o *ήc@sddlmZddlmZmZmZddlmZddlm Z m Z m Z m Z m Z ddlmZmZddlmZmZddlmZdd Zd d d d dZdS))combinations_with_replacement)symbolsAddDummy)Rational)cancelComputationFailedparallel_poly_from_exprreducedPoly)Monomial monomial_div) DomainErrorPolificationFailed)debugcCsZt|\}}z t||gddd\}}Wn ty"||YSwt|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)fieldexpand)ras_numer_denomr rr)exprfgQrr=/tmp/pip-target-vg8gfxp4/lib/python/sympy/simplify/ratsimp.pyratsimp s   rTF)quick polynomialcsddlmtd|t|\}}zt||gg|Ri|\}Wn ty1|YSwj} | jr>| _nt d| fdd|ddDt fd d dfd d t |j jd d}t |j jd d}|r||St|j jdt|j jdg\} } } s| rtdt| g} | D]\}}}}||ddd}| ||||fqt| 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. Examples ======== >>> 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) r)solveratsimpmodprimez.Cannot compute rational simplification over %scsg|]}|jqSr)LMorder).0roptrr Ssz#ratsimpmodprime..Ncs|dkrdgSg}tttj|D]'}dgtj|D] }|d7<qtfddDr:|qfdd|D|dS)z Compute all monomials with degree less than ``n`` that are not divisible by any element of ``leading_monomials``. rc3s|] }t|duVqdS)N)r )r"lmgmrr bsz5ratsimpmodprime..staircase..csg|] }t|jjqSr)r as_exprgensr"sr#rrr%fz6ratsimpmodprime..staircase..)rrangelenr-allappend)nSmii)leading_monomialsr$ staircaser)rr:Vs  z"ratsimpmodprime..staircasec s||}}d}||}r|d} n|} ||| kr2||f vr'n ||f | |td||ftdttdtdttd} ttfddttDj | } ttfd dttDj | } t || || j | j d d d} t| j d  }|d d d }|r t dd|Ds | |}| |}|tttdgtt}|tttdgtt}t|j }t|j }|dkrtd|| | |f|||kr|dg}n|d7}|d7}|d7}||| ks|dkrQ||||||\}}}||||||\}}}|||fS)ak Computes a rational simplification of ``a/b`` which minimizes the sum of the total degrees of the numerator and the denominator. Explanation =========== 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:%dcg|] }||qSrrr"r8)CsM1rrr%r0z=ratsimpmodprime.._ratsimpmodprime..cr<rrr=)DsM2rrr%r0T)r!polys)r- particularrcss|]}|dkVqdS)rNrr.rrrr+sz._ratsimpmodprime..zIdeal not prime?) total_degreeaddrrr2rr sumr1r-r r!coeffsr3valuessubsdictlistzip ValueErrorr4)aballsolNDcdstepsmaxdegboundngc_hatd_hatrr6sol)G_ratsimpmodprimer$rrr:tested)r>r@r?rArr_hsd   &&  ..     1 z)ratsimpmodprime.._ratsimpmodprime)r!r')domainz*Looking for best minimal solution. Got: %sTFrCcSs t|dt|dS)Nrr')r2terms)xrrrs z!ratsimpmodprime..)key)convert)rr)sympy.solvers.solversrrrrr rrahas_assoc_Field get_fieldrsetr r-r!r r2r4rKminis_Field clear_denomsrqp)rr^rrr-argsnumdenomrBrarUrVrRnewsolr[r\r6rZr]cndnrr)r^r_r9r$rrr:r`rrsL  &  ^ "  rN) itertoolsr sympy.corerrrsympy.core.numbersr sympy.polysrrr r r sympy.polys.monomialsr r sympy.polys.polyerrorsrrsympy.utilities.miscrrrrrrrs