() q = HolonomicSequence(n + Sn**2 + (n**2 + 7*n + 12)*Sn**4, [2, 3, -1, -1/2]) assert p == q p = HolonomicFunction(x**3*Dx**5 + 1 + Dx, x).to_sequence() q = HolonomicSequence(1 + (n + 1)*Sn + (n**5 - 5*n**3 + 4*n)*Sn**2) assert p == q
def test_to_Sequence_Initial_Coniditons(): x = symbols('x') R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx') n = symbols('n', integer=True) _, Sn = RecurrenceOperators(QQ.old_poly_ring(n), 'Sn') p = HolonomicFunction(Dx - 1, x, 0, [1]).to_sequence() q = [(HolonomicSequence(-1 + (n + 1)*Sn, 1), 0)] assert p == q p = HolonomicFunction(Dx**2 + 1, x, 0, [0, 1]).to_sequence() q = [(HolonomicSequence(1 + (n**2 + 3*n + 2)*Sn**2, [0, 1]), 0)] assert p == q p = HolonomicFunction(Dx**2 + 1 + x**3*Dx, x, 0, [2, 3]).to_sequence() q = [(HolonomicSequence(n + Sn**2 + (n**2 + 7*n + 12)*Sn**4, [2, 3, -1, -1/2, 1/12]), 1)] assert p == q p = HolonomicFunction(x**3*Dx**5 + 1 + Dx, x).to_sequence() q = [(HolonomicSequence(1 + (n + 1)*Sn + (n**5 - 5*n**3 + 4*n)*Sn**2), 0, 3)] 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)) q = [(HolonomicSequence(n**2 + (n**2 + 2*n)*Sn**2, [0, 0, C_2]), 0, 1)] assert p.to_sequence() == q p = p.diff() q = [(HolonomicSequence((n + 2) + (n + 2)*Sn**2, [C_0, 0]), 1, 0)] assert p.to_sequence() == q p = expr_to_holonomic(erf(x) + x).to_sequence() q = [(HolonomicSequence((2*n**2 - 2*n) + (n**3 + 2*n**2 - n - 2)*Sn**2, [0, 1 + 2/sqrt(pi), 0, C_3]), 0, 2)] 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.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_beta(): a, b, x = symbols("a b x", positive=True) e = x**(a - 1) * (-x + 1)**(b - 1) / beta(a, b) Q = QQ[a, b].get_field() h1 = expr_to_holonomic(e, x, domain=Q) _, Dx = DifferentialOperators(Q.old_poly_ring(x), 'Dx') h2 = HolonomicFunction((a + x * (-a - b + 2) - 1) + (x**2 - x) * Dx, x) assert h1 == h2
def test_gamma(): a, b, x = symbols("a b x", positive=True) e = b**(-a) * x**(a - 1) * exp(-x / b) / gamma(a) Q = QQ[a, b].get_field() h1 = expr_to_holonomic(e, x, domain=Q) _, Dx = DifferentialOperators(Q.old_poly_ring(x), 'Dx') h2 = HolonomicFunction((-a + 1 + x / b) + (x) * Dx, x) assert h1 == h2
def test_gaussian(): mu, x = symbols("mu x") sd = symbols("sd", positive=True) Q = QQ[mu, sd].get_field() e = sqrt(2) * exp(-(-mu + x)**2 / (2 * sd**2)) / (2 * sqrt(pi) * sd) h1 = expr_to_holonomic(e, x, domain=Q) _, Dx = DifferentialOperators(Q.old_poly_ring(x), 'Dx') h2 = HolonomicFunction((-mu / sd**2 + x / sd**2) + (1) * Dx, x) assert h1 == h2
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_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_extended_domain_in_expr_to_holonomic(): x = symbols('x') p = expr_to_holonomic(1.2 * cos(3.1 * x), domain=RR) assert p.to_expr() == 1.2 * cos(3.1 * x) assert sstr(p.integrate(x).to_expr()) == '0.387096774193548*sin(3.1*x)' _, Dx = DifferentialOperators(RR.old_poly_ring(x), 'Dx') p = expr_to_holonomic(1.1329138213 * x, domain=RR, lenics=2) q = HolonomicFunction((-1.1329138213) + (1.1329138213 * x) * Dx, x, 