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_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 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_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 test_dmp_add_term(): assert dmp_add_term([ZZ(1), ZZ(1), ZZ(1)], ZZ(1), 2, 0, ZZ) == \ dup_add_term([ZZ(1), ZZ(1), ZZ(1)], ZZ(1), 2, ZZ) assert dmp_add_term(f_0, [[]], 3, 2, ZZ) == f_0 assert dmp_add_term(F_0, [[]], 3, 2, QQ) == F_0