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(xrange(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 xrange(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 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(xrange(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 xrange(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 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 xrange(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
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 xrange(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
def test_dmp_factor_list(): assert dmp_factor_list([[]], 1, ZZ) == (ZZ(0), []) assert dmp_factor_list([[]], 1, QQ) == (QQ(0), []) assert dmp_factor_list([[]], 1, ZZ['y']) == (DMP([], ZZ), []) assert dmp_factor_list([[]], 1, QQ['y']) == (DMP([], QQ), []) assert dmp_factor_list([[]], 1, ZZ, include=True) == [([[]], 1)] assert dmp_factor_list([[ZZ(7)]], 1, ZZ) == (ZZ(7), []) assert dmp_factor_list([[QQ(1, 7)]], 1, QQ) == (QQ(1, 7), []) assert dmp_factor_list([[DMP([ZZ(7)], ZZ)]], 1, ZZ['y']) == (DMP([ZZ(7)], ZZ), []) assert dmp_factor_list([[DMP([QQ(1, 7)], QQ)]], 1, QQ['y']) == (DMP([QQ(1, 7)], QQ), []) assert dmp_factor_list([[ZZ(7)]], 1, ZZ, include=True) == [([[ZZ(7)]], 1)] f, g = [ZZ(1), ZZ(2), ZZ(1)], [ZZ(1), ZZ(1)] assert dmp_factor_list(dmp_nest(f, 200, ZZ), 200, ZZ) == \ (ZZ(1), [(dmp_nest(g, 200, ZZ), 2)]) assert dmp_factor_list(dmp_raise(f, 200, 0, ZZ), 200, ZZ) == \ (ZZ(1), [(dmp_raise(g, 200, 0, ZZ), 2)]) assert dmp_factor_list([ZZ(1),ZZ(2),ZZ(1)], 0, ZZ) == \ (ZZ(1), [([ZZ(1), ZZ(1)], 2)]) assert dmp_factor_list([QQ(1,2),QQ(1),QQ(1,2)], 0, QQ) == \ (QQ(1,2), [([QQ(1),QQ(1)], 2)]) assert dmp_factor_list([[ZZ(1)],[ZZ(2)],[ZZ(1)]], 1, ZZ) == \ (ZZ(1), [([[ZZ(1)], [ZZ(1)]], 2)]) assert dmp_factor_list([[QQ(1,2)],[QQ(1)],[QQ(1,2)]], 1, QQ) == \ (QQ(1,2), [([[QQ(1)],[QQ(1)]], 2)]) f = [[ZZ(4), ZZ(0)], [ZZ(4), ZZ(0), ZZ(0)], []] assert dmp_factor_list(f, 1, ZZ) == \ (ZZ(4), [([[ZZ(1)],[]], 1), ([[ZZ(1),ZZ(0)]], 1), ([[ZZ(1)],[ZZ(1),ZZ(0)]], 1)]) assert dmp_factor_list(f, 1, ZZ, include=True) == \ [([[ZZ(4)],[]], 1), ([[ZZ(1),ZZ(0)]], 1), ([[ZZ(1)],[ZZ(1),ZZ(0)]], 1)] f = [[QQ(1, 2), QQ(0)], [QQ(1, 2), QQ(0), QQ(0)], []] assert dmp_factor_list(f, 1, QQ) == \ (QQ(1,2), [([[QQ(1)],[]], 1), ([[QQ(1),QQ(0)]], 1), ([[QQ(1)],[QQ(1),QQ(0)]], 1)]) f = [[RR(2.0)], [], [-RR(8.0), RR(0.0), RR(0.0)]] assert dmp_factor_list(f, 1, RR) == \ (RR(2.0), [([[RR(1.0)],[-RR(2.0),RR(0.0)]], 1), ([[RR(1.0)],[ RR(2.0),RR(0.0)]], 1)]) f = [[DMP([ZZ(4), ZZ(0)], ZZ)], [DMP([ZZ(4), ZZ(0), ZZ(0)], ZZ)], [DMP([], ZZ)]] assert dmp_factor_list(f, 1, ZZ['y']) == \ (DMP([ZZ(4)],ZZ), [([[DMP([ZZ(1)],ZZ)],[]], 1), ([[DMP([ZZ(1),ZZ(0)],ZZ)]], 1), ([[DMP([ZZ(1)],ZZ)],[DMP([ZZ(1),ZZ(0)],ZZ)]], 1)]) f = [[DMP([QQ(1, 2), QQ(0)], ZZ)], [DMP([QQ(1, 2), QQ(0), QQ(0)], ZZ)], [DMP([], ZZ)]] assert dmp_factor_list(f, 1, QQ['y']) == \ (DMP([QQ(1,2)],QQ), [([[DMP([QQ(1)],QQ)],[]], 1), ([[DMP([QQ(1),QQ(0)],QQ)]], 1), ([[DMP([QQ(1)],QQ)],[DMP([QQ(1),QQ(0)],QQ)]], 1)]) raises(DomainError, "dmp_factor_list([[EX(sin(1))]], 1, EX)")
