\c@ sdZddlmZmZddlmZddlmZddlm Z m Z m Z ddl m Z de fdYZdd lmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZm Z m!Z!m"Z"m#Z#m$Z$m%Z%m&Z&m'Z'm(Z(m)Z)m*Z*m+Z+m,Z,m-Z-m.Z.m/Z/m0Z0dd l1m2Z2m3Z3m4Z4m5Z5m6Z6m7Z7m8Z8m9Z9m:Z:m;Z;m<Z<m=Z=m>Z>m?Z?m@Z@mAZAmBZBmCZCmDZDmEZEmFZFmGZGmHZHmIZImJZJmKZKmLZLmMZMmNZNmOZOdd lPmQZQmRZRmSZSmTZTmUZUmVZVmWZWmXZXmYZYmZZZm[Z[m\Z\m]Z]m^Z^dd l_m`Z`maZambZbmcZcmdZdmeZemfZfmgZgmhZhmiZidd ljmkZkmlZlmmZmmnZnmoZompZpmqZqddlrmsZsmtZtmuZumvZvddlwmxZxmyZymzZzm{Z{m|Z|m}Z}m~Z~mZddlmZmZdZde efdYZdZde efdYZdZde efdYZdS(s1OO layer for several polynomial representations. i(tprint_functiontdivision(too(t CantSympify(tCoercionFailedt NotReversiblet NotInvertible(tPicklableWithSlotst GenericPolycB s8eZdZdZdZdZedZRS(s5Base class for low-level polynomial representations. cC s|j|jjS(sMake the ground domain a ring. (t set_domaintdomtget_ring(tf((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytground_to_ringscC s|j|jjS(s Make the ground domain a field. (R R t get_field(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytground_to_fieldscC s|j|jjS(sMake the ground domain exact. (R R t get_exact(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytground_to_exactscC s^|r|\}}n|}g|D]\}}|||f^q"}|rV||fS|SdS(N((tpertresulttincludetcoefftfactorstgtk((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt_perify_factorss+ (t__name__t __module__t__doc__R RRt classmethodR(((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyR s    (!t dmp_validatet dup_normalt dmp_normalt dup_convertt dmp_converttdmp_from_sympyt dup_stript dup_degreet dmp_degree_intdmp_degree_listtdmp_negative_ptdup_LCt dmp_ground_LCtdup_TCt dmp_ground_TCtdmp_ground_nthtdmp_onet dmp_groundt dmp_zero_pt dmp_one_pt dmp_ground_pt dup_from_dictt dmp_from_dictt dmp_to_dictt dmp_deflatet dmp_injectt dmp_ejectt dmp_terms_gcdtdmp_list_termst dmp_excludet dmp_slice_int dmp_permutet dmp_to_tuple(tdmp_add_groundtdmp_sub_groundtdmp_mul_groundtdmp_quo_groundtdmp_exquo_groundtdmp_abstdup_negtdmp_negtdup_addtdmp_addtdup_subtdmp_subtdup_multdmp_multdmp_sqrtdup_powtdmp_powtdmp_pdivtdmp_premtdmp_pquot dmp_pexquotdmp_divtdup_remtdmp_remtdmp_quot dmp_exquot dmp_add_mult dmp_sub_mult dmp_max_normt dmp_l1_norm(tdmp_clear_denomstdmp_integrate_int dmp_diff_int dmp_eval_int dup_reverttdmp_ground_trunctdmp_ground_contenttdmp_ground_primitivetdmp_ground_monict dmp_composet dup_decomposet dup_shiftt dup_transformtdmp_lift( tdup_half_gcdext dup_gcdext dup_inverttdmp_subresultantst dmp_resultanttdmp_discriminantt dmp_inner_gcdtdmp_gcdtdmp_lcmt dmp_cancel(t dup_gff_listtdmp_normt dmp_sqf_pt dmp_sqf_normt dmp_sqf_partt dmp_sqf_listtdmp_sqf_list_include(tdup_cyclotomic_ptdmp_irreducible_ptdmp_factor_listtdmp_factor_list_include(tdup_isolate_real_roots_sqftdup_isolate_real_rootstdup_isolate_all_roots_sqftdup_isolate_all_rootstdup_refine_real_roottdup_count_real_rootstdup_count_complex_rootst dup_sturm(tUnificationFailedtPolynomialErrorcC stt|||||S(N(tDMPR (treptlevR ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytinit_normal_DMPsRcB seZdZddddgZdddZdZdZdZde dd Z e dd Z e dd Z e d Ze d Ze dZe dZdZdZdZe dZe ddZdZdZdZdZddZddZddZddZdZdZ d Z!d!Z"d"Z#e d#Z$e d$Z%d%Z&d&Z'd'Z(d(Z)d)Z*d*Z+d+Z,d,Z-d-Z.d.Z/d/Z0d0Z1d1Z2d2Z3d3Z4d4Z5d5Z6d6Z7d7Z8d8Z9d9Z:d:Z;d;Z<dd<Z=d=Z>d>Z?d?Z@d@ZAdAZBdBZCdCZDdDZEdEZFdFZGdGddHZHdGddIZIddJZJdKZKdLZLdMZMdNZNdOZOe dPZPdQZQdRZRdSZSdTZTeUdUZVdVZWdWZXdXZYdYZZdZZ[d[Z\d\Z]d]Z^d^Z_d_Z`d`ZadaZbdbZce dcZde ddZedeZfdfZge ddde e dgZhdde dhZidddiZjdddjZkeldkZmeldlZneldmZoeldnZpeldoZqeldpZreldqZseldrZteldsZueldtZvelduZweldvZxdwZydxZzdyZ{dzZ|d{Z}d|Z~d}Zd~ZdZeZeZdZdZdZdZdZdZdZe dZe dZdZdZdZdZdZdZeZRS(s)Dense Multivariate Polynomials over `K`. RRR tringcC s|dk rct|tkr3t|||}qut|tk rut|j||}qunt|\}}||_||_ ||_ ||_ dS(N( tNonettypetdictR4tlistR/tconvertRRRR R(tselfRR RR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt__init__s    cC s#d|jj|j|j|jfS(Ns%s(%s, %s, %s)(t __class__RRR R(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt__repr__scC s.t|jj|j|j|j|jfS(N(thashRRtto_tupleRR R(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt__hash__sc sJt|t s"|j|jkr;td||fn|j|jkr|j|jkr|j|j|j|j|jfS|j|jj|j}}|j|jdk rdk rj|jq|jnt |j||j|}t |j||j|}||t fd}|||||fSdS(s7Unify representations of two multivariate polynomials. scan't unify %s with %sc s0|r|s|S|d8}nt|||S(Ni(R(RR Rtkill(R(s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRs  N( t isinstanceRRRR RRRtunifyRR"tFalse(R RRR tFtGR((Rs6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRs"$"    cC si|j}|r&|s|S|d8}n|dkr>|j}n|dkrV|j}nt||||S(s.Create