def _dmp_inner_gcd(f, g, u, K): """Helper function for `dmp_inner_gcd()`. """ if not K.is_Exact: try: exact = K.get_exact() except DomainError: return dmp_one(u, K), f, g f = dmp_convert(f, u, K, exact) g = dmp_convert(g, u, K, exact) h, cff, cfg = _dmp_inner_gcd(f, g, u, exact) h = dmp_convert(h, u, exact, K) cff = dmp_convert(cff, u, exact, K) cfg = dmp_convert(cfg, u, exact, K) return h, cff, cfg elif K.has_Field: if K.is_QQ and query('USE_HEU_GCD'): try: return dmp_qq_heu_gcd(f, g, u, K) except HeuristicGCDFailed: pass return dmp_ff_prs_gcd(f, g, u, K) else: if K.is_ZZ and query('USE_HEU_GCD'): try: return dmp_zz_heu_gcd(f, g, u, K) except HeuristicGCDFailed: pass return dmp_rr_prs_gcd(f, g, u, K)
def test_dmp_convert(): K0, K1 = ZZ.poly_ring('x'), ZZ assert dmp_convert([K0(1), K0(2)], 0, K0, K1) == [ZZ(1), ZZ(2)] assert dmp_convert([K1(1), K1(2)], 0, K1, K0) == [K0(1), K0(2)] f = [K0(1), K0(2), K0(0), K0(3)] assert dmp_convert(f, 0, K0, K1) == [ZZ(1), ZZ(2), ZZ(0), ZZ(3)] f = [[K0(1)], [K0(2)], [], [K0(3)]] assert dmp_convert(f, 1, K0, K1) == [[ZZ(1)], [ZZ(2)], [], [ZZ(3)]]
def dmp_cancel(f, g, u, K, include=True): """ Cancel common factors in a rational function `f/g`. Examples ======== >>> from diofant.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_cancel(2*x**2 - 2, x**2 - 2*x + 1) (2*x + 2, x - 1) """ K0 = None if K.has_Field and K.has_assoc_Ring: K0, K = K, K.get_ring() cq, f = dmp_clear_denoms(f, u, K0, K, convert=True) cp, g = dmp_clear_denoms(g, u, K0, K, convert=True) else: cp, cq = K.one, K.one _, p, q = dmp_inner_gcd(f, g, u, K) if K0 is not None: _, cp, cq = K.cofactors(cp, cq) p = dmp_convert(p, u, K, K0) q = dmp_convert(q, u, K, K0) K = K0 p_neg = K.is_negative(dmp_ground_LC(p, u, K)) q_neg = K.is_negative(dmp_ground_LC(q, u, K)) if p_neg and q_neg: p, q = dmp_neg(p, u, K), dmp_neg(q, u, K) elif p_neg: cp, p = -cp, dmp_neg(p, u, K) elif q_neg: cp, q = -cp, dmp_neg(q, u, K) if not include: return cp, cq, p, q p = dmp_mul_ground(p, cp, u, K) q = dmp_mul_ground(q, cq, u, K) return p, q
def test_dmp_convert(): K0, K1 = ZZ['x'], ZZ f = [[K0(1)], [K0(2)], [], [K0(3)]] assert dmp_convert(f, 1, K0, K1) == \ [[ZZ(1)], [ZZ(2)], [], [ZZ(3)]]
def dmp_clear_denoms(f, u, K0, K1=None, convert=False): """ Clear denominators, i.e. transform ``K_0`` to ``K_1``. Examples ======== >>> from diofant.polys import ring, QQ >>> R, x,y = ring("x,y", QQ) >>> f = QQ(1,2)*x + QQ(1,3)*y + 1 >>> R.dmp_clear_denoms(f, convert=False) (6, 3*x + 2*y + 6) >>> R.dmp_clear_denoms(f, convert=True) (6, 3*x + 2*y + 6) """ if not u: return dup_clear_denoms(f, K0, K1, convert=convert) if K1 is None: if K0.has_assoc_Ring: K1 = K0.get_ring() else: K1 = K0 common = _rec_clear_denoms(f, u, K0, K1) if not K1.is_one(common): f = dmp_mul_ground(f, common, u, K0) if not convert: return common, f else: return common, dmp_convert(f, u, K0, K1)