0, [0, 1.1329138213]) assert p == q assert p.to_expr() == 1.1329138213 * x assert sstr(p.integrate((x, 1, 2))) == sstr((1.1329138213 * x).integrate( (x, 1, 2))) y, z = symbols('y, z') p = expr_to_holonomic(sin(x * y * z), x=x, domain=ZZ[y, z]) assert p.to_expr() == sin(x * y * z) assert p.integrate(x).to_expr() == (-cos(x * y * z) + 1) / (y * z) p = expr_to_holonomic(sin(x * y + z), x=x, domain=ZZ[y, z]).integrate(x).to_expr() q = (cos(z) - cos(x * y + z)) / y 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_to_hyper(): x = symbols('x') R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx') p = HolonomicFunction(Dx - 2, x, 0, 3).to_hyper() q = 3 * hyper([], [], 2 * x) assert p == q p = hyperexpand(HolonomicFunction((1 + x) * Dx - 3, x, 0, 2).to_hyper()).expand() q = 2 * x**3 + 6 * x**2 + 6 * x + 2 assert p == q p = HolonomicFunction((1 + x) * Dx**2 + Dx, x, 0, [0, 1]).to_hyper() q = -x**2 * hyper((2, 2, 1), (2, 3), -x) / 2 + x assert p == q p = HolonomicFunction(2 * x * Dx + Dx**2, x, 0, [0, 2 / sqrt(pi)]).to_hyper() q = 2 * x * hyper((1 / 2, ), (3 / 2, ), -x**2) / sqrt(pi) assert p == q p = hyperexpand( HolonomicFunction(2 * x * Dx + Dx**2, x, 0, [1, -2 / sqrt(pi)]).to_hyper()) q = erfc(x) assert p.rewrite(erfc) == q p = hyperexpand( HolonomicFunction((x**2 - 1) + x * Dx + x**2 * Dx**2, x, 0, [0, S(1) / 2]).to_hyper()) q = besselj(1, x) assert p == q p = hyperexpand( HolonomicFunction(x * Dx**2 + Dx + x, x, 0, [1, 0]).to_hyper()) q = besselj(0, x) assert p == q
def test_expr_to_holonomic(): x = symbols('x') R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx') p = expr_to_holonomic((sin(x) / x)**2) q = HolonomicFunction(8*x + (4*x**2 + 6)*Dx + 6*x*Dx**2 + x**2*Dx**3, x, 0, \ [1, 0, -2/3]) assert p == q p = expr_to_holonomic(1 / (1 + x**2)**2) q = HolonomicFunction(4 * x + (x**2 + 1) * Dx, x, 0, 1) assert p == q p = expr_to_holonomic(exp(x) * sin(x) + x * log(1 + x)) q = HolonomicFunction((2*x**3 + 10*x**2 + 20*x + 18) + (-2*x**4 - 10*x**3 - 20*x**2 \ - 18*x)*Dx + (2*x**5 + 6*x**4 + 7*x**3 + 8*x**2 + 10*x - 4)*Dx**2 + \ (-2*x**5 - 5*x**4 - 2*x**3 + 2*x**2 - x + 4)*Dx**3 + (x**5 + 2*x**4 - x**3 - \ 7*x**2/2 + x + 5/2)*Dx**4, x, 0, [0, 1, 4, -1]) assert p == q p = expr_to_holonomic(x * exp(x) + cos(x) + 1) q = HolonomicFunction((-x - 3)*Dx + (x + 2)*Dx**2 + (-x - 3)*Dx**3 + (x + 2)*Dx**4, x, \ 0, [2, 1, 1, 3]) assert p == q assert (x * exp(x) + cos(x) + 1).series(n=10) == p.series(n=10) p = expr_to_holonomic(log(1 + x)**2 + 1) q = HolonomicFunction( Dx + (3 * x + 3) * Dx**2 + (x**2 + 2 * x + 1) * Dx**3, x, 0, [1, 0, 2]) assert p == q p = expr_to_holonomic(erf(x)**2 + x) q = HolonomicFunction((8*x**4 - 2*x**2 + 2)*Dx**2 + (6*x**3 - x/2)*Dx**3 + \ (x**2+ 1/4)*Dx**4, x, 0, [0, 1, 8/pi, 0]) assert p == q p = expr_to_holonomic(cosh(x) * x) q = HolonomicFunction((-x**2 + 2) - 2 * x * Dx + x**2 * Dx**2, x, 0, [0, 1]) assert p == q p = expr_to_holonomic(besselj(2, x)) q = HolonomicFunction((x**2 - 4) + x * Dx + x**2 * Dx**2, x, 0, [0, 0]) assert p == q