a DMP out of the given representation. iN(RRR RR(R RR RRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRs      cC std|||S(Ni(R(tclsRR R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytzeroscC std|||S(Ni(R(RRR R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytonescC s|t||d|||S(sCCreate an instance of ``cls`` given a list of native coefficients. N(R"R(RRRR ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt from_listscC s|t|||||S(sBCreate an instance of ``cls`` given a list of SymPy coefficients. (R#(RRRR ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytfrom_sympy_listscC st|j|j|jd|S(sAConvert ``f`` to a dict representation with native coefficients. R(R5RRR (R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytto_dictscC sXt|j|j|jd|}x0|jD]"\}}|jj|||RR(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyR scC s|t|||||S(sBConstruct and instance of ``cls`` from a ``dict`` representation. (R4(RRRR ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt from_dictscC s(tttt|||||S(N(RRRtzip(RtmonomstcoeffsRR R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytfrom_monoms_coeffsscC s|j|jjS(sMake the ground domain a ring. (RR R (R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytto_ringscC s|j|jjS(s Make the ground domain a field. (RR R(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytto_fieldscC s|j|jjS(sMake the ground domain exact. (RR R(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytto_exact#scC sB|j|kr|Stt|j|j|j|||jSdS(s$Convert the ground domain of ``f``. N(R RR"RR(R R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyR'sicC s+|jt|j||||j|jS(s1Take a continuous subsequence of terms of ``f``. (RR<RRR (R tmtntj((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytslice.scC s8gt|j|j|jd|D]\}}|^q"S(s;Returns all non-zero coefficients from ``f`` in lex order. torder(R:RRR (R Rt_tc((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyR2scC s8gt|j|j|jd|D]\}}|^q"S(s8Returns all non-zero monomials from ``f`` in lex order. R(R:RRR (R RRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyR6scC st|j|j|jd|S(s4Returns all non-zero terms from ``f`` in lex order. R(R:RRR (R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytterms:scC sI|js9|s|jjgSg|jD] }|^q&Sn tddS(s%Returns all coefficients from ``f``. s&multivariate polynomials not supportedN(RR RRR(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt all_coeffs>s   cC sk|js[t|j}|dkr+dgSgt|jD]\}}||f^q;Sn tddS(s"Returns all monomials from ``f``. is&multivariate polynomials not supportedN(i(RR%Rt enumerateR(R RtiR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt all_monomsHs   0cC s}|jsmt|j}|dkr7d|jjfgSgt|jD]\}}||f|f^qGSn tddS(s Returns all terms from a ``f``. is&multivariate polynomials not supportedN(i(RR%RR RRR(R RRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt all_termsTs   6cC s.|jt|j|j|jd|jjS(s-Convert algebraic coefficients to rationals. R (RRjRRR (R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytlift`scC s4t|j|j|j\}}||j|fS(s2Reduce degree of `f` by mapping `x_i^m` to `y_i`. (R6RRR R(R tJR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytdeflateds!cC s@t|j|j|jd|\}}|j||jj|S(s,Inject ground domain generators into ``f``. tfront(R7RRR R(R RRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytinjectis'cC sAt|j|j|d|}|j|||jt|jS(s2Eject selected generators into the ground domain. R(R8RRRtlentsymbols(R R RR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytejectnscC s@t|j|j|j\}}}||j||j|fS(so Remove useless generators from ``f``. Returns the removed generators and the new excluded ``f``. Examples ======== >>> from sympy.polys.polyclasses import DMP >>> from sympy.polys.domains import ZZ >>> DMP([[[ZZ(1)]], [[ZZ(1)], [ZZ(2)]]], ZZ).exclude() ([2], DMP([[1], [1, 2]], ZZ, None)) (R;RRR R(R RRtu((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytexcludess$cC s%|jt|j||j|jS(s Returns a polynomial in `K[x_{P(1)}, ..., x_{P(n)}]`. Examples ======== >>> from sympy.polys.polyclasses import DMP >>> from sympy.polys.domains import ZZ >>> DMP([[[ZZ(2)], [ZZ(1), ZZ(0)]], [[]]], ZZ).permute([1, 0, 2]) DMP([[[2], []], [[1, 0], []]], ZZ, None) >>> DMP([[[ZZ(2)], [ZZ(1), ZZ(0)]], [[]]], ZZ).permute([1, 2, 0]) DMP([[[1], []], [[2, 0], []]], ZZ, None) (RR=RRR (R tP((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytpermutescC s4t|j|j|j\}}||j|fS(s/Remove GCD of terms from the polynomial ``f``. (R9RRR R(R RR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt terms_gcds!cC