def dmp_ext_factor(f, u, K): """Factor multivariate polynomials over algebraic number fields. """ if not u: return dup_ext_factor(f, K) lc = dmp_ground_LC(f, u, K) f = dmp_ground_monic(f, u, K) if all(d <= 0 for d in dmp_degree_list(f, u)): return lc, [] f, F = dmp_sqf_part(f, u, K), f s, g, r = dmp_sqf_norm(f, u, K) factors = dmp_factor_list_include(r, u, K.domain) if len(factors) == 1: coeff, factors = lc, [f] else: H = dmp_raise([K.one, s * K.unit], u, 0, K) for i, (factor, _) in enumerate(factors): h = dmp_convert(factor, u, K.domain, K) h, _, g = dmp_inner_gcd(h, g, u, K) h = dmp_compose(h, H, u, K) factors[i] = h return lc, dmp_trial_division(F, factors, u, K)
def dmp_qq_heu_gcd(f, g, u, K0): """ Heuristic polynomial GCD in `Q[X]`. Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``, ``cff = quo(f, h)``, and ``cfg = quo(g, h)``. Examples ======== >>> from diofant.polys import ring, QQ >>> R, x,y, = ring("x,y", QQ) >>> f = QQ(1,4)*x**2 + x*y + y**2 >>> g = QQ(1,2)*x**2 + x*y >>> R.dmp_qq_heu_gcd(f, g) (x + 2*y, 1/4*x + 1/2*y, 1/2*x) """ result = _dmp_ff_trivial_gcd(f, g, u, K0) if result is not None: return result K1 = K0.get_ring() cf, f = dmp_clear_denoms(f, u, K0, K1) cg, g = dmp_clear_denoms(g, u, K0, K1) f = dmp_convert(f, u, K0, K1) g = dmp_convert(g, u, K0, K1) h, cff, cfg = dmp_zz_heu_gcd(f, g, u, K1) h = dmp_convert(h, u, K1, K0) c = dmp_ground_LC(h, u, K0) h = dmp_ground_monic(h, u, K0) cff = dmp_convert(cff, u, K1, K0) cfg = dmp_convert(cfg, u, K1, K0) cff = dmp_mul_ground(cff, K0.quo(c, cf), u, K0) cfg = dmp_mul_ground(cfg, K0.quo(c, cg), u, K0) return h, cff, cfg
def dmp_qq_collins_resultant(f, g, u, K0): """ Collins's modular resultant algorithm in `Q[X]`. Examples ======== >>> from diofant.polys import ring, QQ >>> R, x,y = ring("x,y", QQ) >>> f = QQ(1,2)*x + y + QQ(2,3) >>> g = 2*x*y + x + 3 >>> R.dmp_qq_collins_resultant(f, g) -2*y**2 - 7/3*y + 5/6 """ n = dmp_degree(f, u) m = dmp_degree(g, u) if n < 0 or m < 0: return dmp_zero(u - 1) K1 = K0.get_ring() cf, f = dmp_clear_denoms(f, u, K0, K1) cg, g = dmp_clear_denoms(g, u, K0, K1) f = dmp_convert(f, u, K0, K1) g = dmp_convert(g, u, K0, K1) r = dmp_zz_collins_resultant(f, g, u, K1) r = dmp_convert(r, u - 1, K1, K0) c = K0.convert(cf**m * cg**n, K1) return dmp_quo_ground(r, c, u - 1, K0)
def test_dmp_integrate_in(): f = dmp_convert(f_6, 3, ZZ, QQ) assert dmp_integrate_in(f, 2, 1, 3, QQ) == \ dmp_swap( dmp_integrate(dmp_swap(f, 0, 1, 3, QQ), 2, 3, QQ), 0, 1, 3, QQ) assert dmp_integrate_in(f, 3, 1, 3, QQ) == \ dmp_swap( dmp_integrate(dmp_swap(f, 0, 1, 3, QQ), 3, 3, QQ), 0, 1, 3, QQ) assert dmp_integrate_in(f, 2, 2, 3, QQ) == \ dmp_swap( dmp_integrate(dmp_swap(f, 0, 2, 3, QQ), 2, 3, QQ), 0, 2, 3, QQ) assert dmp_integrate_in(f, 3, 2, 3, QQ) == \ dmp_swap( dmp_integrate(dmp_swap(f, 0, 2, 3, QQ), 3, 3, QQ), 0, 2, 3, QQ)
