\hSrSSKJr SSKJr SSKJr SSKJrJ r SSK J r SSK J r SSKJr SS KJr SS KJrJrJr SS KJrJr SS KJr SS KJr \"SS55rS/rg)z)Implementation of :class:`Domain` class. ) annotations)Any)AlgebraicNumber)Basicsympify)ordered) GROUND_TYPES) DomainElement)lex)UnificationFailedCoercionFailed DomainError) _unify_gens _not_a_coeff)public) is_sequencecD\rSrSr%SrSrS\S'SrS\S'SrS\S'S r S r S r S r S =r rS =rrS =rrS =rrS =rrS =rrS =rrS =rrS =rrS =rr S =r!r"S =r#r$S r%S r&S r'S r(S r)S r*S r+Sr,S \S 'Sr-S \S 'Sr.Sr/Sr0Sr1Sr2\3S5r4Sr5Sr6Sr7ShSjr8Sr9Sr:Sr;SrSr?Sr@S rAS!rBS"rCS#rDS$rES%rFS&rGS'rHS(rIS)rJS*rKS+rLS,rMS-rNS.rOS/rPShS0jrQS1rRS2rSS3rTS4rUS5rVS6rWS7rX\YS8.S9jrZ\YS8.S:jr[S;r\S<r]SS=.S>jr^SiS?jr_SjS@jr`SAraSBrbSCrcSDrdSEreSFrfSGrgSHrhSIriSJrjSKrkSLrlSMrmSNrnSOroSPrpSQrqSRrrSSrsSTrtSUruSVrvSWrwSXrxSYrySZrzS[r{S\r|S]r}S^r~S_rS`rSarShSbjr\rScrSdrShSejrSfrSgrg)kDomaina*Superclass for all domains in the polys domains system. See :ref:`polys-domainsintro` for an introductory explanation of the domains system. The :py:class:`~.Domain` class is an abstract base class for all of the concrete domain types. There are many different :py:class:`~.Domain` subclasses each of which has an associated ``dtype`` which is a class representing the elements of the domain. The coefficients of a :py:class:`~.Poly` are elements of a domain which must be a subclass of :py:class:`~.Domain`. Examples ======== The most common example domains are the integers :ref:`ZZ` and the rationals :ref:`QQ`. >>> from sympy import Poly, symbols, Domain >>> x, y = symbols('x, y') >>> p = Poly(x**2 + y) >>> p Poly(x**2 + y, x, y, domain='ZZ') >>> p.domain ZZ >>> isinstance(p.domain, Domain) True >>> Poly(x**2 + y/2) Poly(x**2 + 1/2*y, x, y, domain='QQ') The domains can be used directly in which case the domain object e.g. (:ref:`ZZ` or :ref:`QQ`) can be used as a constructor for elements of ``dtype``. >>> from sympy import ZZ, QQ >>> ZZ(2) 2 >>> ZZ.dtype # doctest: +SKIP >>> type(ZZ(2)) # doctest: +SKIP >>> QQ(1, 2) 1/2 >>> type(QQ(1, 2)) # doctest: +SKIP The corresponding domain elements can be used with the arithmetic operations ``+,-,*,**`` and depending on the domain some combination of ``/,//,%`` might be usable. For example in :ref:`ZZ` both ``//`` (floor division) and ``%`` (modulo division) can be used but ``/`` (true division) cannot. Since :ref:`QQ` is a :py:class:`~.Field` its elements can be used with ``/`` but ``//`` and ``%`` should not be used. Some domains have a :py:meth:`~.Domain.gcd` method. >>> ZZ(2) + ZZ(3) 5 >>> ZZ(5) // ZZ(2) 2 >>> ZZ(5) % ZZ(2) 1 >>> QQ(1, 2) / QQ(2, 3) 3/4 >>> ZZ.gcd(ZZ(4), ZZ(2)) 2 >>> QQ.gcd(QQ(2,7), QQ(5,3)) 1/21 >>> ZZ.is_Field False >>> QQ.is_Field True There are also many other domains including: 1. :ref:`GF(p)` for finite fields of prime order. 2. :ref:`RR` for real (floating point) numbers. 3. :ref:`CC` for complex (floating point) numbers. 4. :ref:`QQ(a)` for algebraic number fields. 5. :ref:`K[x]` for polynomial rings. 6. :ref:`K(x)` for rational function fields. 