p = expr_to_holonomic(besselj(0, x) + exp(x)) q = HolonomicFunction((-x**2 - x/2 + 1/2) + (x**2 - x/2 - 3/2)*Dx + (-x**2 + x/2 + 1)*Dx**2 +\ (x**2 + x/2)*Dx**3, x, 0, [2, 1, 1/2]) assert p == q p = expr_to_holonomic(sin(x)**2 / x) q = HolonomicFunction(4 + 4 * x * Dx + 3 * Dx**2 + x * Dx**3, x, 0, [0, 1, 0]) assert p == q p = expr_to_holonomic(sin(x)**2 / x, x0=2) q = HolonomicFunction((4) + (4 * x) * Dx + (3) * Dx**2 + (x) * Dx**3, x, 2, [ sin(2)**2 / 2, sin(2) * cos(2) - sin(2)**2 / 4, -3 * sin(2)**2 / 4 + cos(2)**2 - sin(2) * cos(2) ]) assert p == q p = expr_to_holonomic(log(x) / 2 - Ci(2 * x) / 2 + Ci(2) / 2) q = HolonomicFunction(4*Dx + 4*x*Dx**2 + 3*Dx**3 + x*Dx**4, x, 0, \ [-log(2)/2 - EulerGamma/2 + Ci(2)/2, 0, 1, 0]) assert p == q p = p.to_expr() q = log(x) / 2 - Ci(2 * x) / 2 + Ci(2) / 2 assert p == q p = expr_to_holonomic(x**(S(1) / 2), x0=1) q = HolonomicFunction(x * Dx - 1 / 2, x, 1, 1) assert p == q p = expr_to_holonomic(sqrt(1 + x**2)) q = HolonomicFunction((-x) + (x**2 + 1) * Dx, x, 0, 1) assert p == q
def test_evalf_rk4(): x = symbols('x') R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx') # log(1+x) p = HolonomicFunction((1 + x) * Dx**2 + Dx, x, 0, [0, 1]) # path taken is a straight line from 0 to 1, on the real axis r = [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1] s = '0.693146363174626' # approx. equal to log(2) i.e. 0.693147180559945 assert sstr(p.evalf(r)[-1]) == s # path taken is a traingle 0-->1+i-->2 r = [0.1 + 0.1 * I] for i in range(9): r.append(r[-1] + 0.1 + 0.1 * I) for i in range(10): r.append(r[-1] + 0.1 - 0.1 * I) # close to the exact solution 1.09861228866811 # imaginary part also close to zero s = '1.09861574485151 + 1.36082967699958e-7*I' assert sstr(p.evalf(r)[-1]) == s # sin(x) p = HolonomicFunction(Dx**2 + 1, x, 0, [0, 1]) s = '0.90929463522785 + 1.52655665885959e-16*I' assert sstr(p.evalf(r)[-1]) == s # computing sin(pi/2) using this method # using a linear path from 0 to pi/2 r = [0.1] for i in range(14): r.append(r[-1] + 0.1) r.append(pi / 2) s = '0.999999895088917' # close to 1.0 (exact solution) assert sstr(p.evalf(r)[-1]) == s # trying different path, a rectangle (0-->i-->pi/2 + i-->pi/2) # computing the same value sin(pi/2) using different path r = [0.1 * I] for i in range(9): r.append(r[-1] + 0.1 * I) for i in range(15): r.append(r[-1] + 0.1) r.append(pi / 2 + I) for i in range(10): r.append(r[-1] - 0.1 * I) # close to 1.0 s = '1.00000003415141 + 6.11940487991086e-16*I' assert sstr(p.evalf(r)[-1]) == s # cos(x) p = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0]) # compute cos(pi) along 0-->pi r = [0.05] for i in range(61): r.append(r[-1] + 0.05) r.append(pi) # close to -1 (exact answer) s = '-0.999999993238714' assert sstr(p.evalf(r)[-1]) == s # a rectangular path (0 -> i -> 2+i -> 2) r = [0.1 * I] for i in range(9): r.append(r[-1] + 0.1 * I) for i in range(20): r.append(r[-1] + 0.1) for i in range(10): r.append(r[-1] - 0.1 * I) p = HolonomicFunction(Dx**2 + 1, x, 0, [1, 1]).evalf(r) s = '0.493152791638442 - 1.41553435639707e-15*I' assert sstr(p[-1]) == s
