def rs_integrate(self, x): """ integrate ``p`` with respect to ``x`` Examples ======== >>> from sympy.polys.domains import QQ >>> from sympy.polys.rings import ring >>> from sympy.polys.ring_series import rs_integrate >>> R, x, y = ring('x, y', QQ) >>> p = x + x**2*y**3 >>> rs_integrate(p, x) 1/3*x**3*y**3 + 1/2*x**2 """ ring = self.ring p1 = ring.zero n = ring.gens.index(x) mn = [0]*ring.ngens mn[n] = 1 mn = tuple(mn) for expv in self: e = monomial_mul(expv, mn) p1[e] = self[expv]/(expv[n] + 1) return p1
def rs_integrate(p, x): """ Integrate ``p`` with respect to ``x`` Examples ======== >>> from sympy.polys.domains import QQ >>> from sympy.polys.rings import ring >>> from sympy.polys.ring_series import rs_integrate >>> R, x, y = ring('x, y', QQ) >>> p = x + x**2*y**3 >>> rs_integrate(p, x) 1/3*x**3*y**3 + 1/2*x**2 """ R = p.ring p1 = R.zero n = R.gens.index(x) mn = [0] * R.ngens mn[n] = 1 mn = tuple(mn) for expv in p: e = monomial_mul(expv, mn) p1[e] = p[expv] / (expv[n] + 1) return p1
def sig_mult(s, m): """ Multiply a signature by a monomial. The product of a signature (m, i) and a monomial n is defined as (m * t, i). """ return sig(monomial_mul(s[0], m), s[1])
def representing_matrix(m): M = [[domain.zero] * len(basis) for _ in xrange(len(basis))] for i, v in enumerate(basis): r = ring.term_new(monomial_mul(m, v), domain.one).rem(G) for monom, coeff in r.terms(): j = basis.index(monom) M[j][i] = coeff return M
def representing_matrix(m): M = [[domain.zero] * len(basis) for _ in range(len(basis))] for i, v in enumerate(basis): r = ring.term_new(monomial_mul(m, v), domain.one).rem(G) for monom, coeff in r.terms(): j = basis.index(monom) M[j][i] = coeff return M
def rs_square(p1, x, prec): """ Square the series modulo ``O(x**prec)`` Examples ======== >>> from sympy.polys.domains import QQ >>> from sympy.polys.rings import ring >>> from sympy.polys.ring_series import rs_square >>> R, x = ring('x', QQ) >>> p = x**2 + 2*x + 1 >>> rs_square(p, x, 3) 6*x**2 + 4*x + 1 """ R = p1.ring p = R.zero iv = R.gens.index(x) get = p.get items = list(p1.items()) items.sort(key=lambda e: e[0][iv]) monomial_mul = R.monomial_mul for i in range(len(items)): exp1, v1 = items[i] for j in range(i): exp2, v2 = items[j] if exp1[iv] + exp2[iv] < prec: exp = monomial_mul(exp1, exp2) p[exp] = get(exp, 0) + v1 * v2 else: break p = p.imul_num(2) get = p.get for expv, v in p1.items(): if 2 * expv[iv] < prec: e2 = monomial_mul(expv, expv) p[e2] = get(e2, 0) + v**2 p.strip_zero() return p
def rs_square(p1, x, prec): """ square modulo ``O(x**prec)`` Examples ======== >>> from sympy.polys.domains import QQ >>> from sympy.polys.rings import ring >>> from sympy.polys.ring_series import rs_square >>> R, x = ring('x', QQ) >>> p = x**2 + 2*x + 1 >>> rs_square(p, x, 3) 6*x**2 + 4*x + 1 """ ring = p1.ring p = ring.zero iv = ring.gens.index(x) get = p.get items = list(p1.items()) items.sort(key=lambda e: e[0][iv]) monomial_mul = ring.monomial_mul for i in range(len(items)): exp1, v1 = items[i] for j in range(i): exp2, v2 = items[j] if exp1[iv] + exp2[iv] < prec: exp = monomial_mul(exp1, exp2) p[exp] = get(exp, 0) + v1*v2 else: break p = p.imul_num(2) get = p.get for expv, v in p1.items(): if 2*expv[iv] < prec: e2 = monomial_mul(expv, expv) p[e2] = get(e2, 0) + v**2 p.strip_zero() return p
