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| from sympy.polys.rings import ring | |
| from sympy.polys.domains import ZZ, QQ, AlgebraicField | |
| from sympy.polys.modulargcd import ( | |
| modgcd_univariate, | |
| modgcd_bivariate, | |
| _chinese_remainder_reconstruction_multivariate, | |
| modgcd_multivariate, | |
| _to_ZZ_poly, | |
| _to_ANP_poly, | |
| func_field_modgcd, | |
| _func_field_modgcd_m) | |
| from sympy.functions.elementary.miscellaneous import sqrt | |
| def test_modgcd_univariate_integers(): | |
| R, x = ring("x", ZZ) | |
| f, g = R.zero, R.zero | |
| assert modgcd_univariate(f, g) == (0, 0, 0) | |
| f, g = R.zero, x | |
| assert modgcd_univariate(f, g) == (x, 0, 1) | |
| assert modgcd_univariate(g, f) == (x, 1, 0) | |
| f, g = R.zero, -x | |
| assert modgcd_univariate(f, g) == (x, 0, -1) | |
| assert modgcd_univariate(g, f) == (x, -1, 0) | |
| f, g = 2*x, R(2) | |
| assert modgcd_univariate(f, g) == (2, x, 1) | |
| f, g = 2*x + 2, 6*x**2 - 6 | |
| assert modgcd_univariate(f, g) == (2*x + 2, 1, 3*x - 3) | |
| f = x**4 + 8*x**3 + 21*x**2 + 22*x + 8 | |
| g = x**3 + 6*x**2 + 11*x + 6 | |
| h = x**2 + 3*x + 2 | |
| cff = x**2 + 5*x + 4 | |
| cfg = x + 3 | |
| assert modgcd_univariate(f, g) == (h, cff, cfg) | |
| f = x**4 - 4 | |
| g = x**4 + 4*x**2 + 4 | |
| h = x**2 + 2 | |
| cff = x**2 - 2 | |
| cfg = x**2 + 2 | |
| assert modgcd_univariate(f, g) == (h, cff, cfg) | |
| f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5 | |
| g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21 | |
| h = 1 | |
| cff = f | |
| cfg = g | |
| assert modgcd_univariate(f, g) == (h, cff, cfg) | |
| f = - 352518131239247345597970242177235495263669787845475025293906825864749649589178600387510272*x**49 \ | |
| + 46818041807522713962450042363465092040687472354933295397472942006618953623327997952*x**42 \ | |
| + 378182690892293941192071663536490788434899030680411695933646320291525827756032*x**35 \ | |
| + 112806468807371824947796775491032386836656074179286744191026149539708928*x**28 \ | |
| - 12278371209708240950316872681744825481125965781519138077173235712*x**21 \ | |
| + 289127344604779611146960547954288113529690984687482920704*x**14 \ | |
| + 19007977035740498977629742919480623972236450681*x**7 \ | |
| + 311973482284542371301330321821976049 | |
| g = 365431878023781158602430064717380211405897160759702125019136*x**21 \ | |
| + 197599133478719444145775798221171663643171734081650688*x**14 \ | |
| - 9504116979659010018253915765478924103928886144*x**7 \ | |
| - 311973482284542371301330321821976049 | |
| assert modgcd_univariate(f, f.diff(x))[0] == g | |
| f = 1317378933230047068160*x + 2945748836994210856960 | |
| g = 120352542776360960*x + 269116466014453760 | |
| h = 120352542776360960*x + 269116466014453760 | |
| cff = 10946 | |
| cfg = 1 | |
| assert modgcd_univariate(f, g) == (h, cff, cfg) | |
| def test_modgcd_bivariate_integers(): | |
| R, x, y = ring("x,y", ZZ) | |
| f, g = R.zero, R.zero | |
| assert modgcd_bivariate(f, g) == (0, 0, 0) | |
| f, g = 2*x, R(2) | |
| assert modgcd_bivariate(f, g) == (2, x, 1) | |
| f, g = x + 2*y, x + y | |
| assert modgcd_bivariate(f, g) == (1, f, g) | |
