o 8Va4@sdZddlmZddlmZddlmZmZmZddl m Z Gddde Z ddl m Z mZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZm 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-dd l.m/Z/m0Z0m1Z1m2Z2m3Z3m4Z4m5Z5m6Z6m7Z7m8Z8m9Z9m:Z:m;Z;mZ>m?Z?m@Z@mAZAmBZBmCZCmDZDmEZEmFZFmGZGmHZHmIZImJZJmKZKmLZLdd lMmNZNmOZOmPZPmQZQmRZRmSZSmTZTmUZUmVZVmWZWmXZXmYZYmZZZm[Z[dd l\m]Z]m^Z^m_Z_m`Z`maZambZbmcZcmdZdmeZemfZfdd lgmhZhmiZimjZjmkZkmlZlmmZmmnZndd lompZpmqZqmrZrmsZsddltmuZumvZvmwZwmxZxmyZymzZzm{Z{m|Z|ddlm}Z}m~Z~ddZGddde eZddZGddde eZddZGddde eZdS)z1OO layer for several polynomial representations. )oo) CantSympify)CoercionFailed NotReversible NotInvertible)PicklableWithSlotsc@s4eZdZdZddZddZddZedd Zd S) GenericPolyz5Base class for low-level polynomial representations. cC||jSzMake the ground domain a ring. ) set_domaindomget_ringfr9/usr/lib/python3/dist-packages/sympy/polys/polyclasses.pyground_to_ring zGenericPoly.ground_to_ringcCr z Make the ground domain a field. )r r get_fieldrrrrground_to_fieldrzGenericPoly.ground_to_fieldcCr zMake the ground domain exact. )r r get_exactrrrrground_to_exactrzGenericPoly.ground_to_exactcs.|r|\}}fdd|D}|r||fS|S)Ncsg|] \}}||fqSrr.0gkperrr sz/GenericPoly._perify_factors..r)rresultincludecoefffactorsrrr_perify_factorss zGenericPoly._perify_factorsN) __name__ __module__ __qualname____doc__rrr classmethodr%rrrrr sr)! dmp_validate dup_normal dmp_normal dup_convert dmp_convertdmp_from_sympy dup_strip dup_degree dmp_degree_indmp_degree_listdmp_negative_pdup_LC dmp_ground_LCdup_TC dmp_ground_TCdmp_ground_nthdmp_one dmp_ground dmp_zero_p dmp_one_p dmp_ground_p dup_from_dict dmp_from_dict dmp_to_dict dmp_deflate dmp_inject dmp_eject dmp_terms_gcddmp_list_terms dmp_exclude dmp_slice_in dmp_permute dmp_to_tuple)dmp_add_grounddmp_sub_grounddmp_mul_grounddmp_quo_grounddmp_exquo_grounddmp_absdup_negdmp_negdup_adddmp_adddup_subdmp_subdup_muldmp_muldmp_sqrdup_powdmp_powdmp_pdivdmp_premdmp_pquo dmp_pexquodmp_divdup_remdmp_remdmp_quo dmp_exquo dmp_add_mul dmp_sub_mul dmp_max_norm dmp_l1_norm)dmp_clear_denomsdmp_integrate_in dmp_diff_in dmp_eval_in dup_revertdmp_ground_truncdmp_ground_contentdmp_ground_primitivedmp_ground_monic dmp_compose dup_decompose dup_shift dup_transformdmp_lift) dup_half_gcdex dup_gcdex dup_invertdmp_subresultants dmp_resultantdmp_discriminant dmp_inner_gcddmp_gcddmp_lcm dmp_cancel) dup_gff_listdmp_norm dmp_sqf_p dmp_sqf_norm dmp_sqf_part dmp_sqf_listdmp_sqf_list_include)dup_cyclotomic_pdmp_irreducible_pdmp_factor_listdmp_factor_list_include)dup_isolate_real_roots_sqfdup_isolate_real_rootsdup_isolate_all_roots_sqfdup_isolate_all_rootsdup_refine_real_rootdup_count_real_rootsdup_count_complex_roots