def dmp_fateman_poly_F_1(n, K): """Fateman's GCD benchmark: trivial GCD """ u = [K(1), K(0)] for i in range(0, n): u = [dmp_one(i, K), u] v = [K(1), K(0), K(0)] for i in range(0, n): v = [dmp_one(i, K), dmp_zero(i), v] m = n - 1 U = dmp_add_term(u, dmp_ground(K(1), m), 0, n, K) V = dmp_add_term(u, dmp_ground(K(2), m), 0, n, K) f = [[-K(3), K(0)], [], [K(1), K(0), -K(1)]] W = dmp_add_term(v, dmp_ground(K(1), m), 0, n, K) Y = dmp_raise(f, m, 1, K) F = dmp_mul(U, V, n, K) G = dmp_mul(W, Y, n, K) H = dmp_one(n, K) return F, G, H
def dmp_compose(f, g, u, K): """ Evaluate functional composition ``f(g)`` in ``K[X]``. Examples ======== >>> from diofant.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_compose(x*y + 2*x + y, y) y**2 + 3*y """ if not u: return dup_compose(f, g, K) if dmp_zero_p(f, u): return f h = [f[0]] for c in f[1:]: h = dmp_mul(h, g, u, K) h = dmp_add_term(h, c, 0, u, K) return h
def dmp_rr_lcm(f, g, u, K): """ Computes polynomial LCM over a ring in `K[X]`. Examples ======== >>> from diofant.polys import ring, ZZ >>> R, x,y, = ring("x,y", ZZ) >>> f = x**2 + 2*x*y + y**2 >>> g = x**2 + x*y >>> R.dmp_rr_lcm(f, g) x**3 + 2*x**2*y + x*y**2 """ fc, f = dmp_ground_primitive(f, u, K) gc, g = dmp_ground_primitive(g, u, K) c = K.lcm(fc, gc) h = dmp_quo(dmp_mul(f, g, u, K), dmp_gcd(f, g, u, K), u, K) return dmp_mul_ground(h, c, u, K)
def dmp_fateman_poly_F_3(n, K): """Fateman's GCD benchmark: sparse inputs (deg f ~ vars f) """ u = dup_from_raw_dict({n + 1: K.one}, K) for i in range(0, n - 1): u = dmp_add_term([u], dmp_one(i, K), n + 1, i + 1, K) v = dmp_add_term(u, dmp_ground(K(2), n - 2), 0, n, K) f = dmp_sqr( dmp_add_term([dmp_neg(v, n - 1, K)], dmp_one(n - 1, K), n + 1, n, K), n, K) g = dmp_sqr(dmp_add_term([v], dmp_one(n - 1, K), n + 1, n, K), n, K) v = dmp_add_term(u, dmp_one(n - 2, K), 0, n - 1, K) h = dmp_sqr(dmp_add_term([v], dmp_one(n - 1, K), n + 1, n, K), n, K) return dmp_mul(f, h, n, K), dmp_mul(g, h, n, K), h
def dmp_fateman_poly_F_2(n, K): """Fateman's GCD benchmark: linearly dense quartic inputs """ u = [K(1), K(0)] for i in range(0, n - 1): u = [dmp_one(i, K), u] m = n - 1 v = dmp_add_term(u, dmp_ground(K(2), m - 1), 0, n, K) f = dmp_sqr([dmp_one(m, K), dmp_neg(v, m, K)], n, K) g = dmp_sqr([dmp_one(m, K), v], n, K) v = dmp_add_term(u, dmp_one(m - 1, K), 0, n, K) h = dmp_sqr([dmp_one(m, K), v], n, K) return dmp_mul(f, h, n, K), dmp_mul(g, h, n, K), h