7. :ref:`EX` for arbitrary expressions. Each domain is represented by a domain object and also an implementation class (``dtype``) for the elements of the domain. For example the :ref:`K[x]` domains are represented by a domain object which is an instance of :py:class:`~.PolynomialRing` and the elements are always instances of :py:class:`~.PolyElement`. The implementation class represents particular types of mathematical expressions in a way that is more efficient than a normal SymPy expression which is of type :py:class:`~.Expr`. The domain methods :py:meth:`~.Domain.from_sympy` and :py:meth:`~.Domain.to_sympy` are used to convert from :py:class:`~.Expr` to a domain element and vice versa. >>> from sympy import Symbol, ZZ, Expr >>> x = Symbol('x') >>> K = ZZ[x] # polynomial ring domain >>> K ZZ[x] >>> type(K) # class of the domain >>> K.dtype # doctest: +SKIP >>> p_expr = x**2 + 1 # Expr >>> p_expr x**2 + 1 >>> type(p_expr) >>> isinstance(p_expr, Expr) True >>> p_domain = K.from_sympy(p_expr) >>> p_domain # domain element x**2 + 1 >>> type(p_domain) >>> K.to_sympy(p_domain) == p_expr True The :py:meth:`~.Domain.convert_from` method is used to convert domain elements from one domain to another. >>> from sympy import ZZ, QQ >>> ez = ZZ(2) >>> eq = QQ.convert_from(ez, ZZ) >>> type(ez) # doctest: +SKIP >>> type(eq) # doctest: +SKIP Elements from different domains should not be mixed in arithmetic or other operations: they should be converted to a common domain first. The domain method :py:meth:`~.Domain.unify` is used to find a domain that can represent all the elements of two given domains. >>> from sympy import ZZ, QQ, symbols >>> x, y = symbols('x, y') >>> ZZ.unify(QQ) QQ >>> ZZ[x].unify(QQ) QQ[x] >>> ZZ[x].unify(QQ[y]) QQ[x,y] If a domain is a :py:class:`~.Ring` then is might have an associated :py:class:`~.Field` and vice versa. The :py:meth:`~.Domain.get_field` and :py:meth:`~.Domain.get_ring` methods will find or create the associated domain. >>> from sympy import ZZ, QQ, Symbol >>> x = Symbol('x') >>> ZZ.has_assoc_Field True >>> ZZ.get_field() QQ >>> QQ.has_assoc_Ring True >>> QQ.get_ring() ZZ >>> K = QQ[x] >>> K QQ[x] >>> K.get_field() QQ(x) See also ======== DomainElement: abstract base class for domain elements construct_domain: construct a minimal domain for some expressions Nz type | NonedtyperzerooneFTz str | Nonerepaliasc[eNNotImplementedErrorselfs R/var/www/auris/envauris/lib/python3.13/site-packages/sympy/polys/domains/domain.py__init__Domain.__init__gs!!cUR$r)rrs r!__str__Domain.__str__js xxr$c[U5$r)strrs r!__repr__Domain.__repr__ms 4yr$cX[URRUR45$r)hash __class____name__rrs r!__hash__Domain.__hash__ps T^^,,djj9::r$c UR"U6$rrr argss r!new Domain.newszz4  r$cUR$)z#Alias for :py:attr:`~.Domain.dtype`r3rs r!tp Domain.tpvszzr$c UR"U6$)z7Construct an element of ``self`` domain from ``args``. )r6r4s r!__call__Domain.__call__{sxxr$c UR"U6$rr3r4s r!normal Domain.normalr8r$c URbSUR-nOSURR-n[X5nUb U"X5nUbU$[ SU<S[ U5<SU<SU<35e)z=Convert ``element`` to ``self.dtype`` given the base domain. from_Cannot convert of type z from  to )rr.r/getattrr type)r elementbasemethod_convertresults r! convert_fromDomain.convert_fromsp :: !tzz)Ft~~666F4(  g,F! WVZ[bVceikopqqr$cXUb/[U5(a[SU-5eURX5$URU5(aU$[U5(a[SU-5eSSKJnJnJnJn URU5(aURX5$[U[5(aURU"U5U5$[S:waV[XR5(aURX5$[XR5(aURX5$[U[5(aU"5nURU"U5U5$[U[5(aU"5nURU"U5U5$[U5R S:XaU"5nURU"U5U5$[U5R S:XaU"5nURU"U5U5$[U["5(aURXR%55$UR&(a1[)USS5(aUR+UR-55$[U[.5(aUR1U5$[7U5(d2[9US S 9n[U[.5(aUR1U5$[S U<S [U5<S U<35e![2[44a N2f=f![2[44a NHf=f)z'Convert ``element`` to ``self.dtype``. z%s is not in any domainr)ZZQQ RealField ComplexFieldpythonmpfmpc is_groundFT)strictrDrErF)rr rNof_typesympy.polys.domainsrQrRrSrT isinstanceintr r:floatcomplexrHr/r parent is_NumericalrGconvertLCr from_sympy TypeError ValueErrorrr)r rIrJrQrRrSrTr`s r!rbDomain.converts  G$$$%>%HII$$W3 3 << N   !:W!DE EGG ::g  $$W1 1 gs # #$$R["5 5 8 #'55))((55'55))((55 gu % %[F$$VG_f= = gw ' '!