def test_prde_no_cancel(): # b large DE = DifferentialExtension(extension={'D': [Poly(1, x)]}) assert prde_no_cancel_b_large(Poly(1, x), [Poly(x**2, x), Poly(1, x)], 2, DE) == \ ([Poly(x**2 - 2*x + 2, x), Poly(1, x)], Matrix([[1, 0, -1, 0], [0, 1, 0, -1]])) assert prde_no_cancel_b_large(Poly(1, x), [Poly(x**3, x), Poly(1, x)], 3, DE) == \ ([Poly(x**3 - 3*x**2 + 6*x - 6, x), Poly(1, x)], Matrix([[1, 0, -1, 0], [0, 1, 0, -1]])) # b small # XXX: Is there a better example of a monomial with D.degree() > 2? DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**3 + 1, t)]}) # My original q was t**4 + t + 1, but this solution implies q == t**4 # (c1 = 4), with some of the ci for the original q equal to 0. G = [Poly(t**6, t), Poly(x*t**5, t), Poly(t**3, t), Poly(x*t**2, t), Poly(1 + x, t)] assert prde_no_cancel_b_small(Poly(x*t, t), G, 4, DE) == \ ([Poly(t**4/4 - x/12*t**3 + x**2/24*t**2 + (-Rational(11, 12) - x**3/24)*t + x/24, t), Poly(x/3*t**3 - x**2/6*t**2 + (-Rational(1, 3) + x**3/6)*t - x/6, t), Poly(t, t), Poly(0, t), Poly(0, t)], Matrix([[1, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 1, -Rational(1, 4), 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, -1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, -1]]))
def solve_poly_rde(b, cQ, n, DE, parametric=False): """ Solve a Polynomial Risch Differential Equation with degree bound n. This constitutes step 4 of the outline given in the rde.py docstring. For parametric=False, cQ is c, a Poly; for parametric=True, cQ is Q == [q1, ..., qm], a list of Polys. """ from diofant.integrals.prde import (prde_no_cancel_b_large, prde_no_cancel_b_small) # No cancellation if not b.is_zero and (DE.case == 'base' or b.degree(DE.t) > max(0, DE.d.degree(DE.t) - 1)): if parametric: return prde_no_cancel_b_large(b, cQ, n, DE) return no_cancel_b_large(b, cQ, n, DE) elif (b.is_zero or b.degree(DE.t) < DE.d.degree(DE.t) - 1) and \ (DE.case == 'base' or DE.d.degree(DE.t) >= 2): if parametric: return prde_no_cancel_b_small(b, cQ, n, DE) R = no_cancel_b_small(b, cQ, n, DE) if isinstance(R, Poly): return R else: # XXX: Might k be a field? (pg. 209) h, b0, c0 = R with DecrementLevel(DE): b0, c0 = b0.as_poly(DE.t), c0.as_poly(DE.t) if b0 is None: # See above comment raise ValueError("b0 should be a non-Null value") if c0 is None: raise ValueError("c0 should be a non-Null value") y = solve_poly_rde(b0, c0, n, DE).as_poly(DE.t) return h + y elif DE.d.degree(DE.t) >= 2 and b.degree(DE.t) == DE.d.degree(DE.t) - 1 and \ n > -b.as_poly(DE.t).LC()/DE.d.as_poly(DE.t).LC(): # TODO: Is this check necessary, and if so, what should it do if it fails? # b comes from the first element returned from spde() if not b.as_poly(DE.t).LC().is_number: raise TypeError("Result should be a number") if parametric: raise NotImplementedError("prde_no_cancel_b_equal() is not yet " "implemented.") R = no_cancel_equal(b, cQ, n, DE) if isinstance(R, Poly): return R else: h, m, C = R # XXX: Or should it be rischDE()? y = solve_poly_rde(b, C, m, DE) return h + y else: # Cancellation if b.is_zero: raise NotImplementedError( "Remaining cases for Poly (P)RDE are " "not yet implemented (is_deriv_in_field() required).") else: if DE.case == 'exp': if parametric: raise NotImplementedError( "Parametric RDE cancellation " "hyperexponential case is not yet implemented.") return cancel_exp(b, cQ, n, DE) elif DE.case == 'primitive': if parametric: raise NotImplementedError( "Parametric RDE cancellation " "primitive case is not yet implemented.") return cancel_primitive(b, cQ, n, DE) else: raise NotImplementedError( "Other Poly (P)RDE cancellation " "cases are not yet implemented (%s)." % case) if parametric: raise NotImplementedError("Remaining cases for Poly PRDE not yet " "implemented.") raise NotImplementedError("Remaining cases for Poly RDE not yet " "implemented.")