def test_evalf_euler(): x = symbols('x') R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx') # log(1+x) p = HolonomicFunction((1 + x) * Dx**2 + Dx, x, 0, [0, 1]) # path taken is a straight line from 0 to 1, on the real axis r = [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1] s = '0.699525841805253' # approx. equal to log(2) i.e. 0.693147180559945 assert sstr(p.evalf(r, method='Euler')[-1]) == s # path taken is a traingle 0-->1+i-->2 r = [0.1 + 0.1 * I] for i in range(9): r.append(r[-1] + 0.1 + 0.1 * I) for i in range(10): r.append(r[-1] + 0.1 - 0.1 * I) # close to the exact solution 1.09861228866811 # imaginary part also close to zero s = '1.07530466271334 - 0.0251200594793912*I' assert sstr(p.evalf(r, method='Euler')[-1]) == s # sin(x) p = HolonomicFunction(Dx**2 + 1, x, 0, [0, 1]) s = '0.905546532085401 - 6.93889390390723e-18*I' assert sstr(p.evalf(r, method='Euler')[-1]) == s # computing sin(pi/2) using this method # using a linear path from 0 to pi/2 r = [0.1] for i in range(14): r.append(r[-1] + 0.1) r.append(pi / 2) s = '1.08016557252834' # close to 1.0 (exact solution) assert sstr(p.evalf(r, method='Euler')[-1]) == s # trying different path, a rectangle (0-->i-->pi/2 + i-->pi/2) # computing the same value sin(pi/2) using different path r = [0.1 * I] for i in range(9): r.append(r[-1] + 0.1 * I) for i in range(15): r.append(r[-1] + 0.1) r.append(pi / 2 + I) for i in range(10): r.append(r[-1] - 0.1 * I) # close to 1.0 s = '0.976882381836257 - 1.65557671738537e-16*I' assert sstr(p.evalf(r, method='Euler')[-1]) == s # cos(x) p = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0]) # compute cos(pi) along 0-->pi r = [0.05] for i in range(61): r.append(r[-1] + 0.05) r.append(pi) # close to -1 (exact answer) s = '-1.08140824719196' assert sstr(p.evalf(r, method='Euler')[-1]) == s # a rectangular path (0 -> i -> 2+i -> 2) r = [0.1 * I] for i in range(9): r.append(r[-1] + 0.1 * I) for i in range(20): r.append(r[-1] + 0.1) for i in range(10): r.append(r[-1] - 0.1 * I) p = HolonomicFunction(Dx**2 + 1, x, 0, [1, 1]).evalf(r, method='Euler') s = '0.501421652861245 - 3.88578058618805e-16*I' assert sstr(p[-1]) == s
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, singular_ics=[ (-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)
def test_evalf_rk4(): x = symbols('x') R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx') # log(1+x) p = HolonomicFunction((1 + x)*Dx**2 + Dx, x, 0, [0, 1]) # path taken is a straight line from 0 to 1, on the real axis r = [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1] s = '0.693146363174626' # approx. equal to log(2) i.e. 0.693147180559945 assert sstr(p.evalf(r)[-1]) == s # path taken is a traingle 0-->1+i-->2 r = [0.1 + 0.1*I] for i in range(9): r.append(r[-1]+0.1+0.1*I) for i in range(10): r.append(r[-1]+0.1-0.1*I) # close to the exact solution 1.09861228866811 # imaginary part also close to zero s = '1.09861574485151 + 