def test_dmp_nest(): assert dmp_nest(ZZ(1), 2, ZZ) == [[[1]]] assert dmp_nest([[1]], 0, ZZ) == [[1]] assert dmp_nest([[1]], 1, ZZ) == [[[1]]] assert dmp_nest([[1]], 2, ZZ) == [[[[1]]]]
def test_dmp_factor_list(): assert dmp_factor_list([[]], 1, ZZ) == (ZZ(0), []) assert dmp_factor_list([[]], 1, QQ) == (QQ(0), []) assert dmp_factor_list([[]], 1, ZZ['y']) == (DMP([],ZZ), []) assert dmp_factor_list([[]], 1, QQ['y']) == (DMP([],QQ), []) assert dmp_factor_list_include([[]], 1, ZZ) == [([[]], 1)] assert dmp_factor_list([[ZZ(7)]], 1, ZZ) == (ZZ(7), []) assert dmp_factor_list([[QQ(1,7)]], 1, QQ) == (QQ(1,7), []) assert dmp_factor_list([[DMP([ZZ(7)],ZZ)]], 1, ZZ['y']) == (DMP([ZZ(7)],ZZ), []) assert dmp_factor_list([[DMP([QQ(1,7)],QQ)]], 1, QQ['y']) == (DMP([QQ(1,7)],QQ), []) assert dmp_factor_list_include([[ZZ(7)]], 1, ZZ) == [([[ZZ(7)]], 1)] f, g = [ZZ(1),ZZ(2),ZZ(1)], [ZZ(1),ZZ(1)] assert dmp_factor_list(dmp_nest(f, 200, ZZ), 200, ZZ) == \ (ZZ(1), [(dmp_nest(g, 200, ZZ), 2)]) assert dmp_factor_list(dmp_raise(f, 200, 0, ZZ), 200, ZZ) == \ (ZZ(1), [(dmp_raise(g, 200, 0, ZZ), 2)]) assert dmp_factor_list([ZZ(1),ZZ(2),ZZ(1)], 0, ZZ) == \ (ZZ(1), [([ZZ(1), ZZ(1)], 2)]) assert dmp_factor_list([QQ(1,2),QQ(1),QQ(1,2)], 0, QQ) == \ (QQ(1,2), [([QQ(1),QQ(1)], 2)]) assert dmp_factor_list([[ZZ(1)],[ZZ(2)],[ZZ(1)]], 1, ZZ) == \ (ZZ(1), [([[ZZ(1)], [ZZ(1)]], 2)]) assert dmp_factor_list([[QQ(1,2)],[QQ(1)],[QQ(1,2)]], 1, QQ) == \ (QQ(1,2), [([[QQ(1)],[QQ(1)]], 2)]) f = [[ZZ(4),ZZ(0)],[ZZ(4),ZZ(0),ZZ(0)],[]] assert dmp_factor_list(f, 1, ZZ) == \ (ZZ(4), [([[ZZ(1),ZZ(0)]], 1), ([[ZZ(1)],[]], 1), ([[ZZ(1)],[ZZ(1),ZZ(0)]], 1)]) assert dmp_factor_list_include(f, 1, ZZ) == \ [([[ZZ(4),ZZ(0)]], 1), ([[ZZ(1)],[]], 1), ([[ZZ(1)],[ZZ(1),ZZ(0)]], 1)] f = [[QQ(1,2),QQ(0)],[QQ(1,2),QQ(0),QQ(0)],[]] assert dmp_factor_list(f, 1, QQ) == \ (QQ(1,2), [([[QQ(1),QQ(0)]], 1), ([[QQ(1)],[]], 1), ([[QQ(1)],[QQ(1),QQ(0)]], 1)]) f = [[RR(2.0)],[],[-RR(8.0),RR(0.0),RR(0.0)]] assert dmp_factor_list(f, 1, RR) == \ (RR(2.0), [([[RR(1.0)],[-RR(2.0),RR(0.0)]], 1), ([[RR(1.0)],[ RR(2.0),RR(0.0)]], 1)]) f = [[DMP([ZZ(4),ZZ(0)],ZZ)],[DMP([ZZ(4),ZZ(0),ZZ(0)],ZZ)],[DMP([],ZZ)]] assert dmp_factor_list(f, 1, ZZ['y']) == \ (DMP([ZZ(4)],ZZ), [([[DMP([ZZ(1),ZZ(0)],ZZ)]], 1), ([[DMP([ZZ(1)],ZZ)],[]], 1), ([[DMP([ZZ(1)],ZZ)],[DMP([ZZ(1),ZZ(0)],ZZ)]], 1)]) f = [[DMP([QQ(1,2),QQ(0)],ZZ)],[DMP([QQ(1,2),QQ(0),QQ(0)],ZZ)],[DMP([],ZZ)]] assert dmp_factor_list(f, 1, QQ['y']) == \ (DMP([QQ(1,2)],QQ), [([[DMP([QQ(1),QQ(0)],QQ)]], 1), ([[DMP([QQ(1)],QQ)],[]], 1), ([[DMP([QQ(1)],QQ)],[DMP([QQ(1),QQ(0)],QQ)]], 1)]) K = FF(2) raises(DomainError, "dmp_factor_list([[K(1)],[],[K(1),K(0),K(0)]], 1, K)") raises(DomainError, "dmp_factor_list([[EX(sin(1))]], 1, EX)")