dup_sturm)UnificationFailedPolynomialErrorcCstt|||||SN)DMPr-)replevr rrrinit_normal_DMPrc@s8eZdZdZdZdddZddZdd Zd d Zdd dZ e dddZ e dddZ e ddZ e ddZdddZdddZddZddZdd Ze d!d"Ze dd#d$Zd%d&Zd'd(Zd)d*Zd+d,Zdd.d/Zdd0d1Zdd2d3Zdd4d5Zd6d7Zd8d9Zd:d;Z dd?Z"dd@dAZ#ddBdCZ$dDdEZ%dFdGZ&dHdIZ'dJdKZ(dLdMZ)dNdOZ*dPdQZ+dRdSZ,dTdUZ-dVdWZ.dXdYZ/dZd[Z0d\d]Z1d^d_Z2d`daZ3dbdcZ4dddeZ5dfdgZ6dhdiZ7djdkZ8dldmZ9dndoZ:dpdqZ;ddrdsZdxdyZ?dzd{Z@d|d}ZAd~dZBddZCddZDddZEddZFdddZGdddZHdddZIddZJddZKddZLddZMddZNdddZOddZPddZQddZRddZSdddZTddZUddZVddZWddZXddZYddZZddZ[ddZ\ddZ]ddZ^ddZ_ddZ`ddZadddZbdddÄZcddńZdddDŽZedddɄZfd dd˄Zgddd̈́ZhdddτZiejddфZkejddӄZlejddՄZmejddׄZnejddلZoejddۄZpejdd݄Zqejdd߄ZrejddZsejddZtejddZuejddZvddZwddZxddZyddZzddZ{ddZ|ddZ}ddZ~ddZddZddZddZddZddZddZddZddd Zdd d Zd d ZddZddZddZddZddZdS(!rz)Dense Multivariate Polynomials over `K`. )rrr ringNcCsh|dur t|turt|||}nt|turt|||}nt|\}}||_||_||_ ||_ dSr) typedictrAlistr<convertr+rrr r)selfrr rrrrr__init__s    z DMP.__init__cCd|jj|j|j|jfSNz%s(%s, %s, %s)) __class__r&rr rrrrr__repr__z DMP.__repr__cCs t|jj||j|j|jfSr)hashrr&to_tuplerr rrrrr__hash__s z DMP.__hash__cst|tr |j|jkrtd||f|j|jkr+|j|jkr+|j|j|j|j|jfS|j|j|j}}|j|jdurLdurI|jn|jt |j||j|}t |j||j|}||dffdd }|||||fS)z7Unify representations of two multivariate polynomials. can't unify %s with %sNFcs"|r |s|S|d8}t|||SNr)rr rkillrrrrs zDMP.unify..per) isinstancerrrr rrrunifyr/)rrrr FGrrrrrs  z DMP.unifyFcCsD|j}|r |s |S|d8}|dur|j}|dur|j}t||||S).Create a DMP out of the given representation. rN)rr rr)rrr rrrrrrrszDMP.percCtd|||SNrrclsrr rrrrzerozDMP.zerocCrrrrrrronerzDMP.onecCs|t||d|||S)zCCreate an instance of ``cls`` given a list of native coefficients. N)r/rrrr rrr from_listz DMP.from_listcC|t|||||S)zBCreate an instance of ``cls`` given a list of SymPy coefficients. )r0rrrrfrom_sympy_listzDMP.from_sympy_listcCt|j|j|j|dS)AConvert ``f`` to a dict representation with native coefficients. r)rBrrr )rrrrrto_dictz DMP.to_dictcCs<t|j|j|j|d}|D] \}}|j|||<q|S)@Convert ``f`` to a dict representation with SymPy coefficients. r)rBrrr itemsto_sympy)rrrrvrrr to_sympy_dictszDMP.to_sympy_dictcC|jSzAConvert ``f`` to a list representation with native coefficients. rrrrrto_listz DMP.to_listcsfddjS)@Convert ``f`` to a list representation with SymPy coefficients. cs>g}|D]}t|tr||q|j|q|Sr)rrappendr r)routvalrsympify_nested_listrrrs  z.DMP.to_sympy_list..sympify_nested_listrrrrr to_sympy_lists zDMP.to_sympy_listcCt|j|jS)x Convert ``f`` to a tuple representation with native coefficients. This is needed for hashing. )rKrrrrrrrsz DMP.to_tuplecCr)zBConstruct and instance of ``cls`` from a ``dict`` representation. )rArrrr from_dictrz DMP.from_dictcCstttt|||||Sr)rrrzip)rmonomscoeffsrr rrrrfrom_monoms_coeffszDMP.from_monoms_coeffscCr