1.36082967699958e-7*I' assert sstr(p.evalf(r)[-1]) == s # sin(x) p = HolonomicFunction(Dx**2 + 1, x, 0, [0, 1]) s = '0.90929463522785 + 1.52655665885959e-16*I' assert sstr(p.evalf(r)[-1]) == s # computing sin(pi/2) using this method # using a linear path from 0 to pi/2 r = [0.1] for i in range(14): r.append(r[-1] + 0.1) r.append(pi/2) s = '0.999999895088917' # close to 1.0 (exact solution) assert sstr(p.evalf(r)[-1]) == s # trying different path, a rectangle (0-->i-->pi/2 + i-->pi/2) # computing the same value sin(pi/2) using different path r = [0.1*I] for i in range(9): r.append(r[-1]+0.1*I) for i in range(15): r.append(r[-1]+0.1) r.append(pi/2+I) for i in range(10): r.append(r[-1]-0.1*I) # close to 1.0 s = '1.00000003415141 + 6.11940487991086e-16*I' assert sstr(p.evalf(r)[-1]) == s # cos(x) p = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0]) # compute cos(pi) along 0-->pi r = [0.05] for i in range(61): r.append(r[-1]+0.05) r.append(pi) # close to -1 (exact answer) s = '-0.999999993238714' assert sstr(p.evalf(r)[-1]) == s # a rectangular path (0 -> i -> 2+i -> 2) r = [0.1*I] for i in range(9): r.append(r[-1]+0.1*I) for i in range(20): r.append(r[-1]+0.1) for i in range(10): r.append(r[-1]-0.1*I) p = HolonomicFunction(Dx**2 + 1, x, 0, [1,1]).evalf(r) s = '0.493152791638442 - 1.41553435639707e-15*I' assert sstr(p[-1]) == s
def test_to_Sequence_Initial_Coniditons(): x = symbols('x') R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx') n = symbols('n', integer=True) _, Sn = RecurrenceOperators(QQ.old_poly_ring(n), 'Sn') p = HolonomicFunction(Dx - 1, x, 0, 1).to_sequence() q = [(HolonomicSequence(-1 + (n + 1) * Sn, 1), 0)] assert p == q p = HolonomicFunction(Dx**2 + 1, x, 0, [0, 1]).to_sequence() q = [(HolonomicSequence(1 + (n**2 + 3 * n + 2) * Sn**2, [0, 1]), 0)] assert p == q p = HolonomicFunction(Dx**2 + 1 + x**3 * Dx, x, 0, [2, 3]).to_sequence() q = [(HolonomicSequence(n + Sn**2 + (n**2 + 7 * n + 12) * Sn**4, [2, 3, -1, -1 / 2, 1 / 12]), 1)] assert p == q p = HolonomicFunction(x**3 * Dx**5 + 1 + Dx, x).to_sequence() q = [(HolonomicSequence(1 + (n + 1) * Sn + (n**5 - 5 * n**3 + 4 * n) * Sn**2), 0, 3)] 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)) q = [(HolonomicSequence(n**2 + (n**2 + 2 * n) * Sn**2, [0, 0, C_2]), 0, 1)] assert p.to_sequence() == q p = p.diff() q = [(HolonomicSequence((n + 2) + (n + 2) * Sn**2, [C_0, 0]), 1, 0)] assert p.to_sequence() == q p = expr_to_holonomic(erf(x) + x).to_sequence() q = [(HolonomicSequence( (2 * n**2 - 2 * n) + (n**3 + 2 * n**2 - n - 2) * Sn**2, [0, 1 + 2 / sqrt(pi), 0, C_3]), 0, 2)] assert p == q