^F$$VG_f= = = ! !U *[F$$VG_f= = = ! !U *!^F$$VG_f= = g} - -$$Wnn.>? ?   +u!E!E<< - - gu % % w//w''%gd;G!'511#w772 WdSZm]abccz*  ":.s$ L//LLLL)(L)c,[XR5$)z%Check if ``a`` is of type ``dtype``. )r\r:)r rIs r!rZDomain.of_types'77++r$ct[U5(a[eURU5 g![a gf=f)z'Check if ``a`` belongs to this domain. FT)rr rbr as r! __contains__Domain.__contains__s: A$$ LLO  s '* 77c[e)aConvert domain element *a* to a SymPy expression (Expr). Explanation =========== Convert a :py:class:`~.Domain` element *a* to :py:class:`~.Expr`. Most public SymPy functions work with objects of type :py:class:`~.Expr`. The elements of a :py:class:`~.Domain` have a different internal representation. It is not possible to mix domain elements with :py:class:`~.Expr` so each domain has :py:meth:`~.Domain.to_sympy` and :py:meth:`~.Domain.from_sympy` methods to convert its domain elements to and from :py:class:`~.Expr`. Parameters ========== a: domain element An element of this :py:class:`~.Domain`. Returns ======= expr: Expr A normal SymPy expression of type :py:class:`~.Expr`. Examples ======== Construct an element of the :ref:`QQ` domain and then convert it to :py:class:`~.Expr`. >>> from sympy import QQ, Expr >>> q_domain = QQ(2) >>> q_domain 2 >>> q_expr = QQ.to_sympy(q_domain) >>> q_expr 2 Although the printed forms look similar these objects are not of the same type. >>> isinstance(q_domain, Expr) False >>> isinstance(q_expr, Expr) True Construct an element of :ref:`K[x]` and convert to :py:class:`~.Expr`. >>> from sympy import Symbol >>> x = Symbol('x') >>> K = QQ[x] >>> x_domain = K.gens[0] # generator x as a domain element >>> p_domain = x_domain**2/3 + 1 >>> p_domain 1/3*x**2 + 1 >>> p_expr = K.to_sympy(p_domain) >>> p_expr x**2/3 + 1 The :py:meth:`~.Domain.from_sympy` method is used for the opposite conversion from a normal SymPy expression to a domain element. >>> p_domain == p_expr False >>> K.from_sympy(p_expr) == p_domain True >>> K.to_sympy(p_domain) == p_expr True >>> K.from_sympy(K.to_sympy(p_domain)) == p_domain True >>> K.to_sympy(K.from_sympy(p_expr)) == p_expr True The :py:meth:`~.Domain.from_sympy` method makes it easier to construct domain elements interactively. >>> from sympy import Symbol >>> x = Symbol('x') >>> K = QQ[x] >>> K.from_sympy(x**2/3 + 1) 1/3*x**2 + 1 See also ======== from_sympy convert_from rrks r!to_sympyDomain.to_sympys v"!r$c[e)ajConvert a SymPy expression to an element of this domain. Explanation =========== See :py:meth:`~.Domain.to_sympy` for explanation and examples. Parameters ========== expr: Expr A normal SymPy expression of type :py:class:`~.Expr`. Returns ======= a: domain element An element of this :py:class:`~.Domain`. See also ======== to_sympy convert_from rrks r!rdDomain.from_sympyBs 4"!r$c([XRS9$)N)start)sumrr4s r!rv Domain.sum^s4yy))r$cgz.Convert ``ModularInteger(int)`` to ``dtype``. NK1rlK0s r!from_FFDomain.from_FFar$cgryrzr{s r!from_FF_pythonDomain.from_FF_pythonerr$cg)z.Convert a Python ``int`` object to ``dtype``. Nrzr{s