def dmp_zz_wang_lead_coeffs(f, T, cs, E, H, A, u, K): """Wang/EEZ: Compute correct leading coefficients. """ C, J, v = [], [0] * len(E), u - 1 for h in H: c = dmp_one(v, K) d = dup_LC(h, K) * cs for i in reversed(range(len(E))): k, e, (t, _) = 0, E[i], T[i] while not (d % e): d, k = d // e, k + 1 if k != 0: c, J[i] = dmp_mul(c, dmp_pow(t, k, v, K), v, K), 1 C.append(c) if any(not j for j in J): raise ExtraneousFactors # pragma: no cover CC, HH = [], [] for c, h in zip(C, H): d = dmp_eval_tail(c, A, v, K) lc = dup_LC(h, K) if K.is_one(cs): cc = lc // d else: g = K.gcd(lc, d) d, cc = d // g, lc // g h, cs = dup_mul_ground(h, d, K), cs // d c = dmp_mul_ground(c, cc, v, K) CC.append(c) HH.append(h) if K.is_one(cs): return f, HH, CC CCC, HHH = [], [] for c, h in zip(CC, HH): CCC.append(dmp_mul_ground(c, cs, v, K)) HHH.append(dmp_mul_ground(h, cs, 0, K)) f = dmp_mul_ground(f, cs**(len(H) - 1), u, K) return f, HHH, CCC
def dmp_zz_wang_hensel_lifting(f, H, LC, A, p, u, K): """Wang/EEZ: Parallel Hensel lifting algorithm. """ S, n, v = [f], len(A), u - 1 H = list(H) for i, a in enumerate(reversed(A[1:])): s = dmp_eval_in(S[0], a, n - i, u - i, K) S.insert(0, dmp_ground_trunc(s, p, v - i, K)) d = max(dmp_degree_list(f, u)[1:]) for j, s, a in zip(range(2, n + 2), S, A): G, w = list(H), j - 1 I, J = A[:j - 2], A[j - 1:] for i, (h, lc) in enumerate(zip(H, LC)): lc = dmp_ground_trunc(dmp_eval_tail(lc, J, v, K), p, w - 1, K) H[i] = [lc] + dmp_raise(h[1:], 1, w - 1, K) m = dmp_nest([K.one, -a], w, K) M = dmp_one(w, K) c = dmp_sub(s, dmp_expand(H, w, K), w, K) dj = dmp_degree_in(s, w, w) for k in K.map(range(0, dj)): if dmp_zero_p(c, w): break M = dmp_mul(M, m, w, K) C = dmp_diff_eval_in(c, k + 1, a, w, w, K) if not dmp_zero_p(C, w - 1): C = dmp_quo_ground(C, K.factorial(k + 1), w - 1, K) T = dmp_zz_diophantine(G, C, I, d, p, w - 1, K) for i, (h, t) in enumerate(zip(H, T)): h = dmp_add_mul(h, dmp_raise(t, 1, w - 1, K), M, w, K) H[i] = dmp_ground_trunc(h, p, w, K) h = dmp_sub(s, dmp_expand(H, w, K), w, K) c = dmp_ground_trunc(h, p, w, K) if dmp_expand(H, u, K) != f: raise ExtraneousFactors # pragma: no cover else: return H
def dup_real_imag(f, K): """ Return bivariate polynomials ``f1`` and ``f2``, such that ``f = f1 + f2*I``. Examples ======== >>> from diofant.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dup_real_imag(x**3 + x**2 + x + 1) (x**3 + x**2 - 3*x*y**2 + x - y**2 + 1, 3*x**2*y + 2*x*y - y**3 + y) """ if not K.is_ZZ and not K.is_QQ: raise DomainError( "computing real and imaginary parts is not supported over %s" % K) f1 = dmp_zero(1) f2 = dmp_zero(1) if not f: return f1, f2 g = [[[K.one, K.zero]], [[K.one], []]] h = dmp_ground(f[0], 2) for c in f[1:]: h = dmp_mul(h, g, 2, K) h = dmp_add_term(h, dmp_ground(c, 1), 0, 2, K) H = dup_to_raw_dict(h) for k, h in H.items(): m = k % 4 if not m: f1 = dmp_add(f1, h, 1, K) elif m == 1: f2 = dmp_add(f2, h, 1, K) elif m == 2: f1 = dmp_sub(f1, h, 1, K) else: f2 = dmp_sub(f2, h, 1, K) return f1, f2
