def dmp_ground_extract(f, g, u, K): """ Extract common content from a pair of polynomials in ``K[X]``. **Examples** >>> from sympy.polys.domains import ZZ >>> from sympy.polys.densetools import dmp_ground_extract >>> f = ZZ.map([[6, 12], [18]]) >>> g = ZZ.map([[4, 8], [12]]) >>> dmp_ground_extract(f, g, 1, ZZ) (2, [[3, 6], [9]], [[2, 4], [6]]) """ fc = dmp_ground_content(f, u, K) gc = dmp_ground_content(g, u, K) gcd = K.gcd(fc, gc) if not K.is_one(gcd): f = dmp_exquo_ground(f, gcd, u, K) g = dmp_exquo_ground(g, gcd, u, K) return gcd, f, g
def test_dmp_exquo_ground(): f = dmp_normal([[6],[2],[8]], 1, ZZ) assert dmp_exquo_ground(f, ZZ(1), 1, ZZ) == f assert dmp_exquo_ground(f, ZZ(2), 1, ZZ) == dmp_normal([[3],[1],[4]], 1, ZZ) assert dmp_normal(dmp_exquo_ground(f, ZZ(3), 1, ZZ), 1, ZZ) == dmp_normal([[2],[],[2]], 1, ZZ)
def dmp_ground_primitive(f, u, K): """ Compute content and the primitive form of ``f`` in ``K[X]``. **Examples** >>> from sympy.polys.domains import ZZ, QQ >>> from sympy.polys.densetools import dmp_ground_primitive >>> f = ZZ.map([[2, 6], [4, 12]]) >>> g = QQ.map([[2, 6], [4, 12]]) >>> dmp_ground_primitive(f, 1, ZZ) (2, [[1, 3], [2, 6]]) >>> dmp_ground_primitive(g, 1, QQ) (1/1, [[2/1, 6/1], [4/1, 12/1]]) """ if not u: return dup_primitive(f, K) if dmp_zero_p(f, u): return K.zero, f cont = dmp_ground_content(f, u, K) if K.is_one(cont): return cont, f else: return cont, dmp_exquo_ground(f, cont, u, K)
def dmp_ground_monic(f, u, K): """ Divide all coefficients by ``LC(f)`` in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ, QQ >>> R, x,y = ring("x,y", ZZ) >>> f = 3*x**2*y + 6*x**2 + 3*x*y + 9*y + 3 >>> R.dmp_ground_monic(f) x**2*y + 2*x**2 + x*y + 3*y + 1 >>> R, x,y = ring("x,y", QQ) >>> f = 3*x**2*y + 8*x**2 + 5*x*y + 6*x + 2*y + 3 >>> R.dmp_ground_monic(f) x**2*y + 8/3*x**2 + 5/3*x*y + 2*x + 2/3*y + 1 """ if not u: return dup_monic(f, K) if dmp_zero_p(f, u): return f lc = dmp_ground_LC(f, u, K) if K.is_one(lc): return f else: return dmp_exquo_ground(f, lc, u, K)
def dmp_ground_monic(f, u, K): """ Divides all coefficients by ``LC(f)`` in ``K[X]``. Examples ======== >>> from sympy.polys.domains import ZZ, QQ >>> from sympy.polys.densetools import dmp_ground_monic >>> f = ZZ.map([[3, 6], [3, 0], [9, 3]]) >>> g = QQ.map([[3, 8], [5, 6], [2, 3]]) >>> dmp_ground_monic(f, 1, ZZ) [[1, 2], [1, 0], [3, 1]] >>> dmp_ground_monic(g, 1, QQ) [[1/1, 8/3], [5/3, 2/1], [2/3, 1/1]] """ if not u: return dup_monic(f, K) if dmp_zero_p(f, u): return f lc = dmp_ground_LC(f, u, K) if K.is_one(lc): return f else: return dmp_exquo_ground(f, lc, u, K)
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_exquo_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_exquo_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_gcd_interpolate(h, x, v, K): """Interpolate polynomial GCD from integer GCD. """ f = [] while not dmp_zero_p(h, v): g = dmp_ground_trunc(h, x, v, K) f.insert(0, g) h = dmp_sub(h, g, v, K) h = dmp_exquo_ground(h, x, v, K) if K.is_negative(dmp_ground_LC(f, v+1, K)): return dmp_neg(f, v+1, K) else: return f
def _dmp_zz_gcd_interpolate(h, x, v, K): """Interpolate polynomial GCD from integer GCD. """ f = [] while not dmp_zero_p(h, v): g = dmp_ground_trunc(h, x, v, K) f.insert(0, g) h = dmp_sub(h, g, v, K) h = dmp_exquo_ground(h, x, v, K) if K.is_negative(dmp_ground_LC(f, v + 1, K)): return dmp_neg(f, v + 1, K) else: return f
def dmp_qq_collins_resultant(f, g, u, K0): """ Collins's modular resultant algorithm in ``Q[X]``. **Examples** >>> from sympy.polys.domains import QQ >>> from sympy.polys.euclidtools import dmp_qq_collins_resultant >>> f = [[QQ(1,2)], [QQ(1), QQ(2,3)]] >>> g = [[QQ(2), QQ(1)], [QQ(3)]] >>> dmp_qq_collins_resultant(f, g, 1, QQ) [-2/1, -7/3, 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_exquo_ground(r, c, u-1, K0)
def dmp_qq_collins_resultant(f, g, u, K0): """ Collins's modular resultant algorithm in ``Q[X]``. **Examples** >>> from sympy.polys.domains import QQ >>> from sympy.polys.euclidtools import dmp_qq_collins_resultant >>> f = [[QQ(1,2)], [QQ(1), QQ(2,3)]] >>> g = [[QQ(2), QQ(1)], [QQ(3)]] >>> dmp_qq_collins_resultant(f, g, 1, QQ) [-2/1, -7/3, 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_exquo_ground(r, c, u - 1, K0)
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_exquo(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_exquo_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 exquo_ground(f, c): """Exact quotient of ``f`` by a an element of the ground domain. """ return f.per(dmp_exquo_ground(f.rep, f.dom.convert(c), f.lev, f.dom))
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_exquo(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_exquo_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