r )rr r rrrrto_ringrz DMP.to_ringcCr r)rr rrrrrto_fieldrz DMP.to_fieldcCr r)rr rrrrrto_exact rz DMP.to_exactcCs,|j|kr|Stt|j|j|j|||jS)z$Convert the ground domain of ``f``. )r rr/rr)rr rrrr$s z DMP.convertrc Cs|t|j||||j|jS)z1Take a continuous subsequence of terms of ``f``. )rrIrrr )rmnjrrrslice+sz DMP.slicecC ddt|j|j|j|dDS)z;Returns all non-zero coefficients from ``f`` in lex order. cSsg|]\}}|qSrr)r_crrrr 1zDMP.coeffs..orderrGrrr rrrrrr/ z DMP.coeffscCr)z8Returns all non-zero monomials from ``f`` in lex order. cSsg|]\}}|qSrr)rrrrrrr 5rzDMP.monoms..rrrrrrr3rz DMP.monomscCr)z4Returns all non-zero terms from ``f`` in lex order. rrrrrrterms7rz DMP.termscCs,|js|s |jjgSdd|jDStd)z%Returns all coefficients from ``f``. cSsg|]}|qSrrrrrrrr Asz"DMP.all_coeffs..&multivariate polynomials not supported)rr rrrrrrr all_coeffs;s  zDMP.all_coeffscs>|jst|jdkrdgSfddt|jDStd)z"Returns all monomials from ``f``. rrcsg|] \}}|fqSrrrirrrrr Msz"DMP.all_monoms..r)rr2r enumeraterrrrr all_monomsEs  zDMP.all_monomscsF|jst|jdkrd|jjfgSfddt|jDStd)z Returns all terms from a ``f``. rrcsg|] \}}|f|fqSrrrrrrr Yz!DMP.all_terms..r)rr2rr rrrrrrr all_termsQs  z DMP.all_termscCs |jt|j|j|j|jjdS)z-Convert algebraic coefficients to rationals. r )rrwrrr rrrrlift]rzDMP.liftcC$t|j|j|j\}}|||fS)z2Reduce degree of `f` by mapping `x_i^m` to `y_i`. )rCrrr rrJrrrrdeflateaz DMP.deflatecCs,t|j|j|j|d\}}|||jj|S)z,Inject ground domain generators into ``f``. front)rDrrr r)rrrrrrrinjectfsz DMP.injectcCs.t|j|j||d}||||jt|jS)z2Eject selected generators into the ground domain. r)rErrrlensymbols)rr rrrrrejectksz DMP.ejectcCs,t|j|j|j\}}}||||j|fS)ao 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)) )rHrrr r)rrrurrrexcludepsz DMP.excludecCs|t|j||j|jS)a 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) )rrJrrr )rPrrrpermutesz DMP.permutecCr)z/Remove GCD of terms from the polynomial ``f``. )rFrrr rrrrr terms_gcdrz DMP.terms_gcdcC"|t|j|j||j|jS)z.Add an element of the ground domain to ``f``. )rrLrr rrrrrrr add_ground"zDMP.add_groundcCr)z5Subtract an element of the ground domain from ``f``. )rrMrr rrrrrr sub_groundrzDMP.sub_groundcCr)z5Multiply ``f`` by a an element of the ground domain. )rrNrr rrrrrr mul_groundrzDMP.mul_groundcCr)z8Quotient of ``f`` by a an element of the ground domain. )rrOrr rrrrrr quo_groundrzDMP.quo_groundcCr)z>Exact quotient of ``f`` by a an element of the ground domain. )rrPrr rrrrrr exquo_groundrzDMP.exquo_groundcC|t|j|j|jS)z)Make all coefficients in ``f`` positive. )rrQrrr rrrrabszDMP.abscCr)"Negate all coefficients in ``f``. )rrSrrr rrrrnegrzDMP.negcC&||\}}}}}|t||||S)z2Add two multivariate polynomials ``f`` and ``g``. )rrUrrrr rrrrrraddzDMP.addcCr)z7Subtract two