s1|jt|j|jj||j|jS(s.Add an element of the ground domain to ``f``. (RR?RR RR(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt add_groundscC s1|jt|j|jj||j|jS(s5Subtract an element of the ground domain from ``f``. (RR@RR RR(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt sub_groundscC s1|jt|j|jj||j|jS(s5Multiply ``f`` by a an element of the ground domain. (RRARR RR(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt mul_groundscC s1|jt|j|jj||j|jS(s8Quotient of ``f`` by a an element of the ground domain. (RRBRR RR(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt quo_groundscC s1|jt|j|jj||j|jS(s>Exact quotient of ``f`` by a an element of the ground domain. (RRCRR RR(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt exquo_groundscC s"|jt|j|j|jS(s)Make all coefficients in ``f`` positive. (RRDRRR (R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytabsscC s"|jt|j|j|jS(s"Negate all coefficients in ``f``. (RRFRRR (R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytnegscC s7|j|\}}}}}|t||||S(s2Add two multivariate polynomials ``f`` and ``g``. (RRH(R RRR RRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytaddscC s7|j|\}}}}}|t||||S(s7Subtract two multivariate polynomials ``f`` and ``g``. (RRJ(R RRR RRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytsubscC s7|j|\}}}}}|t||||S(s7Multiply two multivariate polynomials ``f`` and ``g``. (RRL(R RRR RRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytmulscC s"|jt|j|j|jS(s(Square a multivariate polynomial ``f``. (RRMRRR (R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytsqrscC sNt|tr4|jt|j||j|jStdt|dS(s+Raise ``f`` to a non-negative power ``n``. s``int`` expected, got %sN( RtintRRORRR t TypeErrorR(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytpows%c C sO|j|\}}}}}t||||\}}||||fS(s/Polynomial pseudo-division of ``f`` and ``g``. (RRP( R RRR RRRtqtr((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytpdivscC s7|j|\}}}}}|t||||S(s0Polynomial pseudo-remainder of ``f`` and ``g``. (RRQ(R RRR RRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytpremscC s7|j|\}}}}}|t||||S(s/Polynomial pseudo-quotient of ``f`` and ``g``. (RRR(R RRR RRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytpquoscC s7|j|\}}}}}|t||||S(s5Polynomial exact pseudo-quotient of ``f`` and ``g``. (RRS(R RRR RRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytpexquosc C sO|j|\}}}}}t||||\}}||||fS(s7Polynomial division with remainder of ``f`` and ``g``. (RRT( R RRR RRRRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytdivscC s7|j|\}}}}}|t||||S(s2Computes polynomial remainder of ``f`` and ``g``. (RRV(R RRR RRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytremscC s7|j|\}}}}}|t||||S(s1Computes polynomial quotient of ``f`` and ``g``. (RRW(R RRR RRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytquosc C s}|j|\}}}}}|t||||}|jry||jkryddlm}||||jn|S(s7Computes polynomial exact quotient of ``f`` and ``g``. i(tExactQuotientFailed(RRXRtsympy.polys.polyerrorsR( R RRR RRRtresR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytexquos cC s?t|tr%t|j||jStdt|dS(s0Returns the leading degree of ``f`` in ``x_j``. s``int`` expected, got %sN(RRR&RRRR(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytdegreescC st|j|jS(s$Returns a list of degrees of ``f``. (R'RR(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt degree_list scC std|jDS(s#Returns the total degree of ``f``. cs s|]}t|VqdS(N(tsum(t.0R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pys s(tmaxR(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt total_degree sc C s|j}i}|t|jddk}x|jD]}t|d}||krn||}nd}|r|d||d|f>ssa sequence of integers expectedN(tallR-RRR R(R tN((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytnth<scC st|j|j|jS(sReturns maximum norm of ``f``. (R[RRR (R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytmax_normCscC st|j|j|jS(sReturns l1 norm of ``f``. (R\RRR (R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytl1_normGscC s4t|j|j|j\}}||j|fS(s0Clear denominators, but keep the ground domain. (R]RRR R(R RR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt clear_denomsKs!icC sxt|ts(tdt|nt|tsPtdt|n|jt|j|||j|jS(sEComputes the ``m``-th order indefinite integral of ``f`` in ``x_j``. s``int`` expected, got %s( RRRRRR^RRR (R RR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt integratePs cC sxt|ts(tdt|nt|tsPtdt|n|jt|j|||j|jS(s<Computes the ``m``-th order derivative of ``f`` in ``x_j``. s``int`` expected, got %s( RRRRRR_RRR (R RR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytdiffZs cC sbt|ts(tdt|n|jt|j|jj|||j |jdt S(s5Evaluates ``f`` at the given point ``a`` in ``x_j``. s``int`` expected, got %sR( RRRRRR`RR RRtTrue(R taR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytevaldsc C sb|j|\}}}}}|sRt|||\}}||||fStddS(s2Half extended Euclidean algorithm, if univariate. sunivariate polynomial expectedN(RRkt ValueError( R RRR RRRRth((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt half_gcdexls c C sn|j|\}}}}}|s^t|||\}}} |||||| fStddS(s-Extended Euclidean algorithm, if univariate. sunivariate polynomial expectedN(RRlR( R RRR RRRRttR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytgcdexvs cC sJ|j|\}}}}}|s:|t|||StddS(s(Invert ``f`` modulo ``g``, if possible. sunivariate polynomial expectedN(RRmR(R RRR RRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytinvertscC s8|js(|jt|j||jStddS(s"Compute ``f**(-1)`` mod ``x**n``. sunivariate polynomial expectedN(RRRaRR R(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytreverts cC sF|j|\}}}}}t||||}tt||S(s7Computes subresultant PRS sequence of ``f`` and ``g``. (RRnRtmap(R RRR RRRtR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt subresultantssc C s|j|\}}}}}|rjt||||d|\}} ||dttt|| fS|t||||dtS(s/Computes resultant of ``f`` and ``g`` via PRS. t includePRSR(RRoR RR( R RRRR RRRRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt resultants !%cC s(|jt|j|j|jdtS(s Computes discriminant of ``f``. R(RRpRRR R (R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt discriminantsc C s[|j|\}}}}}t||||\}}} |||||| fS(s4Returns GCD of ``f`` and ``g`` and their cofactors. (RRq( R RRR RRRRtcfftcfg((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt cofactorsscC s7|j|\}}}}}|t||||S(s+Returns polynomial GCD of ``f`` and ``g``. (RRr(R RRR RRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytgcdscC s7|j|\}}}}}|t||||S(s+Returns polynomial LCM of ``f`` and ``g``. (RRs(R RRR RRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytlcmsc C s|j|\}}}}}|rHt||||dt\}}n't||||dt\}} }}||||}}|r||fS|| ||fSdS(s6Cancel common factors in a rational function ``f/g``. RN(RRtR R( R RRRR RRRtcFtcG((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytcancels$' cC s1|jt|j|jj||j|jS(s&Reduce ``f`` modulo a constant ``p``. (RRbRR RR(R tp((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyttruncscC s"|jt|j|j|jS(s'Divides all coefficients by ``LC(f)``. (RReRRR (R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytmonicscC st|j|j|jS(s(Returns GCD of polynomial coefficients. (RcRRR (R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytcontentscC s4t|j|j|j\}}||j|fS(s/Returns content and a primitive form of ``f``. (RdRRR R(R tcontR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt primitives!cC s7|j|\}}}}}|t||||S(s4Computes functional composition of ``f`` and ``g``. (RRf(R RRR RRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytcomposescC s>|js.tt|jt|j|jStddS(s,Computes functional decomposition of ``f``. sunivariate polynomial expectedN(RRRRRgRR R(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt decomposes %cC sD|js4|jt|j|jj||jStddS(s/Efficiently compute Taylor shift ``f(x + a)``. sunivariate polynomial expectedN(RRRhRR RR(R R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytshifts +c C s|jrtdn|j|\}}}}}|j||||\}}}}}||||j||||\}}}}}|s|t||||StddS(s5Evaluate functional transformation ``q**n * f(p/q)``.sunivariate polynomial expectedN(RRRRi( R R#RRR RRtQR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt transforms *6cC s>|js.tt|jt|j|jStddS(s&Computes the Sturm sequence of ``f``. sunivariate polynomial expectedN(RRRRRRR R(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytsturms %cC sT|jsDgt|j|jD]!\}}|j||f^qStddS(s4Computes greatest factorial factorization of ``f``. sunivariate polynomial expectedN(RRuRR RR(R RR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytgff_lists ;cC s4t|j|j|j}|j|d|jjS(sComputes ``Norm(f)``.R (RvRRR R(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytnormscC sOt|j|j|j\}}}||j||j|d|jjfS(s$Computes square-free norm of ``f``. R (RxRRR R(R RRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytsqf_norms$cC s"|jt|j|j|jS(s$Computes square-free part of ``f``. (RRyRRR (R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytsqf_part scC sVt|j|j|j|\}}|g|D]!\}}|j||f^q.fS(s0Returns a list of square-free factors of ``f``. (RzRRR R(R RRRRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytsqf_lists$cC sJt|j|j|j|}g|D]!\}}|j||f^q%S(s0Returns a list of square-free factors of ``f``. (R{RRR R(R RRRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytsqf_list_includescC sSt|j|j|j\}}|g|D]!