def test_expr_to_holonomic(): x = symbols('x') R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx') p = expr_to_holonomic((sin(x) / x)**2) q = HolonomicFunction(8*x + (4*x**2 + 6)*Dx + 6*x*Dx**2 + x**2*Dx**3, x, 0, \ [1, 0, Rational(-2, 3)]) assert p == q p = expr_to_holonomic(1 / (1 + x**2)**2) q = HolonomicFunction(4 * x + (x**2 + 1) * Dx, x, 0, [1]) assert p == q p = expr_to_holonomic(exp(x) * sin(x) + x * log(1 + x)) q = HolonomicFunction((2*x**3 + 10*x**2 + 20*x + 18) + (-2*x**4 - 10*x**3 - 20*x**2 \ - 18*x)*Dx + (2*x**5 + 6*x**4 + 7*x**3 + 8*x**2 + 10*x - 4)*Dx**2 + \ (-2*x**5 - 5*x**4 - 2*x**3 + 2*x**2 - x + 4)*Dx**3 + (x**5 + 2*x**4 - x**3 - \ 7*x**2/2 + x + Rational(5, 2))*Dx**4, x, 0, [0, 1, 4, -1]) assert p == q p = expr_to_holonomic(x * exp(x) + cos(x) + 1) q = HolonomicFunction((-x - 3)*Dx + (x + 2)*Dx**2 + (-x - 3)*Dx**3 + (x + 2)*Dx**4, x, \ 0, [2, 1, 1, 3]) assert p == q assert (x * exp(x) + cos(x) + 1).series(n=10) == p.series(n=10) p = expr_to_holonomic(log(1 + x)**2 + 1) q = HolonomicFunction( Dx + (3 * x + 3) * Dx**2 + (x**2 + 2 * x + 1) * Dx**3, x, 0, [1, 0, 2]) assert p == q p = expr_to_holonomic(erf(x)**2 + x) q = HolonomicFunction((8*x**4 - 2*x**2 + 2)*Dx**2 + (6*x**3 - x/2)*Dx**3 + \ (x**2+ Rational(1, 4))*Dx**4, x, 0, [0, 1, 8/pi, 0]) assert p == q p = expr_to_holonomic(cosh(x) * x) q = HolonomicFunction((-x**2 + 2) - 2 * x * Dx + x**2 * Dx**2, x, 0, [0, 1]) assert p == q p = expr_to_holonomic(besselj(2, x)) q = HolonomicFunction((x**2 - 4) + x * Dx + x**2 * Dx**2, x, 0, [0, 0]) assert p == q p = expr_to_holonomic(besselj(0, x) + exp(x)) q = HolonomicFunction((-x**2 - x/2 + S.Half) + (x**2 - x/2 - Rational(3, 2))*Dx + (-x**2 + x/2 + 1)*Dx**2 +\ (x**2 + x/2)*Dx**3, x, 0, [2, 1, S.Half]) assert p == q p = expr_to_holonomic(sin(x)**2 / x) q = HolonomicFunction(4 + 4 * x * Dx + 3 * Dx**2 + x * Dx**3, x, 0, [0, 1, 0]) assert p == q p = expr_to_holonomic(sin(x)**2 / x, x0=2) q = HolonomicFunction((4) + (4 * x) * Dx + (3) * Dx**2 + (x) * Dx**3, x, 2, [ sin(2)**2 / 2, sin(2) * cos(2) - sin(2)**2 / 4, -3 * sin(2)**2 / 4 + cos(2)**2 - sin(2) * cos(2) ]) assert p == q p = expr_to_holonomic(log(x) / 2 - Ci(2 * x) / 2 + Ci(2) / 2) q = HolonomicFunction(4*Dx + 4*x*Dx**2 + 3*Dx**3 + x*Dx**4, x, 0, \ [-log(2)/2 - EulerGamma/2 + Ci(2)/2, 0, 1, 0]) assert p == q p = p.to_expr() q = log(x) / 2 - Ci(2 * x) / 2 + Ci(2) / 2 assert p == q p = expr_to_holonomic(x**S.Half, x0=1) q = HolonomicFunction(x * Dx - S.Half, x, 1, [1]) assert p == q p = expr_to_holonomic(sqrt(1 + x**2)) q = HolonomicFunction((-x) + (x**2 + 1) * Dx, x, 0, [1]) assert p == q assert (expr_to_holonomic(sqrt(x) + sqrt(2*x)).to_expr()-\ (sqrt(x) + sqrt(2*x))).simplify() == 0 assert expr_to_holonomic(3 * x + 2 * sqrt(x)).to_expr() == 3 * x + 2 * sqrt(x) p = expr_to_holonomic((x**4 + x**3 + 5 * x**2 + 3 * x + 2) / x**2, lenics=3) q = HolonomicFunction((-2*x**4 - x**3 + 3*x + 4) + (x**5 + x**4 + 5*x**3 + 3*x**2 + \ 2*x)*Dx, x, 0, {-2: [2, 3, 5]}) assert p == q p = expr_to_holonomic(1 / (x - 1)**2, lenics=3, x0=1) q = HolonomicFunction((2) + (x - 1) * Dx, x, 1, {-2: [1, 0, 0]}) assert p == q a = symbols("a") p = expr_to_holonomic(sqrt(a * x), x=x) assert p.to_expr() == sqrt(a) * sqrt(x)
