a RG5d-@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@mAZAmBZBmCZCmDZDmEZEmFZFmGZGmHZHmIZImJZJmKZKmLZLmMZMdd lNmOZOmPZPmQZQmRZRmSZSmTZTmUZUmVZVmWZWmXZXmYZYmZZZm[Z[m\Z\dd l]m^Z^m_Z_m`Z`maZambZbmcZcmdZdmeZemfZfmgZgdd lhmiZimjZjmkZkmlZlmmZmmnZnmoZodd lpmqZqmrZrmsZsmtZtddlumvZvmwZwmxZxmyZymzZzm{Z{m|Z|m}Z}m~Z~mZmZddlmZmZddZGddde 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. cCs||jSzMake the ground domain a ring. ) set_domaindomget_ringfrS/var/www/html/django/DPS/env/lib/python3.9/site-packages/sympy/polys/polyclasses.pyground_to_ring szGenericPoly.ground_to_ringcCs||jSz Make the ground domain a field. )r r get_fieldr rrrground_to_fieldszGenericPoly.ground_to_fieldcCs||jSzMake the ground domain exact. )r r get_exactr rrrground_to_exactszGenericPoly.ground_to_exactcs2|r |\}}fdd|D}|r*||fS|SdS)Ncsg|]\}}||fqSrr.0gkperrr z/GenericPoly._perify_factors..r)rresultincludecoefffactorsrrr_perify_factorss zGenericPoly._perify_factorsN) __name__ __module__ __qualname____doc__rrr classmethodr$rrrrr s r)! dmp_validate dup_normal dmp_normal dup_convert dmp_convertdmp_from_sympy dup_strip dup_degree dmp_degree_indmp_degree_listdmp_negative_pdup_LC dmp_ground_LCdup_TC dmp_ground_TCdmp_ground_nthdmp_one dmp_ground dmp_zero_p dmp_one_p dmp_ground_p dup_from_dict dmp_from_dict dmp_to_dict dmp_deflate dmp_inject dmp_eject dmp_terms_gcddmp_list_terms dmp_exclude dmp_slice_in dmp_permute dmp_to_tuple)dmp_add_grounddmp_sub_grounddmp_mul_grounddmp_quo_grounddmp_exquo_grounddmp_absdup_negdmp_negdup_adddmp_adddup_subdmp_subdup_muldmp_muldmp_sqrdup_powdmp_powdmp_pdivdmp_premdmp_pquo dmp_pexquodmp_divdup_remdmp_remdmp_quo dmp_exquo dmp_add_mul dmp_sub_mul dmp_max_norm dmp_l1_normdmp_l2_norm_squared)dmp_clear_denomsdmp_integrate_in dmp_diff_in dmp_eval_in dup_revertdmp_ground_truncdmp_ground_contentdmp_ground_primitivedmp_ground_monic dmp_compose dup_decompose dup_shift dup_transformdmp_lift) dup_half_gcdex dup_gcdex dup_invertdmp_subresultants dmp_resultantdmp_discriminant dmp_inner_gcddmp_gcddmp_lcm dmp_cancel) dup_gff_listdmp_norm dmp_sqf_p dmp_sqf_norm dmp_sqf_part dmp_sqf_listdmp_sqf_list_include)dup_cyclotomic_pdmp_irreducible_pdmp_factor_listdmp_factor_list_include) dup_isolate_real_roots_sqfdup_isolate_real_rootsdup_isolate_all_roots_sqfdup_isolate_all_rootsdup_refine_real_rootdup_count_real_rootsdup_count_complex_roots dup_sturmdup_cauchy_upper_bounddup_cauchy_lower_bounddup_mignotte_sep_bound_squared)UnificationFailedPolynomialErrorcCstt|||||SN)DMPr,)replevr rrrinit_normal_DMPsrc@sheZdZdZdZd ddZddZdd Zd d Zd!d