\}}|j||f^q+fS(s0Returns a list of irreducible factors of ``f``. (R~RRR R(R RRRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt factor_lists!cC sGt|j|j|j}g|D]!\}}|j||f^q"S(s0Returns a list of irreducible factors of ``f``. (RRRR R(R RRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytfactor_list_includesc C s|js|sn|s@t|j|jd|d|d|d|St|j|jd|d|d|d|Sq|st|j|jd|d|d|d|St|j|jd|d|d|d|Sn tddS(s0Compute isolating intervals for roots of ``f``. tepstinftsuptfasts0can't isolate roots of a multivariate polynomialN(RRRR RRRR(R RR7R8R9R:tsqf((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt intervals$s +.+.c C sD|js4t|j|||jd|d|d|StddS(su Refine an isolating interval to the given precision. ``eps`` should be a rational number. R7tstepsR:s0can't refine a root of a multivariate polynomialN(RRRR R(R RRR7R=R:((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt refine_root5s +cC st|j|jd|d|S(s<Return the number of real roots of ``f`` in ``[inf, sup]``. R8R9(RRR (R R8R9((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytcount_real_rootsBscC st|j|jd|d|S(s?Return the number of complex roots of ``f`` in ``[inf, sup]``. R8R9(RRR (R R8R9((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytcount_complex_rootsFscC st|j|jS(s0Returns ``True`` if ``f`` is a zero polynomial. (R0RR(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRJscC st|j|j|jS(s0Returns ``True`` if ``f`` is a unit polynomial. (R1RRR (R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytis_oneOscC st|jd|jS(s>Returns ``True`` if ``f`` is an element of the ground domain. N(R2RRR(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt is_groundTscC st|j|j|jS(s7Returns ``True`` if ``f`` is a square-free polynomial. (RwRRR (R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytis_sqfYscC s%|jjt|j|j|jS(s=Returns ``True`` if the leading coefficient of ``f`` is one. (R RAR*RR(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytis_monic^scC s%|jjt|j|j|jS(sAReturns ``True`` if the GCD of the coefficients of ``f`` is one. (R RARcRR(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt is_primitivecscC s/tdt|j|j|jjDS(s:Returns ``True`` if ``f`` is linear in all its variables. cs s!|]}t|dkVqdS(iN(R(RR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pys ks(RR5RRR tkeys(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt is_linearhscC s/tdt|j|j|jjDS(s=Returns ``True`` if ``f`` is quadratic in all its variables. cs s!|]}t|dkVqdS(iN(R(RR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pys ps(RR5RRR RF(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt is_quadraticmscC st|jdkS(s8Returns ``True`` if ``f`` is zero or has only one term. i(RR(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt is_monomialrscC s|jdk S(s7Returns ``True`` if ``f`` is a homogeneous polynomial. N(RR(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytis_homogeneouswscC st|j|j|jS(s:Returns ``True`` if ``f`` has no factors over its domain. (R}RRR (R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytis_irreducible|scC s$|jst|j|jStSdS(s6Returns ``True`` if ``f`` is a cyclotomic polynomial. N(RR|RR R(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt is_cyclotomics cC s |jS(N(R(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt__abs__scC s |jS(N(R(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt__neg__scC st|tsy+|jt|jj||j}Wqtk rNtSt t fk r|j dk ry|j j|}Wqt t fk rtSXqqXn|j |S(N(RRRR/R RRRtNotImplementedRtNotImplementedErrorRRR(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt__add__s+ cC s |j|S(N(RQ(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt__radd__scC st|tsy+|jt|jj||j}Wqtk rNtSt t fk r|j dk ry|j j|}Wqt t fk rtSXqqXn|j |S(N(RRRR/R RRRRORRPRRR(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt__sub__s+ cC s| j|S(N(RQ(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt__rsub__scC st|tr|j|Sy|j|SWnrtk rAtSttfk r|jdk ry|j|jj |SWqttfk rqXntSXdS(N( RRRRRRORRPRRR(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt__mul__s  cC st|tr|j|Sy|j|SWnrtk rAtSttfk r|jdk ry|j|jj |SWqttfk rqXntSXdS(N( RRRRRRORRPRRR(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt__div__s  cC sit|tr|j|S|jdk rey|jj|j|SWqettfk raqeXntS(N( RRRRRRRRPRO(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt__rdiv__s cC s |j|S(N(RU(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt__rmul__scC s |j|S(N(R(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt__pow__scC s |j|S(N(R(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt __divmod__scC s |j|S(N(R(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt__mod__scC sFt|tr|j|Sy|j|SWntk rAtSXdS(N(RRRRRRO(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt __floordiv__s   cC sVy>|j|\}}}}}|j|jkr=||kSWntk rQnXtS(N(RRRR(R RRRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt__eq__s cC s ||k S(N((R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt__ne__scC s!