def dmp_ff_lcm(f, g, u, K): """ Computes polynomial LCM over a field in `K[X]`. 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_ff_lcm(f, g) x**3 + 4*x**2*y + 4*x*y**2 """ h = dmp_quo(dmp_mul(f, g, u, K), dmp_gcd(f, g, u, K), u, K) return dmp_ground_monic(h, u, K)
def dmp_zz_modular_resultant(f, g, p, u, K): """ Compute resultant of `f` and `g` modulo a prime `p`. Examples ======== >>> from diofant.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = x + y + 2 >>> g = 2*x*y + x + 3 >>> R.dmp_zz_modular_resultant(f, g, 5) -2*y**2 + 1 """ if not u: return gf_int(dup_prs_resultant(f, g, K)[0] % p, p) v = u - 1 n = dmp_degree(f, u) m = dmp_degree(g, u) N = dmp_degree_in(f, 1, u) M = dmp_degree_in(g, 1, u) B = n * M + m * N D, a = [K.one], -K.one r = dmp_zero(v) while dup_degree(D) <= B: while True: a += K.one if a == p: raise HomomorphismFailed('no luck') F = dmp_eval_in(f, gf_int(a, p), 1, u, K) if dmp_degree(F, v) == n: G = dmp_eval_in(g, gf_int(a, p), 1, u, K) if dmp_degree(G, v) == m: break R = dmp_zz_modular_resultant(F, G, p, v, K) e = dmp_eval(r, a, v, K) if not v: R = dup_strip([R]) e = dup_strip([e]) else: R = [R] e = [e] d = K.invert(dup_eval(D, a, K), p) d = dup_mul_ground(D, d, K) d = dmp_raise(d, v, 0, K) c = dmp_mul(d, dmp_sub(R, e, v, K), v, K) r = dmp_add(r, c, v, K) r = dmp_ground_trunc(r, p, v, K) D = dup_mul(D, [K.one, -a], K) D = dup_trunc(D, p, K) return r
def dmp_inner_subresultants(f, g, u, K): """ Subresultant PRS algorithm in `K[X]`. Examples ======== >>> from diofant.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = 3*x**2*y - y**3 - 4 >>> g = x**2 + x*y**3 - 9 >>> a = 3*x*y**4 + y**3 - 27*y + 4 >>> b = -3*y**10 - 12*y**7 + y**6 - 54*y**4 + 8*y**3 + 729*y**2 - 216*y + 16 >>> prs = [f, g, a, b] >>> sres = [[1], [1], [3, 0, 0, 0, 0], [-3, 0, 0, -12, 1, 0, -54, 8, 729, -216, 16]] >>> R.dmp_inner_subresultants(f, g) == (prs, sres) True """ if not u: return dup_inner_subresultants(f, g, K) n = dmp_degree(f, u) m = dmp_degree(g, u) if n < m: f, g = g, f n, m = m, n if dmp_zero_p(f, u): return [], [] v = u - 1 if dmp_zero_p(g, u): return [f], [dmp_ground(K.one, v)] R = [f, g] d = n - m b = dmp_pow(dmp_ground(-K.one, v), d + 1, v, K) h = dmp_prem(f, g, u, K) h = dmp_mul_term(h, b, 0, u, K) lc = dmp_LC(g, K) c = dmp_pow(lc, d, v, K) S = [dmp_ground(K.one, v), c] c = dmp_neg(c, v, K) while not dmp_zero_p(h, u): k = dmp_degree(h, u) R.append(h) f, g, m, d = g, h, k, m - k b = dmp_mul(dmp_neg(lc, v, K), dmp_pow(c, d, v, K), v, K) h = dmp_prem(f, g, u, K) h = [dmp_quo(ch, b, v, K) for ch in h] lc = dmp_LC(g, K) if d > 1: p = dmp_pow(dmp_neg(lc, v, K), d, v, K) q = dmp_pow(c, d - 1, v, K) c = dmp_quo(p, q, v, K) else: c = dmp_neg(lc, v, K) S.append(dmp_neg(c, v, K)) return R, S