| f, g = x**2 + 2*x*y + y**2, x**3 + y**3 | |
| assert modgcd_bivariate(f, g) == (x + y, x + y, x**2 - x*y + y**2) | |
| f, g = x*y**2 + 2*x*y + x, x*y**3 + x | |
| assert modgcd_bivariate(f, g) == (x*y + x, y + 1, y**2 - y + 1) | |
| f, g = x**2*y**2 + x**2*y + 1, x*y**2 + x*y + 1 | |
| assert modgcd_bivariate(f, g) == (1, f, g) | |
| f = 2*x*y**2 + 4*x*y + 2*x + y**2 + 2*y + 1 | |
| g = 2*x*y**3 + 2*x + y**3 + 1 | |
| assert modgcd_bivariate(f, g) == (2*x*y + 2*x + y + 1, y + 1, y**2 - y + 1) | |
| f, g = 2*x**2 + 4*x + 2, x + 1 | |
| assert modgcd_bivariate(f, g) == (x + 1, 2*x + 2, 1) | |
| f, g = x + 1, 2*x**2 + 4*x + 2 | |
| assert modgcd_bivariate(f, g) == (x + 1, 1, 2*x + 2) | |
| f = 2*x**2 + 4*x*y - 2*x - 4*y | |
| g = x**2 + x - 2 | |
| assert modgcd_bivariate(f, g) == (x - 1, 2*x + 4*y, x + 2) | |
| f = 2*x**2 + 2*x*y - 3*x - 3*y | |
| g = 4*x*y - 2*x + 4*y**2 - 2*y | |
| assert modgcd_bivariate(f, g) == (x + y, 2*x - 3, 4*y - 2) | |
| def test_chinese_remainder(): | |
| R, x, y = ring("x, y", ZZ) | |
| p, q = 3, 5 | |
| hp = x**3*y - x**2 - 1 | |
| hq = -x**3*y - 2*x*y**2 + 2 | |
| hpq = _chinese_remainder_reconstruction_multivariate(hp, hq, p, q) | |
| assert hpq.trunc_ground(p) == hp | |
| assert hpq.trunc_ground(q) == hq | |
| T, z = ring("z", R) | |
| p, q = 3, 7 | |
| hp = (x*y + 1)*z**2 + x | |
| hq = (x**2 - 3*y)*z + 2 | |
| hpq = _chinese_remainder_reconstruction_multivariate(hp, hq, p, q) | |
| assert hpq.trunc_ground(p) == hp | |
| assert hpq.trunc_ground(q) == hq | |
| def test_modgcd_multivariate_integers(): | |
| R, x, y = ring("x,y", ZZ) | |
| f, g = R.zero, R.zero | |
| assert modgcd_multivariate(f, g) == (0, 0, 0) | |
| f, g = 2*x**2 + 4*x + 2, x + 1 | |
| assert modgcd_multivariate(f, g) == (x + 1, 2*x + 2, 1) | |
| f, g = x + 1, 2*x**2 + 4*x + 2 | |
| assert modgcd_multivariate(f, g) == (x + 1, 1, 2*x + 2) | |
| f = 2*x**2 + 2*x*y - 3*x - 3*y | |
| g = 4*x*y - 2*x + 4*y**2 - 2*y | |
| assert modgcd_multivariate(f, g) == (x + y, 2*x - 3, 4*y - 2) | |
| f, g = x*y**2 + 2*x*y + x, x*y**3 + x | |
| assert modgcd_multivariate(f, g) == (x*y + x, y + 1, y**2 - y + 1) | |
| f, g = x**2*y**2 + x**2*y + 1, x*y**2 + x*y + 1 | |
| assert modgcd_multivariate(f, g) == (1, f, g) | |
| f = x**4 + 8*x**3 + 21*x**2 + 22*x + 8 | |
| g = x**3 + 6*x**2 + 11*x + 6 | |
| h = x**2 + 3*x + 2 | |
| cff = x**2 + 5*x + 4 | |
| cfg = x + 3 | |
| assert modgcd_multivariate(f, g) == (h, cff, cfg) | |
| R, x, y, z, u = ring("x,y,z,u", ZZ) | |
| f, g = x + y + z, -x - y - z - u | |
| assert modgcd_multivariate(f, g) == (1, f, g) | |
| f, g = u**2 + 2*u + 1, 2*u + 2 | |
| assert modgcd_multivariate(f, g) == (u + 1, u + 1, 2) | |
| f, g = z**2*u**2 + 2*z**2*u + z**2 + z*u + z, u**2 + 2*u + 1 | |
| h, cff, cfg = u + 1, z**2*u + z**2 + z, u + 1 | |
| assert modgcd_multivariate(f, g) == (h, cff, cfg) | |
| assert modgcd_multivariate(g, f) == (h, cfg, cff) | |
| R, x, y, z = ring("x,y,z", ZZ) | |
| f, g = x - y*z, x - y*z | |
| assert modgcd_multivariate(f, g) == (x - y*z, 1, 1) | |
| f, g, h = R.fateman_poly_F_1() | |
| H, cff, cfg = modgcd_multivariate(f, g) | |