|s||kS|j|SdS(N(t _strict_eq(R Rtstrict((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyteqs cC s|j|d| S(NR`(Ra(R RR`((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytne scC sFt||joE|j|jkoE|j|jkoE|j|jkS(N(RRRR R(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyR_ s$cC s(|j|\}}}}}||kS(N(R(R RRRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt__lt__scC s(|j|\}}}}}||kS(N(R(R RRRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt__le__scC s(|j|\}}}}}||kS(N(R(R RRRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt__gt__scC s(|j|\}}}}}||kS(N(R(R RRRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt__ge__scC st|j|j S(N(R0RR(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt __nonzero__!sN(RRRt __slots__RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR R R RRRRRRRRRRR R"R$R%R&R(R)R*R+R-R.R/R0R1R2R3R4R5R6R<R>R?R@tpropertyRRARBRCRDRERGRHRIRJRKRLRMRNRQRRRSRTRURVRWt __truediv__t __rtruediv__RXRYRZR[R\R]R^RaRbR_RcRdReRfRgt__bool__(((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRs                                                                                                      cC s+tt|||t|||||S(N(tDMFR (tnumtdenRR ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytinit_normal_DMF'sRmcB seZdZdddddgZd.d.dZed.d.dZed.dZd Z d Z d Z d Z e ed.d ZedZed.dZed.dZdZdZdZdZdZdZdZdZdZeZe dZedZedZ dZ!dZ"dZ#d Z$d!Z%d"Z&d#Z'd$Z(d%Z)d&Z*e)Z+e*Z,d'Z-d(Z.d)Z/d*Z0d+Z1d,Z2d-Z3e3Z4RS(/s'Dense Multivariate Fractions over `K`. RnRoRR RcC sj|j|||\}}}t||||\}}||_||_||_||_||_dS(N(t_parseRtRnRoRR R(RRR RRRnRo((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyR1s    cC s^|j|||\}}}tj|}||_||_||_||_||_|S(N(Rqtobjectt__new__RnRoRR R(RRR RRRnRotobj((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytnew;s     cC st|tkr;|\}}|dk r{t|tkrQt|||}nt|tkrt|||}qnEt|\}}t|\}}||kr|}n tdt||rtdnt||rt ||}qt |||rt |||}t |||}qn|}|dk rt|tkrtt|||}qt|t k rt |j||}qnt|\}}t ||}|||fS(Nsinconsistent number of levelssfraction denominator(RRRRR4RRR0tZeroDivisionErrorR.R(RFRR/R(RRR RRnRotnum_levtden_lev((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRqIs8      cC s)d|jj|j|j|j|jfS(Ns%s((%s, %s), %s, %s)(RRRnRoR R(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRuscC sIt|jjt|j|jt|j|j|j|j|jfS(N( RRRR>RnRRoR R(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRysc stt|t s"j|jkr;td|fnj|jkrj|jkrjjjjjf|j fSjjj |j}j|jdk rdk rj |jq|jnt j|jt j|jf}t |j ||j}t t|fd}||||fSdS(s0Unify a multivariate fraction and a polynomial. scan't unify %s with %sc sg|r!|s||S|d}n|rEt|||\}}njj||f|dS(NiR(RtRRu(RnRoR"RR(R R R(s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRs N(RRRRR RRRnRoRRRR"R R(R RRRRR((R R Rs6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt poly_unify}s"$+    c st|t s"j|jkr;td|fnj|jkrj|jkrjjjjjf|j|jffSjjj |j}j|jdk rdk rj |jq|jnt j|jt j|jf}t |j||jt |j||jf}t t |fd}||||fSdS(s5Unify representations of two multivariate fractions. scan't unify %s with %sc sg|r!|s||S|d}n|rEt|||\}}njj||f|dS(NiR(RtRRu(RnRoR"RR(R R R(s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRs N(RRmRRR RRRnRoRRR"R R(R RRRRR((R R Rs6lib/python2.7/site-packages/sympy/polys/polyclasses.pyt frac_unifys""$!    cC s|j|j}}|r4|s'||S|d8}n|rXt||||\}}n|dkrp|j}n|jj||f||d|S(s.Create a DMF out of the given representation. iRN(RR RtRRRRu(R RnRoR"RRRR ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRs   cC s9|j}|r&|s|S|d8}nt||j|S(s.Create a DMP out of the given representation. i(RRR (R RRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pythalf_pers   cC s|jd||d|S(NiR(Ru(RRR R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRscC s|jd||d|S(NiR(Ru(RRR R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRscC s|j|jS(s Returns the numerator of ``f``. (R{Rn(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytnumerscC s|j|jS(s"Returns the denominator of ``f``. (R{Ro(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytdenomscC s|j|j|jS(s4Remove common factors from ``f.num`` and ``f.den``. (RRnRo(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyR"scC s.