dZ e d"ddZ e d#ddZ e ddZ e ddZd$ddZd%ddZddZddZdd Ze d!d"Ze d&d#d$Zd%d&Zd'd(Zd)d*Zd+d,Zd'd.d/Zd(d0d1Zd)d2d3Zd*d4d5Zd6d7Zd8d9Zd:d;Z dd?Z"d+d@dAZ#d,dBdCZ$dDdEZ%dFdGZ&dHdIZ'dJdKZ(dLdMZ)dNdOZ*dPdQZ+dRdSZ,dTdUZ-dVdWZ.dXdYZ/dZd[Z0d\d]Z1d^d_Z2d`daZ3dbdcZ4dddeZ5dfdgZ6dhdiZ7djdkZ8dldmZ9dndoZ:dpdqZ;d-drdsZdxdyZ?dzd{Z@d|d}ZAd~dZBddZCddZDddZEddZFddZGd.ddZHd/ddZId0ddZJddZKddZLddZMddZNddZOd1ddZPddZQddZRddZSddZTd2ddZUddZVddZWddZXddZYddZZddZ[ddZ\ddZ]ddZ^ddZ_ddZ`ddZaddZbddÄZcddńZdddDŽZed3ddɄZfd4dd˄Zgdd̈́ZhddτZid5ddфZjd6ddӄZkd7ddՄZld8ddׄZmenddلZoenddۄZpendd݄Zqendd߄ZrenddZsenddZtenddZuenddZvenddZwenddZxenddZyenddZzddZ{ddZ|ddZ}ddZ~ddZddZddZddZddZddZddZddZdd Zd d Zd d ZddZd9ddZd:ddZddZddZddZddZddZddZdS(;rz)Dense Multivariate Polynomials over `K`. )rrr ringNcCsf|dur>t|tur"t|||}qJt|tsJt|||}n t|\}}||_||_ ||_ ||_ dSr) typedictr@ isinstancelistr;convertr*rrr r)selfrr rrrrr__init__s   z DMP.__init__cCsd|jj|j|j|jfSNz%s(%s, %s, %s)) __class__r%rr rr rrr__repr__sz DMP.__repr__cCs t|jj||j|j|jfSr)hashrr%to_tuplerr rr rrr__hash__sz DMP.__hash__cst|tr|j|jkr&td||f|j|jkrV|j|jkrV|j|j|j|j|jfS|j|j|j}}|j|jdurdur|jn|jt |j||j|}t |j||j|}||dffdd }|||||fSdS)z7Unify representations of two multivariate polynomials. Cannot unify %s with %sNFcs"|r|s |S|d8}t|||SNr)rr rkillrrrrs zDMP.unify..per) rrrrr rrrunifyr.)rrrr FGrrrrrs  z DMP.unifyFcCsD|j}|r|s|S|d8}|dur(|j}|dur6|j}t||||S).Create a DMP out of the given representation. rN)rr rr)rrr rrrrrrrszDMP.percCstd|||SNrrclsrr rrrrzeroszDMP.zerocCstd|||SrrrrrroneszDMP.onecCs|t||d|||S)zCCreate an instance of ``cls`` given a list of native coefficients. N)r.rrrr rrr from_listsz DMP.from_listcCs|t|||||S)zBCreate an instance of ``cls`` given a list of SymPy coefficients. )r/rrrrfrom_sympy_listszDMP.from_sympy_listcCst|j|j|j|dS)AConvert ``f`` to a dict representation with native coefficients. r)rArrr )rrrrrto_dictsz DMP.to_dictcCs<t|j|j|j|d}|D]\}}|j|||<q|S)@Convert ``f`` to a dict representation with SymPy coefficients. r)rArrr itemsto_sympy)rrrrvrrr to_sympy_dictszDMP.to_sympy_dictcCs|jSzAConvert ``f`` to a list representation with native coefficients. rr rrrto_listsz DMP.to_listcsfddjS)@Convert ``f`` to a list representation with SymPy coefficients. cs>g}|D]0}t|tr&||q|j|q|Sr)rrappendr r)routvalrsympify_nested_listrrrs  z.DMP.to_sympy_list..sympify_nested_listrr rrr to_sympy_lists zDMP.to_sympy_listcCst|j|jS)x Convert ``f`` to a tuple representation with native coefficients. This is needed for hashing. )rJrrr rrrr sz DMP.to_tuplecCs|t|||||S)zBConstruct and instance of ``cls`` from a ``dict`` representation. )r@rrrr