U Zʹj@sddlmZddlmZmZmZddlmZddlm Z m Z m Z m Z m Z ddlmZmZddlmZmZddlmZmZdd Zd d d d dZdS))combinations_with_replacement)symbolsAddDummy)Rational)cancelComputationFailedparallel_poly_from_exprreducedPoly)Monomial monomial_div) DomainErrorPolificationFailed)debugdebugfcCs\t|\}}zt||gddd\}}Wntk rF||YSXt|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)exprfgQrrU/var/www/html/bwave/python/venv/lib/python3.8/site-packages/sympy/simplify/ratsimp.pyratsimp s rTF)quick polynomialcsddlmtd|t|\}}z t||gf||\}Wntk r^|YSXj} | jrx| _n t 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]8\}}}}||ddd}| ||||fq`t| 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, https://dl.acm.org/doi/pdf/10.1145/1145768.1145809 (specifically, the second algorithm) r)solveratsimpmodprimez.Cannot compute rational simplification over %scsg|]}|jqSr)ZLMorder).0roptrr Ssz#ratsimpmodprime..Ncs|dkrdgSg}tttj|D]N}dgtj|D]}|d7<q>tfddDr&|q&fdd|D|dS)z Compute all monomials with degree less than ``n`` that are not divisible by any element of ``leading_monomials``. rc3s|]}t|dkVqdSN)r )r"Zlmgmrr bsz5ratsimpmodprime..staircase..csg|]}t|jjqSr)r Zas_exprgensr"sr#rrr%fsz6ratsimpmodprime..staircase..)rrangelenr,allappend)nSmii)leading_monomialsr$ staircaser)rr8Vs  z"ratsimpmodprime..staircasec s||}}d}||}r,|d} n|} ||| krZ||f krNqZ ||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|||krZ|dg}qZ|d7}|d7}|d7}q0|dkr||||||\}}}||||||\}}}|||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:%dc3s|]}||VqdSr(rr"r6)CsM1rrr+sz._ratsimpmodprime..c3s|]}||VqdSr(rr:)DsM2rrr+sT)r!polys)r,Z particularrcss|]}|dkVqdS)rNrr-rrrr+szIdeal not prime?)Z total_degreeaddrrr0rr sumr/r,r r!Zcoeffsr1valuessubsdictlistzip ValueErrorr2)aballsolNDcdZstepsZmaxdegboundngc_hatd_hatrr4sol)G_ratsimpmodprimer$rrr8tested)r;r=r<r>rrWhsh     ..      z)ratsimpmodprime.._ratsimpmodprime)r!r')domainz*Looking for best minimal solution. Got: %sTFr@cSs t|dt|dS)Nrr')r0Zterms)xrrrz!ratsimpmodprime..)key)convert)rr)Zsympy.solvers.solversrrrrr rrYZhas_assoc_Field get_fieldrsetr r,r!r rr0r2rEminZis_FieldZ clear_denomsrqp)rrVrrr,argsnumdenomr?rYrOrPrLZnewsolrSrTr4rRrUZcndnrr)rVrWr7r$rrr8rXrr sN     ^    r N) itertoolsrZ sympy.corerrrZsympy.core.numbersrZ sympy.polysrrr r r Zsympy.polys.monomialsr r Zsympy.polys.polyerrorsrrZsympy.utilities.miscrrrr rrrrs