|jt|j|j|j|jdtS(s"Negate all coefficients in ``f``. R"(RRFRnRR RoR(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRsc C st|trU|j|\}}}\}}}t||||||}} n|j|\}}}} }| |\}}\} } tt|| ||t|| ||||}t|| ||} ||| S(s0Add two multivariate fractions ``f`` and ``g``. (RRRyRYRzRHRL( R RRR RtF_numtF_denRRnRoRtG_numtG_den((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRs$"c C st|trU|j|\}}}\}}}t||||||}} n|j|\}}}} }| |\}}\} } tt|| ||t|| ||||}t|| ||} ||| S(s5Subtract two multivariate fractions ``f`` and ``g``. (RRRyRZRzRJRL( R RRR RR~RRRnRoRRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyR s$"c C st|trR|j|\}}}\}}}t|||||}} na|j|\}}}} }| |\}}\} } t|| ||}t|| ||} ||| S(s5Multiply two multivariate fractions ``f`` and ``g``. (RRRyRLRz( R RRR RR~RRRnRoRRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRs$cC sot|trU|jt|j||j|jt|j||j|jdtSt dt |dS(s+Raise ``f`` to a non-negative power ``n``. R"s``int`` expected, got %sN( RRRRORnRR RoRRR(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyR's!%cC s t|trR|j|\}}}\}}}|t||||}} na|j|\}}}} }| |\}}\} } t|| ||}t|| ||} ||| } |jdk r| |jkrddlm}||||jn| S(s0Computes quotient of fractions ``f`` and ``g``. i(RN( RRRyRLRzRRRR(R RRR RR~RRRnRoRRRRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyR/s$cC s_|r=|jdk r=|jj| r=t||jn|j|j|jdt}|S(s&Computes inverse of a fraction ``f``. R"N(RRtis_unitRRRoRnR(R tcheckR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRCs(cC st|j|jS(s.Returns ``True`` if ``f`` is a zero fraction. (R0RnR(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRJscC s4t|j|j|jo3t|j|j|jS(s.Returns ``True`` if ``f`` is a unit fraction. (R1RnRR Ro(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRAOscC s |jS(N(R(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRNUscC st|ttfr"|j|Sy|j|j|SWnrtk rPtSttfk r|j dk ry|j|j j |SWqttfk rqXntSXdS(N( RRRmRR{RRORRPRRR(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRQXs  cC s |j|S(N(RQ(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRRhscC st|ttfr"|j|Sy|j|j|SWnrtk rPtSttfk r|j dk ry|j|j j |SWqttfk rqXntSXdS(N( RRRmRR{RRORRPRRR(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRSks  cC s| j|S(N(RQ(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRT{scC st|ttfr"|j|Sy|j|j|SWnrtk rPtSttfk r|j dk ry|j|j j |SWqttfk rqXntSXdS(N( RRRmRR{RRORRPRRR(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRU~s  cC s |j|S(N(RU(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRXscC s |j|S(N(R(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRYscC st|ttfr"|j|Sy|j|j|SWnrtk rPtSttfk r|j dk ry|j|j j |SWqttfk rqXntSXdS(N( RRRmRR{RRORRPRRR(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRVs  cC sZ|jdt|}|jrV||jkrVddlm}||||jn|S(NRi(R(RRRRR(RRRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRWs cC syt|trm|j|\}}}\}}}|j|jkrt||j|joi||kSn:|j|\}}}}}|j|jkr||kSWntk rnXtS(N( RRRyRR1R RzRR(R RRR~RRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyR]s$% cC syt|trn|j|\}}}\}}}|j|jkrt||j|joi||k Sn:|j|\}}}}}|j|jkr||kSWntk rnXtS(N( RRRyRR1R RzRR (R RRR~RRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyR^s$& cC s(|j|\}}}}}||kS(N(Rz(R RRRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRcscC s(|j|\}}}}}||kS(N(Rz(R RRRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRdscC s(|j|\}}}}}||kS(N(Rz(R RRRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRescC s(|j|\}}}}}||kS(N(Rz(R RRRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRfscC st|j|j S(N(R0RnR(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRgsN(5RRRRhRRRRuRqRRRyRzR RRR{RRR|R}R"RRRRRRRRRiRRARNRQRRRSRTRURXRYRVRWRjRkR]R^RcRdReRfRgRl(((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRm,sd  +   # %                            cC s"tt||t|||S(N(tANPR(RtmodR ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytinit_normal_ANPsRcB seZdZdddgZdZdZdZdZd0d0dZ e d Z e d Z d Z d Zd ZdZdZe dZdZdZdZdZdZdZdZdZeZdZdZedZedZ edZ!dZ"dZ#d Z$d!Z%d"Z&d#Z'd$Z(d%Z)d&Z*d'Z+d(Z,e,Z-d)Z.d*Z/d+Z0d,Z1d-Z2d.Z3d/Z4e4Z5RS(1s1Dense Algebraic Number Polynomials over a field. RRR cC st|tkr't|||_n6t|tk rN|j|g}nt||_t|tr{|j|_ n6t|tkrt|||_ nt||_ ||_ dS(N( RRR3RRRR$RRRR (RRRR ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRscC s#d|jj|j|j|jfS(Ns%s(%s, %s, %s)(RRRRR (R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRscC s1t|jj|jt|jd|jfS(Ni(RRRRR>RR (R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRsc s6t|t s"|j|jkr;td||fn|j|jkro|j|j|j|j|jfS|jj|jt|j|j}t|j|j}|jkr|jkrt|j|jn$|jkr|jn |jfd}|||fS(s0Unify representations of two algebraic numbers. scan't unify %s with %sc st|S(N(R(R(R R(s6lib/python2.7/site-packages/sympy/polys/polyclasses.pytt( RRRRR RRRR!