B [@sddlmZmZddlmZddlmZmZmZddl m Z ddl m Z m Z mZmZmZddlmZmZddlmZddlmZd d Zd d Zd S))print_functiondivision)combinations_with_replacement)symbolsAddDummy)Rational)cancelComputationFailedparallel_poly_from_exprreducedPoly)Monomial monomial_div)PolificationFailed)debugcCsXt|\}}yt||gddd\}}Wntk rB||SXt|t||S)z 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) TF)Zfieldexpand)r as_numer_denomr r r)exprfgQrr5lib/python3.7/site-packages/sympy/simplify/ratsimp.pyratsimp s  rcsddlm|dd|dd}td|t|\}}y t||gf||\}Wntk rr|SXj}|j r| _n t d|fd d |d d Dt fd ddfdd t |jjdd}t |jjdd}|r$||St|jjdt|jjdg\} } } s| rtdt| g} x@| D]8\} }}}||ddd}| | |||fqxWt| ddd\} } |js| jdd\}} | jdd\}} t||}ntd}| |j| |jS)a 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 ========== 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) r)solvequickT polynomialFratsimpmodprimez-can't compute rational simplification over %scsg|]}|jqSr)ZLMorder).0r)optrr Usz#ratsimpmodprime..Ncs|dkrdgSg}xjtttj|D]R}dgtjx|D]}|d7<qBWtfddDr(|q(Wfdd|D|dS)z Compute all monomials with degree less than ``n`` that are not divisible by any element of ``leading_monomials``. rcsg|]}t|dkqS)N)r)r!Zlmg)mrrr#dsz6ratsimpmodprime..staircase..csg|]}t|jjqSr)rZas_exprgens)r!s)r"rrr#hs)rrangelenr'allappend)nSZmii)leading_monomialsr" staircase)r&rr1Xs   z"ratsimpmodprime..staircasec s||}}d}||}r,|d} n|} x*||| kr\||f krPP ||f | |td||ftdttdtdttd} ttfddttDj | } ttfd dttDj | } t || || j | j d d d} t| j d  }|d d d }|rBt dd|DsB| |}| |}|tttdgtt}|tttdgtt}t|j }t|j }|dkrtd|| | |f|||kr@|dg}P|d7}|d7}|d7}q4W|dkr||||||\}}}||||||\}}}|||fS)aB Computes a rational simplification of ``a/b`` which minimizes the sum of the total degrees of the numerator and the denominator. The algorithm proceeds by looking at ``a * d - b * c`` modulo the ideal generated by ``G`` for some ``c`` and ``d`` with degree less than ``a`` and ``b`` respectively. The coefficients of ``c`` and ``d`` are indeterminates and thus the coefficients of the normalform of ``a * d - b * c`` are linear polynomials in these indeterminates. If these linear polynomials, considered as system of equations, have a nontrivial solution, then `\frac{a}{b} \equiv \frac{c}{d}` modulo the ideal generated by ``G``. So, by construction, the degree of ``c`` and ``d`` is less than the degree of ``a`` and ``b``, so a simpler representation has been found. After a simpler representation has been found, the algorithm tries to reduce the degree of the numerator and denominator and returns the result afterwards. As an extension, if quick=False, we look at all possible degrees such that the total degree is less than *or equal to* the best current solution. We retain a list of all solutions of minimal degree, and try to find the best one at the end. rr%z%s / %s: %s, %szc:%d)clszd:%dcsg|]}||qSrr)r!r/)CsM1rrr#sz=ratsimpmodprime.._ratsimpmodprime..csg|]}||qSrr)r!r/)DsM2rrr#sT)r polys)r') particularrcSsg|] }|dkqS)rr)r!r(rrrr#szIdeal not prime?)Z total_degreeaddrrr*rr sumr)r'r r Zcoeffsr+valuessubsdictlistzip ValueErrorr,)aballsolNDcdZstepsZmaxdegZboundngc_hatd_hatrr.sol)G_ratsimpmodprimer"rrr1tested)r3r5r4r6rrNjsX   **  ..      z)ratsimpmodprime.._ratsimpmodprime)r r%)domainz*Looking for best minimal solution. Got: %s)r8rcSs t|dt|dS)Nrr%)r*Zterms)xrrrsz!ratsimpmodprime..)key)Zconvert)rr)Zsympyrpoprr rr rrPZhas_assoc_FieldZ get_fieldZ DomainErrorsetr r'r r r*r,r=minZis_FieldZ clear_denomsrqp)rrMr'argsrZnumZdenomr7rPrGrHrDZnewsolrJrKr.rIrLZcnZdnrr)rMrNr0r"rrr1rOrr!sJ       [ ,   rN)Z __future__rr itertoolsrZ sympy.corerrrZsympy.core.numbersrZ sympy.polysr r r r r Zsympy.polys.monomialsrrZsympy.polys.polyerrorsrZsympy.utilities.miscrrrrrrrs