multivariate polynomials ``f`` and ``g``. )rrWrrrrsubr zDMP.subcCr)z7Multiply two multivariate polynomials ``f`` and ``g``. )rrYrrrrmulr zDMP.mulcCr)z(Square a multivariate polynomial ``f``. )rrZrrr rrrrsqrrzDMP.sqrcCs4t|tr|t|j||j|jStdt|)+Raise ``f`` to a non-negative power ``n``. ``int`` expected, got %s) rintrr\rrr TypeErrorrrrrrrpows zDMP.powc C6||\}}}}}t||||\}}||||fS)z/Polynomial pseudo-division of ``f`` and ``g``. )rr] rrrr rrrqrrrrpdivzDMP.pdivcCr)z0Polynomial pseudo-remainder of ``f`` and ``g``. )rr^rrrrpremr zDMP.premcCr)z/Polynomial pseudo-quotient of ``f`` and ``g``. )rr_rrrrpquor zDMP.pquocCr)z5Polynomial exact pseudo-quotient of ``f`` and ``g``. )rr`rrrrpexquor z DMP.pexquoc Cr*)z7Polynomial division with remainder of ``f`` and ``g``. )rrar+rrrdivr/zDMP.divcCr)z2Computes polynomial remainder of ``f`` and ``g``. )rrcrrrrremr zDMP.remcCr)z1Computes polynomial quotient of ``f`` and ``g``. )rrdrrrrquor zDMP.quoc CsT||\}}}}}|t||||}|jr(||jvr(ddlm}||||j|S)z7Computes polynomial exact quotient of ``f`` and ``g``. rExactQuotientFailed)rrersympy.polys.polyerrorsr7) rrrr rrrresr7rrrexquos  z DMP.exquocCs*t|tr t|j||jStdt|)z0Returns the leading degree of ``f`` in ``x_j``. r%)rr&r3rrr'r)rrrrrdegrees z DMP.degreecCr)z$Returns a list of degrees of ``f``. )r4rrrrrr degree_listrzDMP.degree_listcCstdd|DS)z#Returns the total degree of ``f``. css|]}t|VqdSrsum)rrrrr sz#DMP.total_degree..)maxrrrrr total_degree rzDMP.total_degreec Cs|}i}|t|ddk}|D]7}t|d}||kr'||}nd}|r7|d||d|f<qt|d}|||7<|d|t|<qt||j|jt ||j S)z&Return homogeneous polynomial of ``f``rr) rArrr>rtuplerr rr&r) rsZtdr!Z new_symbolZtermdrlrrr homogenizes    zDMP.homogenizecCsD|jrt S|}t|d}|D] }t|}||krdSq|S)z(Returns the homogeneous order of ``f``. rN)is_zerorrr>)rrZtdegmonomZ_tdegrrrhomogeneous_order!s zDMP.homogeneous_ordercCt|j|j|jSz*Returns the leading coefficient of ``f``. )r7rrr rrrrLC1zDMP.LCcCrJz+Returns the trailing coefficient of ``f``. )r9rrr rrrrTC5rMzDMP.TCcGs.tdd|Drt|j||j|jStd)z+Returns the ``n``-th coefficient of ``f``. css|]}t|tVqdSr)rr&)rrrrrr?;szDMP.nth..za sequence of integers expected)allr:rrr r')rNrrrnth9szDMP.nthcCrJ)zReturns maximum norm of ``f``. )rhrrr rrrrmax_norm@rMz DMP.max_normcCrJ)zReturns l1 norm of ``f``. )rirrr rrrrl1_normDrMz DMP.l1_normcCr)z0Clear denominators, but keep the ground domain. )rjrrr r)rr#rrrr clear_denomsHrzDMP.clear_denomsrcCPt|ts tdt|t|tstdt||t|j|||j|jS)zEComputes the ``m``-th order indefinite integral of ``f`` in ``x_j``. r%) rr&r'rrrkrrr rrrrrr integrateM  z DMP.integratecCrV)z||\}}}}}t||||\}}} |||||| fS)z4Returns GCD of ``f`` and ``g`` and their cofactors. )rr~) rrrr rrrr`ZcffZcfgrrr cofactorssz DMP.cofactorscCr)z+Returns polynomial