def rs_mul(p1, p2, x, prec): """ product of series modulo ``O(x**prec)`` ``x`` is the series variable or its position in the generators. Examples ======== >>> from sympy.polys.domains import QQ >>> from sympy.polys.rings import ring >>> from sympy.polys.ring_series import rs_mul >>> R, x = ring('x', QQ) >>> p1 = x**2 + 2*x + 1 >>> p2 = x + 1 >>> rs_mul(p1, p2, x, 3) 3*x**2 + 3*x + 1 """ ring = p1.ring p = ring.zero if ring.__class__ != p2.ring.__class__ or ring != p2.ring: raise ValueError('p1 and p2 must have the same ring') iv = ring.gens.index(x) if not isinstance(p2, PolyElement): raise ValueError('p1 and p2 must have the same ring') if ring == p2.ring: get = p.get items2 = list(p2.items()) items2.sort(key=lambda e: e[0][iv]) if ring.ngens == 1: for exp1, v1 in p1.items(): for exp2, v2 in items2: exp = exp1[0] + exp2[0] if exp < prec: exp = (exp, ) p[exp] = get(exp, 0) + v1*v2 else: break else: monomial_mul = ring.monomial_mul for exp1, v1 in p1.items(): for exp2, v2 in items2: if exp1[iv] + exp2[iv] < prec: exp = monomial_mul(exp1, exp2) p[exp] = get(exp, 0) + v1*v2 else: break p.strip_zero() return p
def rs_mul(p1, p2, x, prec): """ product of series modulo ``O(x**prec)`` ``x`` is the series variable or its position in the generators. Examples ======== >>> from sympy.polys.domains import QQ >>> from sympy.polys.rings import ring >>> from sympy.polys.ring_series import rs_mul >>> R, x = ring('x', QQ) >>> p1 = x**2 + 2*x + 1 >>> p2 = x + 1 >>> rs_mul(p1, p2, x, 3) 3*x**2 + 3*x + 1 """ ring = p1.ring p = ring.zero if ring.__class__ != p2.ring.__class__ or ring != p2.ring: raise ValueError('p1 and p2 must have the same ring') iv = ring.gens.index(x) if not isinstance(p2, PolyElement): raise ValueError('p1 and p2 must have the same ring') if ring == p2.ring: get = p.get items2 = list(p2.items()) items2.sort(key=lambda e: e[0][iv]) if ring.ngens == 1: for exp1, v1 in p1.items(): for exp2, v2 in items2: exp = exp1[0] + exp2[0] if exp < prec: exp = (exp, ) p[exp] = get(exp, 0) + v1 * v2 else: break else: monomial_mul = ring.monomial_mul for exp1, v1 in p1.items(): for exp2, v2 in items2: if exp1[iv] + exp2[iv] < prec: exp = monomial_mul(exp1, exp2) p[exp] = get(exp, 0) + v1 * v2 else: break p.strip_zero() return p
def sdm_monomial_mul(M, X): """ Multiply tuple ``X`` representing a monomial of `K[X]` into the tuple ``M`` representing a monomial of `F`. Examples ======== Multiplying `xy^3` into `x f_1` yields `x^2 y^3 f_1`: >>> from sympy.polys.distributedmodules import sdm_monomial_mul >>> sdm_monomial_mul((1, 1, 0), (1, 3)) (1, 2, 3) """ return (M[0], ) + monomial_mul(X, M[1:])
