def test_dmp_mul_term(): assert dmp_mul_term([ZZ(1),ZZ(2),ZZ(3)], ZZ(2), 1, 0, ZZ) == \ dup_mul_term([ZZ(1),ZZ(2),ZZ(3)], ZZ(2), 1, ZZ) assert dmp_mul_term([[]], [ZZ(2)], 3, 1, ZZ) == [[]] assert dmp_mul_term([[ZZ(1)]], [], 3, 1, ZZ) == [[]] assert dmp_mul_term([[ZZ(1),ZZ(2)], [ZZ(3)]], [ZZ(2)], 2, 1, ZZ) == \ [[ZZ(2),ZZ(4)], [ZZ(6)], [], []] assert dmp_mul_term([[]], [QQ(2,3)], 3, 1, QQ) == [[]] assert dmp_mul_term([[QQ(1,2)]], [], 3, 1, QQ) == [[]] assert dmp_mul_term([[QQ(1,5),QQ(2,5)], [QQ(3,5)]], [QQ(2,3)], 2, 1, QQ) == \ [[QQ(2,15),QQ(4,15)], [QQ(6,15)], [], []]
def dmp_rr_prs_gcd(f, g, u, K): """ Computes polynomial GCD using subresultants over a ring. Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``, ``cff = quo(f, h)``, and ``cfg = quo(g, h)``. Examples ======== >>> from sympy.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_prs_gcd(f, g) (x + y, x + y, x) """ if not u: return dup_rr_prs_gcd(f, g, K) result = _dmp_rr_trivial_gcd(f, g, u, K) if result is not None: return result fc, F = dmp_primitive(f, u, K) gc, G = dmp_primitive(g, u, K) h = dmp_subresultants(F, G, u, K)[-1] c, _, _ = dmp_rr_prs_gcd(fc, gc, u - 1, K) if K.is_negative(dmp_ground_LC(h, u, K)): h = dmp_neg(h, u, K) _, h = dmp_primitive(h, u, K) h = dmp_mul_term(h, c, 0, u, K) cff = dmp_quo(f, h, u, K) cfg = dmp_quo(g, h, u, K) return h, cff, cfg
def dmp_rr_prs_gcd(f, g, u, K): """ Computes polynomial GCD using subresultants over a ring. Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``, ``cff = quo(f, h)``, and ``cfg = quo(g, h)``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.euclidtools import dmp_rr_prs_gcd >>> f = ZZ.map([[1], [2, 0], [1, 0, 0]]) >>> g = ZZ.map([[1], [1, 0], []]) >>> dmp_rr_prs_gcd(f, g, 1, ZZ) ([[1], [1, 0]], [[1], [1, 0]], [[1], []]) """ if not u: return dup_rr_prs_gcd(f, g, K) result = _dmp_rr_trivial_gcd(f, g, u, K) if result is not None: return result fc, F = dmp_primitive(f, u, K) gc, G = dmp_primitive(g, u, K) h = dmp_subresultants(F, G, u, K)[-1] c, _, _ = dmp_rr_prs_gcd(fc, gc, u-1, K) if K.is_negative(dmp_ground_LC(h, u, K)): h = dmp_neg(h, u, K) _, h = dmp_primitive(h, u, K) h = dmp_mul_term(h, c, 0, u, K) cff = dmp_quo(f, h, u, K) cfg = dmp_quo(g, h, u, K) return h, cff, cfg
def dmp_rr_prs_gcd(f, g, u, K): """ Computes polynomial GCD using subresultants over a ring. Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``, ``cff = quo(f, h)``, and ``cfg = quo(g, h)``. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.euclidtools import dmp_rr_prs_gcd >>> f = ZZ.map([[1], [2, 0], [1, 0, 0]]) >>> g = ZZ.map([[1], [1, 0], []]) >>> dmp_rr_prs_gcd(f, g, 1, ZZ) ([[1], [1, 0]], [[1], [1, 0]], [[1], []]) """ if not u: return dup_rr_prs_gcd(f, g, K) result = _dmp_rr_trivial_gcd(f, g, u, K) if result is not None: return result fc, F = dmp_primitive(f, u, K) gc, G = dmp_primitive(g, u, K) h = dmp_subresultants(F, G, u, K)[-1] c, _, _ = dmp_rr_prs_gcd(fc, gc, u - 1, K) if K.is_negative(dmp_ground_LC(h, u, K)): h = dmp_neg(h, u, K) _, h = dmp_primitive(h, u, K) h = dmp_mul_term(h, c, 0, u, K) cff = dmp_quo(f, h, u, K) cfg = dmp_quo(g, h, u, K) return h, cff, cfg