from_dictsz DMP.from_dictcCstttt|||||Sr)rrrzip)rmonomscoeffsrr rrrrfrom_monoms_coeffsszDMP.from_monoms_coeffscCs||jSr )rr r r rrrto_ringsz DMP.to_ringcCs||jSr)rr rr rrrto_field!sz DMP.to_fieldcCs||jSr)rr rr rrrto_exact%sz DMP.to_exactcCs0|j|kr|Stt|j|j|j|||jSdS)z$Convert the ground domain of ``f``. N)r rr.rr)rr rrrr)s z DMP.convertrc Cs|t|j||||j|jS)z1Take a continuous subsequence of terms of ``f``. )rrHrrr )rmnjrrrslice0sz DMP.slicecCs ddt|j|j|j|dDS)z;Returns all non-zero coefficients from ``f`` in lex order. cSsg|] \}}|qSrr)r_crrrr6rzDMP.coeffs..orderrFrrr rrrrrr4sz DMP.coeffscCs ddt|j|j|j|dDS)z8Returns all non-zero monomials from ``f`` in lex order. cSsg|] \}}|qSrr)rrrrrrr:rzDMP.monoms..rrrrrrr8sz DMP.monomscCst|j|j|j|dS)z4Returns all non-zero terms from ``f`` in lex order. rrrrrrterms<sz DMP.termscCs2|js&|s|jjgSdd|jDSntddS)z%Returns all coefficients from ``f``. cSsg|]}|qSrrrrrrrrFrz"DMP.all_coeffs..&multivariate polynomials not supportedN)rr rrrr rrr all_coeffs@s  zDMP.all_coeffscsD|js8t|jdkrdgSfddt|jDSntddS)z"Returns all monomials from ``f``. rrcsg|]\}}|fqSrrrirrrrrRrz"DMP.all_monoms..rN)rr1r enumeraterr rrr all_monomsJs  zDMP.all_monomscsL|js@t|jdkr&d|jjfgSfddt|jDSntddS)z Returns all terms from a ``f``. rrcsg|]\}}|f|fqSrrrrrrr^rz!DMP.all_terms..rN)rr1rr rrrr rrr all_termsVs  z DMP.all_termscCs |jt|j|j|j|jjdS)z-Convert algebraic coefficients to rationals. r )rrwrrr r rrrliftbszDMP.liftcCs$t|j|j|j\}}|||fS)z2Reduce degree of `f` by mapping `x_i^m` to `y_i`. )rBrrr rrJrrrrdeflatefsz DMP.deflatecCs,t|j|j|j|d\}}|||jj|S)z,Inject ground domain generators into ``f``. front)rCrrr r)rrrrrrrinjectksz DMP.injectcCs.t|j|j||d}||||jt|jS)z2Eject selected generators into the ground domain. r)rDrrrlensymbols)rr rrrrrejectpsz 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)) )rGrrr r)rrrurrrexcludeusz 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) )rrIrrr )rPrrrpermutesz DMP.permutecCs$t|j|j|j\}}|||fS)z/Remove GCD of terms from the polynomial ``f``. )rErrr rrrrr terms_gcdsz DMP.terms_gcdcCs"|t|j|j||j|jS)z.Add an element of the ground domain to ``f``. )rrKrr rrrrrrr add_groundszDMP.add_groundcCs"|t|j|j||j|jS)z5Subtract an element of the ground domain from ``f``. )rrLrr rrrrrr sub_groundszDMP.sub_groundcCs"|t|j|j||j|jS)z5Multiply ``f`` by a an element of the ground domain. )rrMrr rrrrrr mul_groundszDMP.mul_groundcCs"|t|j|j||j|jS)z8Quotient of ``f`` by a an element of the ground domain. )rrNrr rrrrrr quo_groundszDMP.quo_groundcCs"|t|j|j||j|jS)z>Exact quotient of ``f`` by a an element of the ground