EdddlmZddlmZmZmZddlmZddlm Z m Z m Z m Z m Z ddlmZmZddlmZmZddlmZdZd d d d Zd S))combinations_with_replacement)symbolsAddDummy)Rational)cancelComputationFailedparallel_poly_from_exprreducedPoly)Monomial monomial_div) DomainErrorPolificationFailed)debugct|\}} t||gdd\}}n#t$r||z cYSwxYwt |t||z zS)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 r r)exprfgQrs 6lib/python3.11/site-packages/sympy/simplify/ratsimp.pyratsimpr s $<< & & ( (DAqq1#T%88811 s  7VAaC[[  s>AATF)quick polynomialcddlmtd|t|\}} t ||gzg|Ri|\}n#t $r|cYSwxYwj} | jr| _ntd| zfd|ddDtfddfd t|j j d }t|j j d }|r||z St|j j t|j j g\} } } s| rtd t!| zg} | D]S\}}}}||dd}| ||||fTt'| d\} } | jsC| d\}} | d\}} t-||}nt-d }| |jz| |jzz S)a Simplifies a rational expression ``expr`` modulo the prime ideal generated by ``G``. ``G`` should be a Groebner basis of the ideal. Examples ======== >>> from sympy.simplify.ratsimp import ratsimpmodprime >>> from sympy.abc import x, y >>> eq = (x + y**5 + y)/(x - y) >>> ratsimpmodprime(eq, [x*y**5 - x - y], x, y, order='lex') (-x**2 - x*y - x - y)/(-x**2 + x*y) If ``polynomial`` is ``False``, the algorithm computes a rational simplification which minimizes the sum of the total degrees of the numerator and the denominator. If ``polynomial`` is ``True``, this function just brings numerator and denominator into a canonical form. This is much faster, but has potentially worse results. References ========== .. [1] M. Monagan, R. Pearce, Rational Simplification Modulo a Polynomial Ideal, http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.163.6984 (specifically, the second algorithm) r)solveratsimpmodprimez.Cannot compute rational simplification over %scDg|]}|jS)LMorder).0ropts r z#ratsimpmodprime..Ss%<<z5ratsimpmodprime..staircase..bsB&&C<3''4/&&&&&&r)cHg|]}t|jjSr#)r as_exprgens)r&sr's rr(z6ratsimpmodprime..staircase..fs,:::1# #SX.:::r))rrangelenr3allappend)nSmiir/leading_monomialsr' staircases @rr>z"ratsimpmodprime..staircaseVs 6 3J /c#(mm0D0DaHH  BCMM!A  ! &&&&$&&&&&  :::::::YYq1u=M=MMMr)c||}}d}||z}r|dz } n|} ||z| krD||fvrn<||f||t|d|ddtdt zt tdt zt z} t tfd tt Dj | z} t tfd tt Dj | z} t|| z|| zz j | zj d d} t | j  }|zd d }|rtd|DsZ| |}| |}|t!t#t%zdgt t zz}|t!t#t%zdgt t zz}t |j }t |j }|dkrt'd|| | |zf||z|kr |dg}n|dz }|dz }|dz }||z| kD|dkr,||||||z \}}}|||||z |\}}}|||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 / z: z, zc:%d)clszd:%dc2g|]}||zSr#r#)r&r<CsM1s rr(z=ratsimpmodprime.._ratsimpmodprime..%;;;qRURU];;;r)c2g|]}||zSr#r#)r&r<DsM2s rr(z=ratsimpmodprime.._ratsimpmodprime..rDr)T)r%polys)r3 particularrc3"K|] }|dkV dS)rNr#)r&r4s rr0z._ratsimpmodprime..s&<testeds @@@@rrfz)ratsimpmodprime.._ratsimpmodprimehs:!1!!ANN$4$44  QJEEE!eun/ 1v  JJ1v   1B1B qqq!!!RRR4 5 5 5#b'')u555B#b'')u555BbB;;;;;E#b''NN;;;<z!ratsimpmodprime..s3QqTZZ\\):):S1=N=N)Nr))key)convert)rr)sympy.solvers.solversr rrrr rrhhas_assoc_Field get_fieldrsetr r3r%r r6r8rRminis_Field clear_denomsrqp)rrerrr3argsnumdenomrHrhr\r]rYnewsolrbrcr:rardcndnrrfr=r'r r>rgs `` @@@@@@rr!r!s@<,+++++ T""",,..JC,c5\A-=MMMMMM ss  ZF G%%''   > E5!R%2$e<<rs333333**********''''''YYYYYYYYYYYYYY88888888BBBBBBBB&&&&&&!!!,+/5~~~~~~~r)