def test_gf_crt(): U = [49, 76, 65] M = [99, 97, 95] p = 912285 u = 639985 assert gf_crt(U, M, ZZ) == u E = [9215, 9405, 9603] S = [62, 24, 12] assert gf_crt1(M, ZZ) == (p, E, S) assert gf_crt2(U, M, p, E, S, ZZ) == u
def crt(m, v, symmetric=False, check=True): r"""Chinese Remainder Theorem. The moduli in m are assumed to be pairwise coprime. The output is then an integer f, such that f = v_i mod m_i for each pair out of v and m. If ``symmetric`` is False a positive integer will be returned, else \|f\| will be less than or equal to the LCM of the moduli, and thus f may be negative. If the moduli are not co-prime the correct result will be returned if/when the test of the result is found to be incorrect. This result will be None if there is no solution. The keyword ``check`` can be set to False if it is known that the moduli are coprime. As an example consider a set of residues ``U = [49, 76, 65]`` and a set of moduli ``M = [99, 97, 95]``. Then we have:: >>> from sympy.ntheory.modular import crt, solve_congruence >>> crt([99, 97, 95], [49, 76, 65]) (639985, 912285) This is the correct result because:: >>> [639985 % m for m in [99, 97, 95]] [49, 76, 65] If the moduli are not co-prime, you may receive an incorrect result if you use ``check=False``: >>> crt([12, 6, 17], [3, 4, 2], check=False) (954, 1224) >>> [954 % m for m in [12, 6, 17]] [6, 0, 2] >>> crt([12, 6, 17], [3, 4, 2]) is None True >>> crt([3, 6], [2, 5]) (5, 6) Note: the order of gf_crt's arguments is reversed relative to crt, and that solve_congruence takes residue, modulus pairs. Programmer's note: rather than checking that all pairs of moduli share no GCD (an O(n**2) test) and rather than factoring all moduli and seeing that there is no factor in common, a check that the result gives the indicated residuals is performed -- an O(n) operation. See Also ======== solve_congruence sympy.polys.galoistools.gf_crt : low level crt routine used by this routine """ if check: m = list(map(as_int, m)) v = list(map(as_int, v)) result = gf_crt(v, m, ZZ) mm = prod(m) if check: if not all(v % m == result % m for v, m in zip(v, m)): result = solve_congruence(*list(zip(v, m)), check=False, symmetric=symmetric) if result is None: return result result, mm = result if symmetric: return symmetric_residue(result, mm), mm return result, mm
def crt(m, v, symmetric=False, check=True): r"""Chinese Remainder Theorem. The moduli in m are assumed to be pairwise coprime. The output is then an integer f, such that f = v_i mod m_i for each pair out of v and m. If ``symmetric`` is False a positive integer will be returned, else \|f\| will be less than or equal to the LCM of the moduli, and thus f may be negative. If the moduli are not co-prime the correct result will be returned if/when the test of the result is found to be incorrect. This result will be None if there is no solution. The keyword ``check`` can be set to False if it is known that the moduli are coprime. As an example consider a set of residues ``U = [49, 76, 65]`` and a set of moduli ``M = [99, 97, 95]``. Then we have:: >>> from sympy.ntheory.modular import crt, solve_congruence >>> crt([99, 97, 95], [49, 76, 65]) (639985, 912285) This is the correct result because:: >>> [639985 % m for m in [99, 97, 95]] [49, 76, 65] If the moduli are not co-prime, you may receive an incorrect result if you use ``check=False``: >>> crt([12, 6, 17], [3, 4, 2], check=False) (954, 1224) >>> [954 % m for m in [12, 6, 17]] [6, 0, 2] >>> crt([12, 6, 17], [3, 4, 2]) is None True >>> crt([3, 6], [2, 5]) (5, 6) Note: the order of gf_crt's arguments is reversed relative to crt, and that solve_congruence takes residue, modulus pairs. Programmer's note: rather than checking that all pairs of moduli share no GCD (an O(n**2) test) and rather than factoring all moduli and seeing that there is no factor in common, a check that the result gives the indicated residuals is performed -- an O(n) operation. See Also ======== solve_congruence sympy.polys.galoistools.gf_crt : low level crt routine used by this routine """ if check: m = map(as_int, m) v = map(as_int, v) result = gf_crt(v, m, ZZ) mm = prod(m) if check: if not all(v % m == result % m for v, m in zip(v, m)): result = solve_congruence(*zip(v, m), check=False, symmetric=symmetric) if result is None: return result result, mm = result if symmetric: return symmetric_residue(result, mm), mm return result, mm
def _collins_crt(r, R, P, p, K): """Wrapper of CRT for Collins's resultant algorithm. """ return gf_int(gf_crt([r, R], [P, p], K), P*p)