r!from_ZZ_pythonDomain.from_ZZ_pythonirr$cg)z3Convert a Python ``Fraction`` object to ``dtype``. Nrzr{s r!from_QQ_pythonDomain.from_QQ_pythonmrr$cg)z.Convert ``ModularInteger(mpz)`` to ``dtype``. Nrzr{s r! from_FF_gmpyDomain.from_FF_gmpyqrr$cg)z,Convert a GMPY ``mpz`` object to ``dtype``. Nrzr{s r! from_ZZ_gmpyDomain.from_ZZ_gmpyurr$cg)z,Convert a GMPY ``mpq`` object to ``dtype``. Nrzr{s r! from_QQ_gmpyDomain.from_QQ_gmpyyrr$cg)z,Convert a real element object to ``dtype``. Nrzr{s r!from_RealFieldDomain.from_RealField}rr$cg)z(Convert a complex element to ``dtype``. Nrzr{s r!from_ComplexFieldDomain.from_ComplexFieldrr$cg)z*Convert an algebraic number to ``dtype``. Nrzr{s r!from_AlgebraicFieldDomain.from_AlgebraicFieldrr$crUR(a&URURUR5$g)#Convert a polynomial to ``dtype``. N)rXrbrcdomr{s r!from_PolynomialRingDomain.from_PolynomialRings' ;;::addBFF+ + r$cg)z*Convert a rational function to ``dtype``. Nrzr{s r!from_FractionFieldDomain.from_FractionFieldrr$cNURURUR5$)z.Convert an ``ExtensionElement`` to ``dtype``. )rNrringr{s r!from_MonogenicFiniteExtension$Domain.from_MonogenicFiniteExtensionsquubgg..r$c8URUR5$z&Convert a ``EX`` object to ``dtype``. )rdexr{s r!from_ExpressionDomainDomain.from_ExpressionDomains}}QTT""r$c$URU5$r)rdr{s r!from_ExpressionRawDomainDomain.from_ExpressionRawDomains}}Qr$cUR5S::a*URUR5UR5$g)rrN)degreerbrcrr{s r!from_GlobalPolynomialRing Domain.from_GlobalPolynomialRings/ 88:?::addfbff- - r$c$URX5$r)rr{s r!from_GeneralizedPolynomialRing%Domain.from_GeneralizedPolynomialRings$$Q++r$c BUR(a&[UR5[U5-(d7UR(aG[UR5[U5-(a![SU<SU<S[ U5<S35eUR U5$)N Cannot unify  with z, given z generators) is_Compositesetsymbolsr tupleunify)r}r|rs r!unify_with_symbolsDomain.unify_with_symbolssf OORZZ3w,>FOO&&__&F ??BOOr7J7JbNbNb,,C,,C P  &v' '6E**r$c( UbURX5$X:XaU$UR(aUR(dFUR5UR5:wa[SU<SU<35eUR U5$UR (aU$UR (aU$UR (aU$UR (aU$UR(dUR(aUR(aXpUR(aN[[URUR/55SUR:XaXpURU5$URUR5nURRU5nURU5$UR (dUR (aUR U5$UR"(aXpUR"(aVUR"(dUR$(a2UR&UR&:aU$SSKJn U"UR&S9$U$UR$(aXpUR$(aiUR$(aUR&UR&:aU$U$UR,(dUR.(aSSKJn U"UR&S9$U$UR0(aXpUR0(aUR,(aUR35nUR.(aUR55nUR0(aTUR6"UR8RUR85/[;UR<UR<5Q76$U$UR.(aU$UR.(aU$UR,(a#UR>(aUR35nU$UR,(a#UR>(aUR35nU$UR>(aU$UR>(aU$UR@(aU$UR@(aU$SSK!J"n U$)z Construct a minimal domain that contains elements of ``K0`` and ``K1``. Known domains (from smallest to largest): - ``GF(p)`` - ``ZZ`` - ``QQ`` - ``RR(prec, tol)`` - ``CC(prec, tol)`` - ``ALG(a, b, c)`` - ``K[x, y, z]`` - ``K(x, y, z)`` - ``EX`` rrr)rT)prec)EX)#rhas_CharacteristicZerocharacteristicr ris_EXRAWis_EXis_FiniteExtensionlistrmodulus set_domaindropsymbolrrris_ComplexField is_RealField precision sympy.polys.domains.complexfieldrTis_GaussianRingis_GaussianFieldis_AlgebraicField get_fieldas_AlgebraicFieldr.rrorig_extis_RationalFieldis_IntegerRingr[r)r}r|rrTrs r!r Domain.unifys3"  ((5 5 8I))b.G.G  "b&7&7&99'R(LMM %%b) ) ;;I ;;I 88I 88I B$9$9$$B$$RZZ 89:1=K}}R((WWRYY'YY__R(}}R(( ??boo%%b) )    !!R__<<2<</IM'R\\:: ?? ??<<2<</II##r':':I#66    !!\\^""))+##||BFFLL$8a;r{{TVT_T_;`aa  I  I  ""\\^I  ""\\^I  I  I  I  I* r$cb[U[5=(a URUR:H$)z0Returns ``True`` if two domains are equivalent. )r\rrr others r!__eq__ Domain.__eq__Ns#%(FTZZ5;;-FFr$c X:X+$)z1Returns ``False`` if two domains are equivalent. rzrs r!__ne__ Domain.