def dmp_pdiv(f, g, u, K): """ Polynomial pseudo-division in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_pdiv(x**2 + x*y, 2*x + 2) (2*x + 2*y - 2, -4*y + 4) """ if not u: return dup_pdiv(f, g, K) df = dmp_degree(f, u) dg = dmp_degree(g, u) if dg < 0: raise ZeroDivisionError("polynomial division") q, r, dr = dmp_zero(u), f, df if df < dg: return q, r N = df - dg + 1 lc_g = dmp_LC(g, K) while True: lc_r = dmp_LC(r, K) j, N = dr - dg, N - 1 Q = dmp_mul_term(q, lc_g, 0, u, K) q = dmp_add_term(Q, lc_r, j, u, K) R = dmp_mul_term(r, lc_g, 0, u, K) G = dmp_mul_term(g, lc_r, j, u, K) r = dmp_sub(R, G, u, K) _dr, dr = dr, dmp_degree(r, u) if dr < dg: break elif not (dr < _dr): raise PolynomialDivisionFailed(f, g, K) c = dmp_pow(lc_g, N, u - 1, K) q = dmp_mul_term(q, c, 0, u, K) r = dmp_mul_term(r, c, 0, u, K) return q, r
def dmp_ff_div(f, g, u, K): """ Polynomial division with remainder over a field. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x,y = ring("x,y", QQ) >>> R.dmp_ff_div(x**2 + x*y, 2*x + 2) (1/2*x + 1/2*y - 1/2, -y + 1) """ if not u: return dup_ff_div(f, g, K) df = dmp_degree(f, u) dg = dmp_degree(g, u) if dg < 0: raise ZeroDivisionError("polynomial division") q, r, dr = dmp_zero(u), f, df if df < dg: return q, r lc_g, v = dmp_LC(g, K), u - 1 while True: lc_r = dmp_LC(r, K) c, R = dmp_ff_div(lc_r, lc_g, v, K) if not dmp_zero_p(R, v): break j = dr - dg q = dmp_add_term(q, c, j, u, K) h = dmp_mul_term(g, c, j, u, K) r = dmp_sub(r, h, u, K) _dr, dr = dr, dmp_degree(r, u) if dr < dg: break elif not (dr < _dr): raise PolynomialDivisionFailed(f, g, K) return q, r
def dup_pdiv(f, g, K): """ Polynomial pseudo-division in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_pdiv(x**2 + 1, 2*x - 4) (2*x + 4, 20) """ df = dup_degree(f) dg = dup_degree(g) q, r, dr = [], f, df if not g: raise ZeroDivisionError("polynomial division") elif df < dg: return q, r N = df - dg + 1 lc_g = dup_LC(g, K) while True: lc_r = dup_LC(r, K) j, N = dr - dg, N - 1 Q = dup_mul_ground(q, lc_g, K) q = dup_add_term(Q, lc_r, j, K) R = dup_mul_ground(r, lc_g, K) G = dup_mul_term(g, lc_r, j, K) r = dup_sub(R, G, K) _dr, dr = dr, dup_degree(r) if dr < dg: break elif not (dr < _dr): raise PolynomialDivisionFailed(f, g, K) c = lc_g**N q = dup_mul_ground(q, c, K) r = dup_mul_ground(r, c, K) return q, r
def dup_ff_div(f, g, K): """ Polynomial division with remainder over a field. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x = ring("x", QQ) >>> R.dup_ff_div(x**2 + 1, 2*x - 4) (1/2*x + 1, 5) """ df = dup_degree(f) dg = dup_degree(g) q, r, dr = [], f, df if not g: raise ZeroDivisionError("polynomial division") elif df < dg: return q, r lc_g = dup_LC(g, K) while True: lc_r = dup_LC(r, K) c = K.exquo(lc_r, lc_g) j = dr - dg q = dup_add_term(q, c, j, K) h = dup_mul_term(g, c, j, K) r = dup_sub(r, h, K) _dr, dr = dr, dup_degree(r) if dr < dg: break elif dr == _dr and not K.is_Exact: # remove leading term created by rounding error r = dup_strip(r[1:]) dr = dup_degree(r) if dr < dg: break elif not (dr < _dr): raise PolynomialDivisionFailed(f, g, K) return q, r
def dup_rr_div(f, g, K): """ Univariate division with remainder over a ring. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_rr_div(x**2 + 1, 2*x - 4) (0, x**2 + 1) """ df = dup_degree(f) dg = dup_degree(g) q, r, dr = [], f, df if not g: raise ZeroDivisionError("polynomial division") elif df < dg: return q, r lc_g = dup_LC(g, K) while True: lc_r = dup_LC(r, K) if lc_r % lc_g: break c = K.exquo(lc_r, lc_g) j = dr - dg q = dup_add_term(q, c, j, K) h = dup_mul_term(g, c, j, K) r = dup_sub(r, h, K) _dr, dr = dr, dup_degree(r) if dr < dg: break elif not (dr < _dr): raise PolynomialDivisionFailed(f, g, K) return q, r
def dup_prem(f, g, K): """ Polynomial pseudo-remainder in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_prem(x**2 + 1, 2*x - 4) 20 """ df = dup_degree(f) dg = dup_degree(g) r, dr = f, df if not g: raise ZeroDivisionError("polynomial division") elif df < dg: return r N = df - dg + 1 lc_g = dup_LC(g, K) while True: lc_r = dup_LC(r, K) j, N = dr - dg, N - 1 R = dup_mul_ground(r, lc_g, K) G = dup_mul_term(g, lc_r, j, K) r = dup_sub(R, G, K) _dr, dr = dr, dup_degree(r) if dr < dg: break elif not (dr < _dr): raise PolynomialDivisionFailed(f, g, K) return dup_mul_ground(r, lc_g**N, K)