| assert H == h and H*cff == f and H*cfg == g | |
| R, x, y, z, u, v = ring("x,y,z,u,v", ZZ) | |
| f, g, h = R.fateman_poly_F_1() | |
| H, cff, cfg = modgcd_multivariate(f, g) | |
| assert H == h and H*cff == f and H*cfg == g | |
| R, x, y, z, u, v, a, b = ring("x,y,z,u,v,a,b", ZZ) | |
| f, g, h = R.fateman_poly_F_1() | |
| H, cff, cfg = modgcd_multivariate(f, g) | |
| assert H == h and H*cff == f and H*cfg == g | |
| R, x, y, z, u, v, a, b, c, d = ring("x,y,z,u,v,a,b,c,d", ZZ) | |
| f, g, h = R.fateman_poly_F_1() | |
| H, cff, cfg = modgcd_multivariate(f, g) | |
| assert H == h and H*cff == f and H*cfg == g | |
| R, x, y, z = ring("x,y,z", ZZ) | |
| f, g, h = R.fateman_poly_F_2() | |
| H, cff, cfg = modgcd_multivariate(f, g) | |
| assert H == h and H*cff == f and H*cfg == g | |
| f, g, h = R.fateman_poly_F_3() | |
| H, cff, cfg = modgcd_multivariate(f, g) | |
| assert H == h and H*cff == f and H*cfg == g | |
| R, x, y, z, t = ring("x,y,z,t", ZZ) | |
| f, g, h = R.fateman_poly_F_3() | |
| H, cff, cfg = modgcd_multivariate(f, g) | |
| assert H == h and H*cff == f and H*cfg == g | |
| def test_to_ZZ_ANP_poly(): | |
| A = AlgebraicField(QQ, sqrt(2)) | |
| R, x = ring("x", A) | |
| f = x*(sqrt(2) + 1) | |
| T, x_, z_ = ring("x_, z_", ZZ) | |
| f_ = x_*z_ + x_ | |
| assert _to_ZZ_poly(f, T) == f_ | |
| assert _to_ANP_poly(f_, R) == f | |
| R, x, t, s = ring("x, t, s", A) | |
| f = x*t**2 + x*s + sqrt(2) | |
| D, t_, s_ = ring("t_, s_", ZZ) | |
| T, x_, z_ = ring("x_, z_", D) | |
| f_ = (t_**2 + s_)*x_ + z_ | |
| assert _to_ZZ_poly(f, T) == f_ | |
| assert _to_ANP_poly(f_, R) == f | |
| def test_modgcd_algebraic_field(): | |
| A = AlgebraicField(QQ, sqrt(2)) | |
| R, x = ring("x", A) | |
| one = A.one | |
| f, g = 2*x, R(2) | |
| assert func_field_modgcd(f, g) == (one, f, g) | |
| f, g = 2*x, R(sqrt(2)) | |
| assert func_field_modgcd(f, g) == (one, f, g) | |
| f, g = 2*x + 2, 6*x**2 - 6 | |
| assert func_field_modgcd(f, g) == (x + 1, R(2), 6*x - 6) | |
| R, x, y = ring("x, y", A) | |
| f, g = x + sqrt(2)*y, x + y | |
| assert func_field_modgcd(f, g) == (one, f, g) | |
| f, g = x*y + sqrt(2)*y**2, R(sqrt(2))*y | |
| assert func_field_modgcd(f, g) == (y, x + sqrt(2)*y, R(sqrt(2))) | |
| f, g = x**2 + 2*sqrt(2)*x*y + 2*y**2, x + sqrt(2)*y | |
| assert func_field_modgcd(f, g) == (g, g, one) | |
| A = AlgebraicField(QQ, sqrt(2), sqrt(3)) | |
| R, x, y, z = ring("x, y, z", A) | |
| h = x**2*y**7 + sqrt(6)/21*z | |
| f, g = h*(27*y**3 + 1), h*(y + x) | |
| assert func_field_modgcd(f, g) == (h, 27*y**3+1, y+x) | |
| h = x**13*y**3 + 1/2*x**10 + 1/sqrt(2) | |
| f, g = h*(x + 1), h*sqrt(2)/sqrt(3) | |
| assert func_field_modgcd(f, g) == (h, x + 1, R(sqrt(2)/sqrt(3))) | |
| A = AlgebraicField(QQ, sqrt(2)**(-1)*sqrt(3)) | |
| R, x = ring("x", A) | |
| f, g = x + 1, x - 1 | |
| assert func_field_modgcd(f, g) == (A.one, f, g) | |
| # when func_field_modgcd suppors function fields, this test can be changed | |
| def test_modgcd_func_field(): | |
| D, t = ring("t", ZZ) | |
| R, x, z = ring("x, z", D) | |
| minpoly = (z**2*t**2 + z**2*t - 1).drop(0) | |
| f, g = x + 1, x - 1 | |
| assert _func_field_modgcd_m(f, g, minpoly) == R.one | |