def cancel_exp(b, c, n, DE): """ Poly Risch Differential Equation - Cancellation: Hyperexponential case. Given a derivation D on k[t], n either an integer or +oo, b in k, and c in k[t] with Dt/t in k and b != 0, either raise NonElementaryIntegralException, in which case the equation Dq + b*q == c has no solution of degree at most n in k[t], or a solution q in k[t] of this equation with deg(q) <= n. """ from diofant.integrals.prde import parametric_log_deriv eta = DE.d.quo(Poly(DE.t, DE.t)).as_expr() with DecrementLevel(DE): etaa, etad = frac_in(eta, DE.t) ba, bd = frac_in(b, DE.t) A = parametric_log_deriv(ba, bd, etaa, etad, DE) if A is not None: a, m, z = A if a == 1: raise NotImplementedError("is_deriv_in_field() is required to " "solve this problem.") # if c*z*t**m == Dp for p in k<t> and q = p/(z*t**m) in k[t] and # deg(q) <= n: # return q # else: # raise NonElementaryIntegralException if c.is_zero: return c # return 0 if n < c.degree(DE.t): raise NonElementaryIntegralException q = Poly(0, DE.t) while not c.is_zero: m = c.degree(DE.t) if n < m: raise NonElementaryIntegralException # a1 = b + m*Dt/t a1 = b.as_expr() with DecrementLevel(DE): # TODO: Write a dummy function that does this idiom a1a, a1d = frac_in(a1, DE.t) a1a = a1a * etad + etaa * a1d * Poly(m, DE.t) a1d = a1d * etad a2a, a2d = frac_in(c.LC(), DE.t) sa, sd = rischDE(a1a, a1d, a2a, a2d, DE) stm = Poly(sa.as_expr() / sd.as_expr() * DE.t**m, DE.t, expand=False) q += stm n = m - 1 c -= b * stm + derivation(stm, DE) # deg(c) becomes smaller return q
def test_frac_in(): assert frac_in(Poly((x + 1)/x*t, t), x) == \ (Poly(t*x + t, x), Poly(x, x)) assert frac_in((x + 1)/x*t, x) == \ (Poly(t*x + t, x), Poly(x, x)) assert frac_in((Poly((x + 1)/x*t, t), Poly(t + 1, t)), x) == \ (Poly(t*x + t, x), Poly((1 + t)*x, x)) pytest.raises(ValueError, lambda: frac_in((x + 1) / log(x) * t, x)) assert frac_in(Poly((2 + 2*x + x*(1 + x))/(1 + x)**2, t), x, cancel=True) == \ (Poly(x + 2, x), Poly(x + 1, x))
def test_frac_in(): assert frac_in(Poly((x + 1)/x*t, t), x) == \ (Poly(t*x + t, x), Poly(x, x)) assert frac_in((x + 1)/x*t, x) == \ (Poly(t*x + t, x), Poly(x, x)) assert frac_in((Poly((x + 1)/x*t, t), Poly(t + 1, t)), x) == \ (Poly(t*x + t, x), Poly((1 + t)*x, x)) pytest.raises(ValueError, lambda: frac_in((x + 1)/log(x)*t, x)) assert frac_in(Poly((2 + 2*x + x*(1 + x))/(1 + x)**2, t), x, cancel=True) == \ (Poly(x + 2, x), Poly(x + 1, x))
def cancel_primitive(b, c, n, DE): """ Poly Risch Differential Equation - Cancellation: Primitive case. Given a derivation D on k[t], n either an integer or +oo, b in k, and c in k[t] with Dt in k and b != 0, either raise NonElementaryIntegralException, in which case the equation Dq + b*q == c has no solution of degree at most n in k[t], or a solution q in k[t] of this equation with deg(q) <= n. """ from diofant.integrals.prde import is_log_deriv_k_t_radical_in_field with DecrementLevel(DE): ba, bd = frac_in(b, DE.t) A = is_log_deriv_k_t_radical_in_field(ba, bd, DE) if A is not None: n, z = A if n == 1: # b == Dz/z raise NotImplementedError("is_deriv_in_field() is required to " " solve this problem.") # if z*c == Dp for p in k[t] and deg(p) <= n: # return p/z # else: # raise NonElementaryIntegralException if c.is_zero: return c # return 0 if n < c.degree(DE.t): raise NonElementaryIntegralException q = Poly(0, DE.t) while not c.is_zero: m = c.degree(DE.t) if n < m: raise NonElementaryIntegralException with DecrementLevel(DE): a2a, a2d = frac_in(c.LC(), DE.t) sa, sd = rischDE(ba, bd, a2a, a2d, DE) stm = Poly(sa.as_expr() / sd.as_expr() * DE.t**m, DE.t, expand=False) q += stm n = m - 1 c -= b * stm + derivation(stm, DE) return q
return a, B, G, Poly(1, DE.t) else: raise ValueError("case must be one of {'exp', 'tan', 'primitive', " "'base'}, not %s." % case) nb = order_at(ba, p, DE.t) - order_at(bd, p, DE.t) nc = min([order_at(Ga, p, DE.t) - order_at(Gd, p, DE.t) for Ga, Gd in G]) n = min(0, nc - min(0, nb)) if not nb: # Possible cancellation. if case == 'exp': dcoeff = DE.d.quo(Poly(DE.t, DE.t)) # We are guaranteed to not have problems, # because case != 'base'. with DecrementLevel(DE): alphaa, alphad = frac_in(-ba.eval(0)/bd.eval(0)/a.eval(0), DE.t) etaa, etad = frac_in(dcoeff, DE.t) A = parametric_log_deriv(alphaa, alphad, etaa, etad, DE) if A is not None: a, m, z = A if a == 1: n = min(n, m) elif case == 'tan': dcoeff = DE.d.quo(Poly(DE.t**2 + 1, DE.t)) # We are guaranteed to not have problems, # because case != 'base'. with DecrementLevel(DE): betaa, alphaa, alphad = real_imag(ba, bd*a, DE.t) betad = alphad etaa, etad = frac_in(dcoeff, DE.t)
raise ValueError("case must be one of {'exp', 'tan', 'primitive', " "'base'}, not %s." % case) nb = order_at(ba, p, DE.t) - order_at(bd, p, DE.t) nc = order_at(ca, p, DE.t) - order_at(cd, p, DE.t) n = min(0, nc - min(0, nb)) if not nb: # Possible cancellation. if case == 'exp': dcoeff = DE.d.quo(Poly(DE.t, DE.t)) # We are guaranteed to not have problems, # because case != 'base'. with DecrementLevel(DE): alphaa, alphad = frac_in(-ba.eval(0) / bd.eval(0) / a.eval(0), DE.t) etaa, etad = frac_in(dcoeff, DE.t) A = parametric_log_deriv(alphaa, alphad, etaa, etad, DE) if A is not None: a, m, z = A if a == 1: n = min(n, m) elif case == 'tan': dcoeff = DE.d.quo(Poly(DE.t**2 + 1, DE.t)) # We are guaranteed to not have problems, # because case != 'base'. with DecrementLevel(DE): alphaa, alphad = frac_in( im(-ba.eval(sqrt(-1)) / bd.eval(sqrt(-1)) / a.eval(sqrt(-1))), DE.t)