def sdm_monomial_mul(M, X): """ Multiply tuple ``X`` representing a monomial of `K[X]` into the tuple ``M`` representing a monomial of `F`. Examples ======== Multiplying `xy^3` into `x f_1` yields `x^2 y^3 f_1`: >>> from sympy.polys.distributedmodules import sdm_monomial_mul >>> sdm_monomial_mul((1, 1, 0), (1, 3)) (1, 2, 3) """ return (M[0],) + monomial_mul(X, M[1:])
def test_monomial_mul(): assert monomial_mul((3, 4, 1), (1, 2, 0)) == (4, 6, 1)
def update(G, B, ih): # update G using the set of critical pairs B and h # [BW] page 230 h = f[ih] mh = h.LM # filter new pairs (h, g), g in G C = G.copy() D = set() while C: # select a pair (h, g) by popping an element from C ig = C.pop() g = f[ig] mg = g.LM LCMhg = monomial_lcm(mh, mg) def lcm_divides(ip): # LCM(LM(h), LM(p)) divides LCM(LM(h), LM(g)) m = monomial_lcm(mh, f[ip].LM) return monomial_div(LCMhg, m) # HT(h) and HT(g) disjoint: mh*mg == LCMhg if monomial_mul(mh, mg) == LCMhg or (not any( lcm_divides(ipx) for ipx in C) and not any(lcm_divides(pr[1]) for pr in D)): D.add((ih, ig)) E = set() while D: # select h, g from D (h the same as above) ih, ig = D.pop() mg = f[ig].LM LCMhg = monomial_lcm(mh, mg) if not monomial_mul(mh, mg) == LCMhg: E.add((ih, ig)) # filter old pairs B_new = set() while B: # select g1, g2 from B (-> CP) ig1, ig2 = B.pop() mg1 = f[ig1].LM mg2 = f[ig2].LM LCM12 = monomial_lcm(mg1, mg2) # if HT(h) does not divide lcm(HT(g1), HT(g2)) if not monomial_div(LCM12, mh) or \ monomial_lcm(mg1, mh) == LCM12 or \ monomial_lcm(mg2, mh) == LCM12: B_new.add((ig1, ig2)) B_new |= E # filter polynomials G_new = set() while G: ig = G.pop() mg = f[ig].LM if not monomial_div(mg, mh): G_new.add(ig) G_new.add(ih) return G_new, B_new
def update(G, B, ih): # update G using the set of critical pairs B and h # [BW] page 230 h = f[ih] mh = h.LM # filter new pairs (h, g), g in G C = G.copy() D = set() while C: # select a pair (h, g) by popping an element from C ig = C.pop() g = f[ig] mg = g.LM LCMhg = monomial_lcm(mh, mg) def lcm_divides(ip): # LCM(LM(h), LM(p)) divides LCM(LM(h), LM(g)) m = monomial_lcm(mh, f[ip].LM) return monomial_div(LCMhg, m) # HT(h) and HT(g) disjoint: mh*mg == LCMhg if monomial_mul(mh, mg) == LCMhg or ( not any(lcm_divides(ipx) for ipx in C) and not any(lcm_divides(pr[1]) for pr in D)): D.add((ih, ig)) E = set() while D: # select h, g from D (h the same as above) ih, ig = D.pop() mg = f[ig].LM LCMhg = monomial_lcm(mh, mg) if not monomial_mul(mh, mg) == LCMhg: E.add((ih, ig)) # filter old pairs B_new = set() while B: # select g1, g2 from B (-> CP) ig1, ig2 = B.pop() mg1 = f[ig1].LM mg2 = f[ig2].LM LCM12 = monomial_lcm(mg1, mg2) # if HT(h) does not divide lcm(HT(g1), HT(g2)) if not monomial_div(LCM12, mh) or \ monomial_lcm(mg1, mh) == LCM12 or \ monomial_lcm(mg2, mh) == LCM12: B_new.add((ig1, ig2)) B_new |= E # filter polynomials G_new = set() while G: ig = G.pop() mg = f[ig].LM if not monomial_div(mg, mh): G_new.add(ig) G_new.add(ih) return G_new, B_new