def dmp_ff_prs_gcd(f, g, u, K): """ Computes polynomial GCD using subresultants over a field. Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``, ``cff = quo(f, h)``, and ``cfg = quo(g, h)``. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x,y, = ring("x,y", QQ) >>> f = QQ(1,2)*x**2 + x*y + QQ(1,2)*y**2 >>> g = x**2 + x*y >>> R.dmp_ff_prs_gcd(f, g) (x + y, 1/2*x + 1/2*y, x) """ if not u: return dup_ff_prs_gcd(f, g, K) result = _dmp_ff_trivial_gcd(f, g, u, K) if result is not None: return result fc, F = dmp_primitive(f, u, K) gc, G = dmp_primitive(g, u, K) h = dmp_subresultants(F, G, u, K)[-1] c, _, _ = dmp_ff_prs_gcd(fc, gc, u - 1, K) _, h = dmp_primitive(h, u, K) h = dmp_mul_term(h, c, 0, u, K) h = dmp_ground_monic(h, u, K) cff = dmp_quo(f, h, u, K) cfg = dmp_quo(g, h, u, K) return h, cff, cfg
def dmp_ff_prs_gcd(f, g, u, K): """ Computes polynomial GCD using subresultants over a field. Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``, ``cff = quo(f, h)``, and ``cfg = quo(g, h)``. Examples ======== >>> from sympy.polys.domains import QQ >>> from sympy.polys.euclidtools import dmp_ff_prs_gcd >>> f = [[QQ(1,2)], [QQ(1), QQ(0)], [QQ(1,2), QQ(0), QQ(0)]] >>> g = [[QQ(1)], [QQ(1), QQ(0)], []] >>> dmp_ff_prs_gcd(f, g, 1, QQ) ([[1/1], [1/1, 0/1]], [[1/2], [1/2, 0/1]], [[1/1], []]) """ if not u: return dup_ff_prs_gcd(f, g, K) result = _dmp_ff_trivial_gcd(f, g, u, K) if result is not None: return result fc, F = dmp_primitive(f, u, K) gc, G = dmp_primitive(g, u, K) h = dmp_subresultants(F, G, u, K)[-1] c, _, _ = dmp_ff_prs_gcd(fc, gc, u-1, K) _, h = dmp_primitive(h, u, K) h = dmp_mul_term(h, c, 0, u, K) h = dmp_ground_monic(h, u, K) cff = dmp_quo(f, h, u, K) cfg = dmp_quo(g, h, u, K) return h, cff, cfg
def dmp_ff_prs_gcd(f, g, u, K): """ Computes polynomial GCD using subresultants over a field. Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``, ``cff = quo(f, h)``, and ``cfg = quo(g, h)``. Examples ======== >>> from sympy.polys.domains import QQ >>> from sympy.polys.euclidtools import dmp_ff_prs_gcd >>> f = [[QQ(1,2)], [QQ(1), QQ(0)], [QQ(1,2), QQ(0), QQ(0)]] >>> g = [[QQ(1)], [QQ(1), QQ(0)], []] >>> dmp_ff_prs_gcd(f, g, 1, QQ) ([[1/1], [1/1, 0/1]], [[1/2], [1/2, 0/1]], [[1/1], []]) """ if not u: return dup_ff_prs_gcd(f, g, K) result = _dmp_ff_trivial_gcd(f, g, u, K) if result is not None: return result fc, F = dmp_primitive(f, u, K) gc, G = dmp_primitive(g, u, K) h = dmp_subresultants(F, G, u, K)[-1] c, _, _ = dmp_ff_prs_gcd(fc, gc, u - 1, K) _, h = dmp_primitive(h, u, K) h = dmp_mul_term(h, c, 0, u, K) h = dmp_ground_monic(h, u, K) cff = dmp_quo(f, h, u, K) cfg = dmp_quo(g, h, u, K) return h, cff, cfg
