ó ¡¼™\c@ sÌddlmZmZddlmZddlmZmZmZddl m Z ddl m Z m Z mZmZmZddlmZmZddlmZmZddlmZd „Zd „Zd S( iÿÿÿÿ(tprint_functiontdivision(tcombinations_with_replacement(tsymbolstAddtDummy(tRational(tcanceltComputationFailedtparallel_poly_from_exprtreducedtPoly(tMonomialt monomial_div(t DomainErrortPolificationFailed(tdebugcC sqt|ƒjƒ\}}y(t||gdtdtƒ\}}Wntk rX||SXt|Œt||ƒS(s× Put an expression over a common denominator, cancel and reduce. Examples ======== >>> from sympy import ratsimp >>> from sympy.abc import x, y >>> ratsimp(1/x + 1/y) (x + y)/(x*y) tfieldtexpand(Rtas_numer_denomR tTruetFalseRR(texprtftgtQtr((s5lib/python2.7/site-packages/sympy/simplify/ratsimp.pytratsimp s (  c  sâddlm‰|jdtƒ‰|jdtƒ}td|ƒt|ƒjƒ\}}y&t||gˆ||Ž\}‰Wnt k r“|SXˆj }|j r¸|j ƒˆ_ nt d|ƒ‚g|dD]} | jˆjƒ^qÓ‰tƒ‰‡‡‡fd†‰d d ‡‡‡‡‡‡‡fd †‰t|ˆˆjd ˆjƒd }t|ˆˆjd ˆjƒd }|rŽ||jƒSˆt|ˆjd ˆj ƒt|ˆjd ˆj ƒgƒ\} } } ˆ ru| rutdt| ƒƒg} xZ| D]R\}}}}ˆ||dtdtƒ}| j|j|ƒ|j|ƒfƒqWt| dd„ƒ\} } n|jsÀ| jdtƒ\}} | jdtƒ\}} t||ƒ}n td ƒ}| |j| |jS(s¾ Simplifies a rational expression ``expr`` modulo the prime ideal generated by ``G``. ``G`` should be a Groebner basis of the ideal. >>> from sympy.simplify.ratsimp import ratsimpmodprime >>> from sympy.abc import x, y >>> eq = (x + y**5 + y)/(x - y) >>> ratsimpmodprime(eq, [x*y**5 - x - y], x, y, order='lex') (x**2 + x*y + x + y)/(x**2 - x*y) If ``polynomial`` is False, the algorithm computes a rational simplification which minimizes the sum of the total degrees of the numerator and the denominator. If ``polynomial`` is True, this function just brings numerator and denominator into a canonical form. This is much faster, but has potentially worse results. References ========== .. [1] M. Monagan, R. Pearce, Rational Simplification Modulo a Polynomial Ideal, http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.163.6984 (specifically, the second algorithm) iÿÿÿÿ(tsolvetquickt polynomialtratsimpmodprimes-can't compute rational simplification over %sic sî|dkrdgSg}x›tttˆjƒƒ|ƒD]{}dgtˆjƒ}x|D]}||cd7Óttconvert(tsympyRtpopRRRRRR RRTthas_assoc_Fieldt get_fieldRtLMR2tsetR R"R R!R%R:tmintis_Fieldt clear_denomsRtqtp(RRQR"targsRtnumtdenomR3RTRRDRERAtnewsolRNROR(RMRPtcntdnR((RQRRR.R/RRR0RSs5lib/python2.7/site-packages/sympy/simplify/ratsimp.pyR!sJ &    ) '[""B )  N(t __future__RRt itertoolsRt sympy.coreRRRtsympy.core.numbersRt sympy.polysRRR R R tsympy.polys.monomialsR R tsympy.polys.polyerrorsRRtsympy.utilities.miscRRR(((s5lib/python2.7/site-packages/sympy/simplify/ratsimp.pyts(