(R RRRR((R Rs6lib/python2.7/site-packages/sympy/polys/polyclasses.pyR s""  cC s"t||p|j|p|jS(N(RRR (R RRR ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyR"scC std||S(Ni(R(RRR ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyR%scC std||S(Ni(R(RRR ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyR)scC st|jd|jS(sAConvert ``f`` to a dict representation with native coefficients. i(R5RR (R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyR-scC sOt|jd|j}x0|jD]"\}}|jj|||scC st|jdS(sx Convert ``f`` to a tuple representation with native coefficients. This is needed for hashing. i(R>R(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRBscC s(tttt|j|||S(N(RR$RRR(RRRR ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRJscC s|jt|j|jS(N(RRERR (R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRNscC s4|j|\}}}}}|t|||S(N(RRG(R RR RRRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRQscC s4|j|\}}}}}|t|||S(N(RRI(R RR RRRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRUscC s@|j|\}}}}}|tt|||||S(N(RRURK(R RR RRRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRYscC st|trx|dkrAt|j|j|j| }}n |j}|jtt|||j|j|jSt dt |dS(s+Raise ``f`` to a non-negative power ``n``. is``int`` expected, got %sN( RRRmRRR RRURNRR(R RR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyR]s  & .cC s^|j|\}}}}}|tt|t|||||||j||fS(N(RRURKRmR(R RR RRRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRiscC sh|j|\}}}}}t|||\}}||jgkrX|j||StddS(Ns zero divisor(RRkRRR(R RR RRRRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRms cC sL|j|\}}}}}|tt|t||||||S(N(RRURKRm(R RR RRRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRwscC st|j|jS(s*Returns the leading coefficient of ``f``. (R)RR (R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyR}scC st|j|jS(s+Returns the trailing coefficient of ``f``. (R+RR (R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRscC s| S(s6Returns ``True`` if ``f`` is a zero algebraic number. ((R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRscC s|j|jjgkS(s6Returns ``True`` if ``f`` is a unit algebraic number. (RR R(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRAscC s|j pt|jdkS(s>Returns ``True`` if ``f`` is an element of the ground domain. i(RR(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRBscC s |jS(N(R(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRNscC sUt|tr|j|Sy|j|j|SWnttfk rPtSXdS(N(RRRRRRRO(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRQs  cC s |j|S(N(RQ(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRRscC sUt|tr|j|Sy|j|j|SWnttfk rPtSXdS(N(RRRRRRRO(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRSs  cC s| j|S(N(RQ(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRTscC sUt|tr|j|Sy|j|j|SWnttfk rPtSXdS(N(RRRRRRRO(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRUs  cC s |j|S(N(RU(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRXscC s |j|S(N(R(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRYscC s |j|S(N(R(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRZscC s |j|S(N(R(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyR[scC sUt|tr|j|Sy|j|j|SWnttfk rPtSXdS(N(RRRRRRRO(R R((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRVs  cC sEy,|j|\}}}}}||kSWntk r@tSXdS(N(RRR(R RRRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyR]s  cC sEy,|j|\}}}}}||kSWntk r@tSXdS(N(RRR (R RRRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyR^s  cC s(|j|\}}}}}||kS(N(R(R RRRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRcscC s(|j|\}}}}}||kS(N(R(R RRRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRdscC s(|j|\}}}}}||kS(N(R(R RRRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRescC s(|j|\}}}}}||kS(N(R(R RRRR((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRfscC s t|jS(N(tboolR(R ((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRgsN(6RRRRhRRRRRRRRRRRRRRRRRRRRRRRRRRRiRRARBRNRQRRRSRTRURXRYRZR[RVRjR]R^RcRdReRfRgRl(((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyRsb                               N(Rt __future__RRtsympyRtsympy.core.sympifyRRRRRtsympy.polys.polyutilsRRtsympy.polys.densebasicRRR R!R"R#R$R%R&R'R(R)R*R+R,R-R.R/R0R1R2R3R4R5R6R7R8R9R:R;R<R=R>tsympy.polys.densearithR?R@RARBRCRDRERFRGRHRIRJRKRLRMRNRORPRQRRRSRTRURVRWRXRYRZR[R\tsympy.polys.densetoolsR]R^R_R`RaRbRcRdReRfRgRhRiRjtsympy.polys.euclidtoolsRkRlRmRnRoRpRqRrRsRttsympy.polys.sqfreetoolsRuRvRwRxRyRzR{tsympy.polys.factortoolsR|R}R~Rtsympy.polys.rootisolationRRRRRRRRRRRRRpRmRR(((s6lib/python2.7/site-packages/sympy/polys/polyclasses.pyts0^F 4":