def test_dmp_integrate_in(): f = dmp_convert(f_6, 3, ZZ, QQ) assert dmp_integrate_in(f, 2, 1, 3, QQ) == \ dmp_swap( dmp_integrate(dmp_swap(f, 0, 1, 3, QQ), 2, 3, QQ), 0, 1, 3, QQ) assert dmp_integrate_in(f, 3, 1, 3, QQ) == \ dmp_swap( dmp_integrate(dmp_swap(f, 0, 1, 3, QQ), 3, 3, QQ), 0, 1, 3, QQ) assert dmp_integrate_in(f, 2, 2, 3, QQ) == \ dmp_swap( dmp_integrate(dmp_swap(f, 0, 2, 3, QQ), 2, 3, QQ), 0, 2, 3, QQ) assert dmp_integrate_in(f, 3, 2, 3, QQ) == \ dmp_swap( dmp_integrate(dmp_swap(f, 0, 2, 3, QQ), 3, 3, QQ), 0, 2, 3, QQ) pytest.raises(IndexError, lambda: dmp_integrate_in(f, 2, -1, 3, QQ)) pytest.raises(IndexError, lambda: dmp_integrate_in(f, 2, 1, 0, QQ))
def dmp_lift(f, u, K): """ Convert algebraic coefficients to integers in ``K[X]``. Examples ======== >>> from diofant.polys import ring, QQ >>> from diofant import I >>> K = QQ.algebraic_field(I) >>> R, x = ring("x", K) >>> f = x**2 + K([QQ(1), QQ(0)])*x + K([QQ(2), QQ(0)]) >>> R.dmp_lift(f) x**8 + 2*x**6 + 9*x**4 - 8*x**2 + 16 """ if not K.is_Algebraic: raise DomainError( 'computation can be done only in an algebraic domain') F, monoms, polys = dmp_to_dict(f, u), [], [] for monom, coeff in F.items(): if not coeff.is_ground: monoms.append(monom) perms = variations([-1, 1], len(monoms), repetition=True) for perm in perms: G = dict(F) for sign, monom in zip(perm, monoms): if sign == -1: G[monom] = -G[monom] polys.append(dmp_from_dict(G, u, K)) return dmp_convert(dmp_expand(polys, u, K), u, K, K.domain)
def dmp_factor_list(f, u, K0): """Factor polynomials into irreducibles in `K[X]`. """ if not u: return dup_factor_list(f, K0) J, f = dmp_terms_gcd(f, u, K0) cont, f = dmp_ground_primitive(f, u, K0) if K0.is_FiniteField: # pragma: no cover coeff, factors = dmp_gf_factor(f, u, K0) elif K0.is_Algebraic: coeff, factors = dmp_ext_factor(f, u, K0) else: if not K0.is_Exact: K0_inexact, K0 = K0, K0.get_exact() f = dmp_convert(f, u, K0_inexact, K0) else: K0_inexact = None if K0.has_Field: K = K0.get_ring() denom, f = dmp_clear_denoms(f, u, K0, K) f = dmp_convert(f, u, K0, K) else: K = K0 if K.is_ZZ: levels, f, v = dmp_exclude(f, u, K) coeff, factors = dmp_zz_factor(f, v, K) for i, (f, k) in enumerate(factors): factors[i] = (dmp_include(f, levels, v, K), k) elif K.is_Poly: f, v = dmp_inject(f, u, K) coeff, factors = dmp_factor_list(f, v, K.domain) for i, (f, k) in enumerate(factors): factors[i] = (dmp_eject(f, v, K), k) coeff = K.convert(coeff, K.domain) else: # pragma: no cover raise DomainError('factorization not supported over %s' % K0) if K0.has_Field: for i, (f, k) in enumerate(factors): factors[i] = (dmp_convert(f, u, K, K0), k) coeff = K0.convert(coeff, K) if K0_inexact is None: coeff = coeff / denom else: for i, (f, k) in enumerate(factors): f = dmp_quo_ground(f, denom, u, K0) f = dmp_convert(f, u, K0, K0_inexact) factors[i] = (f, k) coeff = K0_inexact.convert(coeff * denom**i, K0) K0 = K0_inexact for i, j in enumerate(reversed(J)): if not j: continue term = {(0, ) * (u - i) + (1, ) + (0, ) * i: K0.one} factors.insert(0, (dmp_from_dict(term, u, K0), j)) return coeff * cont, _sort_factors(factors)