GCD of ``f`` and ``g``. )rrrrrrgcdr zDMP.gcdcCr)z+Returns polynomial LCM of ``f`` and ``g``. )rrrrrrlcmr zDMP.lcmTc Cst||\}}}}}|rt||||dd\}}n t||||dd\}} }}||||}}|r4||fS|| ||fS)z6Cancel common factors in a rational function ``f/g``. T)r"F)rr) rrr"rr rrrZcFZcGrrrcancels z DMP.cancelcCr)z&Reduce ``f`` modulo a constant ``p``. )rrorr rr)rprrrtruncrz DMP.trunccCr)z'Divides all coefficients by ``LC(f)``. )rrrrrr rrrrmonicrz DMP.moniccCrJ)z(Returns GCD of polynomial coefficients. )rprrr rrrrcontentrMz DMP.contentcCr)z/Returns content and a primitive form of ``f``. )rqrrr r)rZcontrrrr primitiverz DMP.primitivecCr)z4Computes functional composition of ``f`` and ``g``. )rrsrrrrcomposer z DMP.composecC(|jstt|jt|j|jStd)z,Computes functional decomposition of ``f``. r^)rrrfrrtrr r_rrrr decomposez DMP.decomposecCs,|js|t|j|j||jStd)z/Efficiently compute Taylor shift ``f(x + a)``. r^)rrrurr rr_)rr\rrrshiftsz DMP.shiftc Cs|jrtd||\}}}}}|||||\}}}}}||||||||\}}}}}|s<|t||||Std)z5Evaluate functional transformation ``q**n * f(p/q)``.r^)rr_rrv) rrpr,rr rr Qrrrr transforms$z DMP.transformcCrv)z&Computes the Sturm sequence of ``f``. r^)rrrfrrrr r_rrrrsturmrxz DMP.sturmcs*jsfddtjjDStd)z4Computes greatest factorial factorization of ``f``. cg|] \}}||fqSrrrrrrr rz DMP.gff_list..r^)rrrr r_rrrrgff_listsz DMP.gff_listcCs$t|j|j|j}|j||jjdS)zComputes ``Norm(f)``.r)rrrr r)rr-rrrnormzDMP.normcCs6t|j|j|j\}}}||||j||jjdfS)z$Computes square-free norm of ``f``. r)rrrr r)rrCrr-rrrsqf_normsz DMP.sqf_normcCr)z$Computes square-free part of ``f``. )rrrrr rrrrsqf_part rz DMP.sqf_partcs.tjjj|\}}|fdd|DfS)0Returns a list of square-free factors of ``f``. cr}rrrrrrr rz DMP.sqf_list..)rrrr )rrPr#r$rrrsqf_list sz DMP.sqf_listcs&tjjj|}fdd|DS)rcr}rrrrrrr rz(DMP.sqf_list_include..)rrrr )rrPr$rrrsqf_list_includer zDMP.sqf_list_includecs,tjjj\}}|fdd|DfS)0Returns a list of irreducible factors of ``f``. cr}rrrrrrr rz#DMP.factor_list..)rrrr )rr#r$rrr factor_listszDMP.factor_listcs$tjjj}fdd|DS)rcr}rrrrrrr rz+DMP.factor_list_include..)rrrr )rr$rrrfactor_list_includerzDMP.factor_list_includecCsz|js9|s|st|j|j||||dSt|j|j||||dS|s-t|j|j||||dSt|j|j||||dStd)z0Compute isolating intervals for roots of ``f``. )epsinfsupfastz0can't isolate roots of a multivariate polynomial)rrrr rrrr)rrPrrrrZsqfrrr intervals!sz DMP.intervalsc Cs(|jst|j|||j|||dStd)zu Refine an isolating interval to the given precision. ``eps`` should be a rational number. )rstepsrz0can't refine a root of a multivariate polynomial)rrrr r)rrCrbrrrrrr refine_root2s zDMP.refine_rootcCt|j|j||dS)zrrr rrrris_oneLz DMP.is_onecCt|jd|jS)>Returns ``True`` if ``f`` is an element of the ground domain. 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