def test_dmp_expand(): assert dmp_expand((), 1, ZZ) == [[1]] assert dmp_expand(([[1], [2], [3]], [[1], [2]], [[7], [5], [4], [3]]), 1, ZZ) == \ dmp_mul([[1], [2], [3]], dmp_mul([[1], [2]], [[7], [5], [ 4], [3]], 1, ZZ), 1, ZZ)
def test_dmp_mul(): assert dmp_mul([ZZ(5)], [ZZ(7)], 0, ZZ) == \ dup_mul([ZZ(5)], [ZZ(7)], ZZ) assert dmp_mul([QQ(5, 7)], [QQ(3, 7)], 0, QQ) == \ dup_mul([QQ(5, 7)], [QQ(3, 7)], QQ) assert dmp_mul([[[]]], [[[]]], 2, ZZ) == [[[]]] assert dmp_mul([[[ZZ(1)]]], [[[]]], 2, ZZ) == [[[]]] assert dmp_mul([[[]]], [[[ZZ(1)]]], 2, ZZ) == [[[]]] assert dmp_mul([[[ZZ(2)]]], [[[ZZ(1)]]], 2, ZZ) == [[[ZZ(2)]]] assert dmp_mul([[[ZZ(1)]]], [[[ZZ(2)]]], 2, ZZ) == [[[ZZ(2)]]] assert dmp_mul([[[]]], [[[]]], 2, QQ) == [[[]]] assert dmp_mul([[[QQ(1, 2)]]], [[[]]], 2, QQ) == [[[]]] assert dmp_mul([[[]]], [[[QQ(1, 2)]]], 2, QQ) == [[[]]] assert dmp_mul([[[QQ(2, 7)]]], [[[QQ(1, 3)]]], 2, QQ) == [[[QQ(2, 21)]]] assert dmp_mul([[[QQ(1, 7)]]], [[[QQ(2, 3)]]], 2, QQ) == [[[QQ(2, 21)]]] K = FF(6) assert dmp_mul([[K(2)], [K(1)]], [[K(3)], [K(4)]], 1, K) == [[K(5)], [K(4)]]
def dmp_zz_diophantine(F, c, A, d, p, u, K): """Wang/EEZ: Solve multivariate Diophantine equations. """ if not A: S = [[] for _ in F] n = dup_degree(c) for i, coeff in enumerate(c): if not coeff: continue T = dup_zz_diophantine(F, n - i, p, K) for j, (s, t) in enumerate(zip(S, T)): t = dup_mul_ground(t, coeff, K) S[j] = dup_trunc(dup_add(s, t, K), p, K) else: n = len(A) e = dmp_expand(F, u, K) a, A = A[-1], A[:-1] B, G = [], [] for f in F: B.append(dmp_quo(e, f, u, K)) G.append(dmp_eval_in(f, a, n, u, K)) C = dmp_eval_in(c, a, n, u, K) v = u - 1 S = dmp_zz_diophantine(G, C, A, d, p, v, K) S = [dmp_raise(s, 1, v, K) for s in S] for s, b in zip(S, B): c = dmp_sub_mul(c, s, b, u, K) c = dmp_ground_trunc(c, p, u, K) m = dmp_nest([K.one, -a], n, K) M = dmp_one(n, K) for k in K.map(range(0, d)): if dmp_zero_p(c, u): break M = dmp_mul(M, m, u, K) C = dmp_diff_eval_in(c, k + 1, a, n, u, K) if not dmp_zero_p(C, v): C = dmp_quo_ground(C, K.factorial(k + 1), v, K) T = dmp_zz_diophantine(G, C, A, d, p, v, K) for i, t in enumerate(T): T[i] = dmp_mul(dmp_raise(t, 1, v, K), M, u, K) for i, (s, t) in enumerate(zip(S, T)): S[i] = dmp_add(s, t, u, K) for t, b in zip(T, B): c = dmp_sub_mul(c, t, b, u, K) c = dmp_ground_trunc(c, p, u, K) S = [dmp_ground_trunc(s, p, u, K) for s in S] return S