domain. )rrOrr rrrrrr exquo_groundszDMP.exquo_groundcCs|t|j|j|jS)z)Make all coefficients in ``f`` positive. )rrPrrr r rrrabsszDMP.abscCs|t|j|j|jS)"Negate all coefficients in ``f``. )rrRrrr r rrrnegszDMP.negcCs&||\}}}}}|t||||S)z2Add two multivariate polynomials ``f`` and ``g``. )rrTrrrr rrrrrraddszDMP.addcCs&||\}}}}}|t||||S)z7Subtract two multivariate polynomials ``f`` and ``g``. )rrVrrrrsubszDMP.subcCs&||\}}}}}|t||||S)z7Multiply two multivariate polynomials ``f`` and ``g``. )rrXrrrrmulszDMP.mulcCs|t|j|j|jS)z(Square a multivariate polynomial ``f``. )rrYrrr r rrrsqrszDMP.sqrcCs8t|tr$|t|j||j|jStdt|dS)+Raise ``f`` to a non-negative power ``n``. ``int`` expected, got %sN) rintrr[rrr TypeErrorrrrrrrpows zDMP.powc Cs6||\}}}}}t||||\}}||||fS)z/Polynomial pseudo-division of ``f`` and ``g``. )rr\ rrrr rrrqrrrrpdivszDMP.pdivcCs&||\}}}}}|t||||S)z0Polynomial pseudo-remainder of ``f`` and ``g``. )rr]rrrrpremszDMP.premcCs&||\}}}}}|t||||S)z/Polynomial pseudo-quotient of ``f`` and ``g``. )rr^rrrrpquoszDMP.pquocCs&||\}}}}}|t||||S)z5Polynomial exact pseudo-quotient of ``f`` and ``g``. )rr_rrrrpexquosz DMP.pexquoc Cs6||\}}}}}t||||\}}||||fS)z7Polynomial division with remainder of ``f`` and ``g``. )rr`rrrrdivszDMP.divcCs&||\}}}}}|t||||S)z2Computes polynomial remainder of ``f`` and ``g``. )rrbrrrrremszDMP.remcCs&||\}}}}}|t||||S)z1Computes polynomial quotient of ``f`` and ``g``. )rrcrrrrquoszDMP.quoc CsT||\}}}}}|t||||}|jrP||jvrPddlm}||||j|S)z7Computes polynomial exact quotient of ``f`` and ``g``. rExactQuotientFailed)rrdrsympy.polys.polyerrorsr) rrrr rrrresrrrrexquos  z DMP.exquocCs.t|trt|j||jStdt|dS)z0Returns the leading degree of ``f`` in ``x_j``. rN)rrr2rrrr)rrrrrdegrees z DMP.degreecCst|j|jS)z$Returns a list of degrees of ``f``. )r3rrr rrr degree_list szDMP.degree_listcCstdd|DS)z#Returns the total degree of ``f``. css|]}t|VqdSrsum)rrrrr rz#DMP.total_degree..)maxrr rrr total_degreeszDMP.total_degreec Cs|}i}|t|ddk}|D]n}t|d}||krN||}nd}|rn|d||d|f<q,t|d}|||7<|d|t|<q,t||j|jt ||j S)z&Return homogeneous polynomial of ``f``rr) r(rrr%rtuplerr rrr) rstdr Z new_symboltermdrlrrr homogenizes    zDMP.homogenizecCsD|jr t S|}t|d}|D]}t|}||kr$dSq$|S)z(Returns the homogeneous order of ``f``. rN)is_zerorrr%)rrZtdegmonomZ_tdegrrrhomogeneous_order&s zDMP.homogeneous_ordercCst|j|j|jSz*Returns the leading coefficient of ``f``. )r6rrr r rrrLC6szDMP.LCcCst|j|j|jSz+Returns the trailing coefficient of ``f``. )r8rrr r rrrTC:szDMP.TCcGs2tdd|Dr&t|j||j|jStddS)z+Returns the ``n``-th coefficient of ``f``. css|]}t|tVqdSr)rr)rrrrrr&@rzDMP.nth..za sequence of