def dmp_inner_subresultants(f, g, u, K): """ Subresultant PRS algorithm in ``K[X]``. **Examples** >>> from sympy.polys.domains import ZZ >>> from sympy.polys.euclidtools import dmp_inner_subresultants >>> f = ZZ.map([[3, 0], [], [-1, 0, 0, -4]]) >>> g = ZZ.map([[1], [1, 0, 0, 0], [-9]]) >>> a = [[3, 0, 0, 0, 0], [1, 0, -27, 4]] >>> b = [[-3, 0, 0, -12, 1, 0, -54, 8, 729, -216, 16]] >>> R = ZZ.map([f, g, a, b]) >>> B = ZZ.map([[-1], [1], [9, 0, 0, 0, 0, 0, 0, 0, 0]]) >>> D = ZZ.map([0, 1, 1]) >>> dmp_inner_subresultants(f, g, 1, ZZ) == (R, B, D) 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 R = [f, g] d = n - m v = u - 1 b = dmp_pow(dmp_ground(-K.one, v), d+1, v, K) c = dmp_ground(-K.one, v) B, D = [b], [d] if dmp_zero_p(f, u) or dmp_zero_p(g, u): return R, B, D h = dmp_prem(f, g, u, K) h = dmp_mul_term(h, b, 0, u, K) while not dmp_zero_p(h, u): k = dmp_degree(h, u) R.append(h) lc = dmp_LC(g, K) p = dmp_pow(dmp_neg(lc, v, K), d, v, K) if not d: q = c else: q = dmp_pow(c, d-1, v, K) c = dmp_exquo(p, q, v, K) b = dmp_mul(dmp_neg(lc, v, K), dmp_pow(c, m-k, v, K), v, K) f, g, m, d = g, h, k, m-k B.append(b) D.append(d) h = dmp_prem(f, g, u, K) h = [ dmp_exquo(ch, b, v, K) for ch in h ] return R, B, D
def dmp_inner_subresultants(f, g, u, K): """ Subresultant PRS algorithm in `K[X]`. Examples ======== >>> from sympy.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 dmp_inner_subresultants(f, g, u, K): """ Subresultant PRS algorithm in `K[X]`. Examples ======== >>> from sympy.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] >>> beta = [[-1], [1], [9, 0, 0, 0, 0, 0, 0, 0, 0]] >>> delta = [0, 1, 1] >>> R.dmp_inner_subresultants(f, g) == (prs, beta, delta) 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 R = [f, g] d = n - m v = u - 1 b = dmp_pow(dmp_ground(-K.one, v), d + 1, v, K) c = dmp_ground(-K.one, v) B, D = [b], [d] if dmp_zero_p(f, u) or dmp_zero_p(g, u): return R, B, D h = dmp_prem(f, g, u, K) h = dmp_mul_term(h, b, 0, u, K) while not dmp_zero_p(h, u): k = dmp_degree(h, u) R.append(h) lc = dmp_LC(g, K) p = dmp_pow(dmp_neg(lc, v, K), d, v, K) if not d: q = c else: q = dmp_pow(c, d - 1, v, K) c = dmp_quo(p, q, v, K) b = dmp_mul(dmp_neg(lc, v, K), dmp_pow(c, m - k, v, K), v, K) f, g, m, d = g, h, k, m - k B.append(b) D.append(d) h = dmp_prem(f, g, u, K) h = [ dmp_quo(ch, b, v, K) for ch in h ] return R, B, D
def dmp_inner_subresultants(f, g, u, K): """ Subresultant PRS algorithm in `K[X]`. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy.polys.euclidtools import dmp_inner_subresultants >>> f = ZZ.map([[3, 0], [], [-1, 0, 0, -4]]) >>> g = ZZ.map([[1], [1, 0, 0, 0], [-9]]) >>> a = [[3, 0, 0, 0, 0], [1, 0, -27, 4]] >>> b = [[-3, 0, 0, -12, 1, 0, -54, 8, 729, -216, 16]] >>> R = ZZ.map([f, g, a, b]) >>> B = ZZ.map([[-1], [1], [9, 0, 0, 0, 0, 0, 0, 0, 0]]) >>> D = ZZ.map([0, 1, 1]) >>> dmp_inner_subresultants(f, g, 1, ZZ) == (R, B, D) 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 R = [f, g] d = n - m v = u - 1 b = dmp_pow(dmp_ground(-K.one, v), d + 1, v, K) c = dmp_ground(-K.one, v) B, D = [b], [d] if dmp_zero_p(f, u) or dmp_zero_p(g, u): return R, B, D h = dmp_prem(f, g, u, K) h = dmp_mul_term(h, b, 0, u, K) while not dmp_zero_p(h, u): k = dmp_degree(h, u) R.append(h) lc = dmp_LC(g, K) p = dmp_pow(dmp_neg(lc, v, K), d, v, K) if not d: q = c else: q = dmp_pow(c, d - 1, v, K) c = dmp_quo(p, q, v, K) b = dmp_mul(dmp_neg(lc, v, K), dmp_pow(c, m - k, v, K), v, K) f, g, m, d = g, h, k, m - k B.append(b) D.append(d) h = dmp_prem(f, g, u, K) h = [dmp_quo(ch, b, v, K) for ch in h] return R, B, D