ó ¡¼™\c@ sÈddlmZmZddlmZmZddlmZddlm Z m Z ddl m Z ddl mZddlmZmZmZd„Zeed „Zd „Zed „Zd „Zd S(iÿÿÿÿ(tprint_functiontdivision(tas_inttreduce(tprod(tigcdextigcd(tisprime(tZZ(tgf_crttgf_crt1tgf_crt2cC s||dkr|S||S(sÔReturn the residual mod m such that it is within half of the modulus. >>> from sympy.ntheory.modular import symmetric_residue >>> symmetric_residue(1, 6) 1 >>> symmetric_residue(4, 6) -2 i((tatm((s4lib/python2.7/site-packages/sympy/ntheory/modular.pytsymmetric_residue s c  sè|r3ttt|ƒƒ}ttt|ƒƒ}nt||tƒ‰t|ƒ}|rÅt‡fd†t||ƒDƒƒsÅtdt d|tt||ƒƒŒ‰ˆdkr³ˆSˆ\‰}qÅn|rÞt ˆ|ƒ|fSˆ|fS(s}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. Examples ======== 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 c3 s)|]\}}||ˆ|kVqdS(N((t.0tvR (tresult(s4lib/python2.7/site-packages/sympy/ntheory/modular.pys [stcheckt symmetricN( tlisttmapRR RRtalltziptsolve_congruencetFalsetNoneR(R RRRtmm((Rs4lib/python2.7/site-packages/sympy/ntheory/modular.pytcrts: %! cC s t|tƒS(sÓFirst part of Chinese Remainder Theorem, for multiple application. Examples ======== >>> from sympy.ntheory.modular import crt1 >>> crt1([18, 42, 6]) (4536, [252, 108, 756], [0, 2, 0]) (R R(R ((s4lib/python2.7/site-packages/sympy/ntheory/modular.pytcrt1gs cC s>t|||||tƒ}|r4t||ƒ|fS||fS(sûSecond part of Chinese Remainder Theorem, for multiple application. Examples ======== >>> from sympy.ntheory.modular import crt1, crt2 >>> mm, e, s = crt1([18, 42, 6]) >>> crt2([18, 42, 6], [0, 0, 0], mm, e, s) (0, 4536) (R RR(R RRtetsRR((s4lib/python2.7/site-packages/sympy/ntheory/modular.pytcrt2us c O s¦d„}|}|jdtƒ}|jdtƒr3g|D]$\}}t|ƒt|ƒf^q:}i}xN|D]F\}}||;}||kr­|||krqdSqqn|||>> from sympy.ntheory.modular import solve_congruence What number is 2 mod 3, 3 mod 5 and 2 mod 7? >>> solve_congruence((2, 3), (3, 5), (2, 7)) (23, 105) >>> [23 % m for m in [3, 5, 7]] [2, 3, 2] If you prefer to work with all remainder in one list and all moduli in another, send the arguments like this: >>> solve_congruence(*zip((2, 3, 2), (3, 5, 7))) (23, 105) The moduli need not be co-prime; in this case there may or may not be a solution: >>> solve_congruence((2, 3), (4, 6)) is None True >>> solve_congruence((2, 3), (5, 6)) (5, 6) The symmetric flag will make the result be within 1/2 of the modulus: >>> solve_congruence((2, 3), (5, 6), symmetric=True) (-1, 6) See Also ======== crt : high level routine implementing the Chinese Remainder Theorem cS sÛ|\}}|\}}||||}}}tt|||gƒ} g|||gD]} | | ^qX\}}}|dkr¸t||ƒ\} } } | dkr«dS|| 9}n|||||}} || fS(sôReturn the tuple (a, m) which satisfies the requirement that n = a + i*m satisfy n = a1 + j*m1 and n = a2 = k*m2. References ========== - https://en.wikipedia.org/wiki/Method_of_successive_substitution iN(RRRR(tc1tc2ta1tm1ta2tm2R tbtctgtitinv_at_R ((s4lib/python2.7/site-packages/sympy/ntheory/modular.pytcombine¼s  /   RRcs s!|]\}}t|ƒVqdS(N(R(RtrR ((s4lib/python2.7/site-packages/sympy/ntheory/modular.pys ñsiiN(ii( tgetRtTrueRRtitemsRRRRR( tremainder_modulus_pairsthintR-trmRR.R tuniqtrvtrmitn((s4lib/python2.7/site-packages/sympy/ntheory/modular.pyRˆs84 1   +   N(t __future__RRtsympy.core.compatibilityRRtsympy.core.mulRtsympy.core.numbersRRtsympy.ntheory.primetestRtsympy.polys.domainsRtsympy.polys.galoistoolsR R R RRR0RRR R(((s4lib/python2.7/site-packages/sympy/ntheory/modular.pyts N