integers expectedN)allr9rrr r)rNrrrnth>szDMP.nthcCst|j|j|jS)zReturns maximum norm of ``f``. )rgrrr r rrrmax_normEsz DMP.max_normcCst|j|j|jS)zReturns l1 norm of ``f``. )rhrrr r rrrl1_normIsz DMP.l1_normcCst|j|j|jS)z!Return squared l2 norm of ``f``. )rirrr r rrrl2_norm_squaredMszDMP.l2_norm_squaredcCs$t|j|j|j\}}|||fS)z0Clear denominators, but keep the ground domain. )rjrrr r)rr"rrrr clear_denomsQszDMP.clear_denomsrcCsPt|tstdt|t|ts4tdt||t|j|||j|jS)zEComputes the ``m``-th order indefinite integral of ``f`` in ``x_j``. r) rrrrrrkrrr rrrrrr integrateVs   z DMP.integratecCsPt|tstdt|t|ts4tdt||t|j|||j|jS)zrrrdiff`s   zDMP.diffcCsBt|tstdt||jt|j|j|||j |jddS)z5Evaluates ``f`` at the given point ``a`` in ``x_j``. rTr) rrrrrrmrr rr)rarrrrevaljs  zDMP.evalc CsD||\}}}}}|s8t|||\}}||||fStddS)z2Half extended Euclidean algorithm, if univariate. univariate polynomial expectedN)rrx ValueError) rrrr rrrr*hrrr half_gcdexrs zDMP.half_gcdexc CsL||\}}}}}|s@t|||\}}} |||||| fStddS)z-Extended Euclidean algorithm, if univariate. rDN)rryrE) rrrr rrrr*trFrrrgcdex|s z DMP.gcdexcCs4||\}}}}}|s(|t|||StddS)z(Invert ``f`` modulo ``g``, if possible. rDN)rrzrErrrrinvertsz DMP.invertcCs(|js|t|j||jStddS)z"Compute ``f**(-1)`` mod ``x**n``. rDN)rrrnrr rErrrrrevertsz DMP.revertcCs0||\}}}}}t||||}tt||S)z7Computes subresultant PRS sequence of ``f`` and ``g``. )rr{rmap)rrrr rrrRrrr subresultantsszDMP.subresultantsc Cs^||\}}}}}|rHt|||||d\}} ||ddtt|| fS|t||||ddS)z/Computes resultant of ``f`` and ``g`` via PRS. ) includePRSTrA)rr|rrL) rrrOrr rrrr rMrrr resultants z DMP.resultantcCs|jt|j|j|jddS)z Computes discriminant of ``f``. TrA)rr}rrr r rrr discriminantszDMP.discriminantc Cs>||\}}}}}t||||\}}} |||||| fS)z4Returns GCD of ``f`` and ``g`` and their cofactors. )rr~) rrrr rrrrFcffcfgrrr cofactorssz DMP.cofactorscCs&||\}}}}}|t||||S)z+Returns polynomial GCD of ``f`` and ``g``. )rrrrrrgcdszDMP.gcdcCs&||\}}}}}|t||||S)z+Returns polynomial LCM of ``f`` and ``g``. )rrrrrrlcmszDMP.lcmTc Csx||\}}}}}|r0t||||dd\}}nt||||dd\}} }}||||}}|rh||fS|| ||fSdS)z6Cancel common factors in a rational function ``f/g``. T)r!FN)rr) rrr!rr rrrZcFZcGrrrcancelsz DMP.cancelcCs"|t|j|j||j|jS)z&Reduce ``f`` modulo a constant ``p``. )rrorr rr)rprrrtruncsz DMP.trunccCs|t|j|j|jS)z'Divides all coefficients by ``LC(f)``. )rrrrrr r rrrmonicsz DMP.moniccCst|j|j|jS)z(Returns GCD of polynomial coefficients. )rprrr r rrrcontentsz DMP.contentcCs$t|j|j|j\}}|||fS)z/Returns content and a primitive form of ``f``. )rqrrr r)rcontrrrr primitivesz DMP.primitivecCs&||\}}}}}|t||||S)z4Computes functional composition