def test_addition_initial_condition(): x = symbols('x') R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx') p = HolonomicFunction(Dx - 1, x, 0, [3]) q = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0]) r = HolonomicFunction(-1 + Dx - Dx**2 + Dx**3, x, 0, [4, 3, 2]) assert p + q == r p = HolonomicFunction(Dx - x + Dx**2, x, 0, [1, 2]) q = HolonomicFunction(Dx**2 + x, x, 0, [1, 0]) r = HolonomicFunction((-x**4 - x**3/4 - x**2 + Rational(1, 4)) + (x**3 + x**2/4 + x*Rational(3, 4) + 1)*Dx + \ (x*Rational(-3, 2) + Rational(7, 4))*Dx**2 + (x**2 - x*Rational(7, 4) + Rational(1, 4))*Dx**3 + (x**2 + x/4 + S.Half)*Dx**4, x, 0, [2, 2, -2, 2]) assert p + q == r p = HolonomicFunction(Dx**2 + 4 * x * Dx + x**2, x, 0, [3, 4]) q = HolonomicFunction(Dx**2 + 1, x, 0, [1, 1]) r = HolonomicFunction((x**6 + 2*x**4 - 5*x**2 - 6) + (4*x**5 + 36*x**3 - 32*x)*Dx + \ (x**6 + 3*x**4 + 5*x**2 - 9)*Dx**2 + (4*x**5 + 36*x**3 - 32*x)*Dx**3 + (x**4 + \ 10*x**2 - 3)*Dx**4, x, 0, [4, 5, -1, -17]) assert p + q == r q = HolonomicFunction(Dx**3 + x, x, 2, [3, 0, 1]) p = HolonomicFunction(Dx - 1, x, 2, [1]) r = HolonomicFunction((-x**2 - x + 1) + (x**2 + x)*Dx + (-x - 2)*Dx**3 + \ (x + 1)*Dx**4, x, 2, [4, 1, 2, -5 ]) assert p + q == r p = expr_to_holonomic(sin(x)) q = expr_to_holonomic(1 / x, x0=1) r = HolonomicFunction((x**2 + 6) + (x**3 + 2*x)*Dx + (x**2 + 6)*Dx**2 + (x**3 + 2*x)*Dx**3, \ x, 1, [sin(1) + 1, -1 + cos(1), -sin(1) + 2]) assert p + q == r C_1 = symbols('C_1') p = expr_to_holonomic(sqrt(x)) q = expr_to_holonomic(sqrt(x**2 - x)) r = (p + q).to_expr().subs(C_1, -I / 2).expand() assert r == I * sqrt(x) * sqrt(-x + 1) + sqrt(x)
def test_multiplication_initial_condition(): x = symbols('x') R, Dx = DifferentialOperators(QQ.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((x**8 - 37*x**7/27 - 10*x**6/27 - 164*x**5/9 - 184*x**4/9 + \ 160*x**3/27 + 404*x**2/9 + 8*x + 40/3) + (6*x**7 - 128*x**6/9 - 98*x**5/9 - 28*x**4/9 + \ 8*x**3/9 + 28*x**2 + 40*x/9 - 40)*Dx + (3*x**6 - 82*x**5/9 + 76*x**4/9 + 4*x**3/3 + \ 220*x**2/9 - 80*x/3)*Dx**2 + (-2*x**6 + 128*x**5/27 - 2*x**4/3 -80*x**2/9 + 200/9)*Dx**3 + \ (3*x**5 - 64*x**4/9 - 28*x**3/9 + 6*x**2 - 20*x/9 - 20/3)*Dx**4 + (-4*x**3 + 64*x**2/9 + \ 8*x/3)*Dx**5 + (x**4 - 64*x**3/27 - 4*x**2/3 + 20/9)*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 p = expr_to_holonomic(sin(x)) q = expr_to_holonomic(1 / x) r = HolonomicFunction(x + 2 * Dx + x * Dx**2, x, 1, [sin(1), -sin(1) + cos(1)]) assert p * q == r
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_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_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_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_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_addition_initial_condition(): x = symbols('x') R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx') p = HolonomicFunction(Dx - 1, x, 0, 3) q = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0]) r = HolonomicFunction(-1 + Dx - Dx**2 + Dx**3, x, 0, [4, 3, 2]) assert p + q == r p = HolonomicFunction(Dx - x + Dx**2, x, 0, [1, 2]) q = HolonomicFunction(Dx**2 + x, x, 0, [1, 0]) r = HolonomicFunction((-x**4 - x**3/4 - x**2 + 1/4) + (x**3 + x**2/4 + 3*x/4 + 1)*Dx + \ (-3*x/2 + 7/4)*Dx**2 + (x**2 - 7*x/4 + 1/4)*Dx**3 + (x**2 + x/4 + 1/2)*Dx**4, x, 0, [2, 2, -2, 2]) assert p + q == r p = HolonomicFunction(Dx**2 + 4 * x * Dx + x**2, x, 0, [3, 4]) q = HolonomicFunction(Dx**2 + 1, x, 0, [1, 1]) r = HolonomicFunction((x**6 + 2*x**4 - 5*x**2 - 6) + (4*x**5 + 36*x**3 - 32*x)*Dx + \ (x**6 + 3*x**4 + 5*x**2 - 9)*Dx**2 + (4*x**5 + 36*x**3 - 32*x)*Dx**3 + (x**4 + \ 10*x**2 - 3)*Dx**4, x, 0, [4, 5, -1, -17]) assert p + q == r q = HolonomicFunction(Dx**3 + x, x, 2, [3, 0, 1]) p = HolonomicFunction(Dx - 1, x, 2, [1]) r = HolonomicFunction((-x**2 - x + 1) + (x**2 + x)*Dx + (-x - 2)*Dx**3 + \ (x + 1)*Dx**4, x, 2, [4, 1, 2, -5 ]) assert p + q == r p = expr_to_holonomic(sin(x)) q = expr_to_holonomic(1 / x) r = HolonomicFunction((x**2 + 6) + (x**3 + 2*x)*Dx + (x**2 + 6)*Dx**2 + (x**3 + 2*x)*Dx**3, \ x, 1, [sin(1) + 1, -1 + cos(1), -sin(1) + 2]) assert p + q == r
def test_evalf_euler(): x = symbols('x') R, Dx = DifferentialOperators(QQ.old_poly_ring(x), 'Dx') # log(1+x) p = HolonomicFunction((1 + x)*Dx**2 + Dx, x, 0, [0, 1]) # path taken is a straight line from 0 to 1, on the real axis r = [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1] s = '0.699525841805253' # approx. equal to log(2) i.e. 0.693147180559945 assert sstr(p.evalf(r, method='Euler')[-1]) == s # path taken is a traingle 0-->1+i-->2 r = [0.1 + 0.1*I] for i in range(9): r.append(r[-1]+0.1+0.1*I) for i in range(10): r.append(r[-1]+0.1-0.1*I) # close to the exact solution 1.09861228866811 # imaginary part also close to zero s = '1.07530466271334 - 0.0251200594793912*I' assert sstr(p.evalf(r, method='Euler')[-1]) == s # sin(x) p = HolonomicFunction(Dx**2 + 1, x, 0, [0, 1]) s = '0.905546532085401 - 6.93889390390723e-18*I' assert sstr(p.evalf(r, method='Euler')[-1]) == s # computing sin(pi/2) using this method # using a linear path from 0 to pi/2 r = [0.1] for i in range(14): r.append(r[-1] + 0.1) r.append(pi/2) s = '1.08016557252834' # close to 1.0 (exact solution) assert sstr(p.evalf(r, method='Euler')[-1]) == s # trying different path, a rectangle (0-->i-->pi/2 + i-->pi/2) # computing the same value sin(pi/2) using different path r = [0.1*I] for i in range(9): r.append(r[-1]+0.1*I) for i in range(15): r.append(r[-1]+0.1) r.append(pi/2+I) for i in range(10): r.append(r[-1]-0.1*I) # close to 1.0 s = '0.976882381836257 - 1.65557671738537e-16*I' assert sstr(p.evalf(r, method='Euler')[-1]) == s # cos(x) p = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0]) # compute cos(pi) along 0-->pi r = [0.05] for i in range(61): r.append(r[-1]+0.05) r.append(pi) # close to -1 (exact answer) s = '-1.08140824719196' assert sstr(p.evalf(r, method='Euler')[-1]) == s # a rectangular path (0 -> i -> 2+i -> 2) r = [0.1*I] for i in range(9): r.append(r[-1]+0.1*I) for i in range(20): r.append(r[-1]+0.1) for i in range(10): r.append(r[-1]-0.1*I) p = HolonomicFunction(Dx**2 + 1, x, 0, [1,1]).evalf(r, method='Euler') s = '0.501421652861245 - 3.88578058618805e-16*I' assert sstr(p[-1]) == s