of ``f`` and ``g``. )rrsrrrrcomposesz DMP.composecCs,|js tt|jt|j|jStddS)z,Computes functional decomposition of ``f``. rDN)rrrLrrtrr rEr rrr decomposesz DMP.decomposecCs0|js$|t|j|j||jStddS)z/Efficiently compute Taylor shift ``f(x + a)``. rDN)rrrurr rrE)rrBrrrshiftsz DMP.shiftc Cs|jrtd||\}}}}}|||||\}}}}}||||||||\}}}}}|sx|t||||StddS)z5Evaluate functional transformation ``q**n * f(p/q)``.rDN)rrErrv) rrXrrr rrQrrrr transforms$z DMP.transformcCs,|js tt|jt|j|jStddS)z&Computes the Sturm sequence of ``f``. rDN)rrrLrrrr rEr rrrsturmsz DMP.sturmcCs |jst|j|jStddS)z7Computes the Cauchy upper bound on the roots of ``f``. rDN)rrrr rEr rrrcauchy_upper_boundszDMP.cauchy_upper_boundcCs |jst|j|jStddS)z?Computes the Cauchy lower bound on the nonzero roots of ``f``. rDN)rrrr rEr rrrcauchy_lower_boundszDMP.cauchy_lower_boundcCs |jst|j|jStddS)zBComputes the squared Mignotte bound on root separations of ``f``. rDN)rrrr rEr rrrmignotte_sep_bound_squaredszDMP.mignotte_sep_bound_squaredcs.js"fddtjjDStddS)z4Computes greatest factorial factorization of ``f``. csg|]\}}||fqSrrrr rrrrz DMP.gff_list..rDN)rrrr rEr rr rgff_listsz DMP.gff_listcCs$t|j|j|j}|j||jjdS)zComputes ``Norm(f)``.r)rrrr r)rrrrrnormszDMP.normcCs6t|j|j|j\}}}||||j||jjdfS)z$Computes square-free norm of ``f``. r)rrrr r)rr*rrrrrsqf_norm"sz DMP.sqf_normcCs|t|j|j|jS)z$Computes square-free part of ``f``. )rrrrr r rrrsqf_part'sz DMP.sqf_partcs.tjjj|\}}|fdd|DfS)0Returns a list of square-free factors of ``f``. csg|]\}}||fqSrrrr rrr.rz DMP.sqf_list..)rrrr )rr7r"r#rr rsqf_list+sz DMP.sqf_listcs&tjjj|}fdd|DS)rkcsg|]\}}||fqSrrrr rrr3rz(DMP.sqf_list_include..)rrrr )rr7r#rr rsqf_list_include0szDMP.sqf_list_includecs,tjjj\}}|fdd|DfS)0Returns a list of irreducible factors of ``f``. csg|]\}}||fqSrrrr rrr8rz#DMP.factor_list..)rrrr )rr"r#rr r factor_list5szDMP.factor_listcs$tjjj}fdd|DS)rncsg|]\}}||fqSrrrr rrr=rz+DMP.factor_list_include..)rrrr )rr#rr rfactor_list_include:szDMP.factor_list_includecCs|jsv|s@|s&t|j|j||||dSt|j|j||||dSq~|s\t|j|j||||dSt|j|j||||dSntddS)z0Compute isolating intervals for roots of ``f``. )epsinfsupfastz1Cannot isolate roots of a multivariate polynomialN)rrrr rrrr)rr7rqrrrsrtsqfrrr intervals?sz DMP.intervalsc Cs,|js t|j|||j|||dStddS)zu Refine an isolating interval to the given precision. ``eps`` should be a rational number. )rqstepsrtz1Cannot refine a root of a multivariate polynomialN)rrrr r)rr*rHrqrwrtrrr refine_rootPs zDMP.refine_rootcCst|j|j||dS)zReturns ``True`` if ``f`` is an element of the ground domain. 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