__ne__Ss   r$c/nUHQn[U[5(a"URURU55 M:URU"U55 MS U$)z5Rersively apply ``self`` to all elements of ``seq``. )r\rappendmap)r seqrMelts r!r Domain.mapWsIC#t$$ dhhsm, d3i(   r$c[SU-5e)z)Returns a ring associated with ``self``. z#there is no ring associated with %srrs r!rDomain.get_ringcs?$FGGr$c[SU-5e)z*Returns a field associated with ``self``. z$there is no field associated with %srrs r!rDomain.get_fieldgs@4GHHr$cU$)z2Returns an exact domain associated with ``self``. rzrs r! get_exactDomain.get_exactks r$cd[US5(aUR"U6$URU5$)z0The mathematical way to make a polynomial ring. __iter__)hasattr poly_ringr rs r! __getitem__Domain.__getitem__os- 7J ' '>>7+ +>>'* *r$)rc SSKJn U"XU5$z(Returns a polynomial ring, i.e. `K[X]`. r)PolynomialRing)"sympy.polys.domains.polynomialringr )r rrr s r!rDomain.poly_ringvsEdU33r$c SSKJn U"XU5$z'Returns a fraction field, i.e. `K(X)`. r) FractionField)!sympy.polys.domains.fractionfieldr)r rrrs r! frac_fieldDomain.frac_field{sCTE22r$c&SSKJn U"U/UQ70UD6$r )rr )r rkwargsr s r! old_poly_ringDomain.old_poly_ringsId7W777r$c&SSKJn U"U/UQ70UD6$r)%sympy.polys.domains.old_fractionfieldr)r rrrs r!old_frac_fieldDomain.old_frac_fieldsGT6G6v66r$rc[SU-5e)z6Returns an algebraic field, i.e. `K(\alpha, \ldots)`. z%Cannot create algebraic field over %sr)r r extensions r!algebraic_fieldDomain.algebraic_fieldsADHIIr$cNSSKJn U"X5n[XRS9nURXbS9$)a Convenience method to construct an algebraic extension on a root of a polynomial, chosen by root index. Parameters ========== poly : :py:class:`~.Poly` The polynomial whose root generates the extension. alias : str, optional (default=None) Symbol name for the generator of the extension. E.g. "alpha" or "theta". root_index : int, optional (default=-1) Specifies which root of the polynomial is desired. The ordering is as defined by the :py:class:`~.ComplexRootOf` class. The default of ``-1`` selects the most natural choice in the common cases of quadratic and cyclotomic fields (the square root on the positive real or imaginary axis, resp. $\mathrm{e}^{2\pi i/n}$). Examples ======== >>> from sympy import QQ, Poly >>> from sympy.abc import x >>> f = Poly(x**2 - 2) >>> K = QQ.alg_field_from_poly(f) >>> K.ext.minpoly == f True >>> g = Poly(8*x**3 - 6*x - 1) >>> L = QQ.alg_field_from_poly(g, "alpha") >>> L.ext.minpoly == g True >>> L.to_sympy(L([1, 1, 1])) alpha**2 + alpha + 1 r)CRootOfr)sympy.polys.rootoftoolsr#rr )r polyr root_indexr#rootalphas r!alg_field_from_polyDomain.alg_field_from_polys0J 4t(2##E#77r$cfSSKJn U(aU[U5- nURU"X5UUS9$)a Convenience method to construct a cyclotomic field. Parameters ========== n : int Construct the nth cyclotomic field. ss : boolean, optional (default=False) If True, append *n* as a subscript on the alias string. alias : str, optional (default="zeta") Symbol name for the generator. gen : :py:class:`~.Symbol`, optional (default=None) Desired variable for the cyclotomic polynomial that defines the field. If ``None``, a dummy variable will be used. root_index : int, optional (default=-1) Specifies which root of the polynomial is desired. The ordering is as defined by the :py:class:`~.ComplexRootOf` class. The default of ``-1`` selects the root $\mathrm{e}^{2\pi i/n}$. Examples ======== >>> from sympy import QQ, latex >>> K = QQ.cyclotomic_field(5) >>> K.to_sympy(K([-1, 1])) 1 - zeta >>> L = QQ.cyclotomic_field(7, True) >>> a = L.to_sympy(L([-1, 1])) >>> print(a) 1 - zeta7 >>> print(latex(a)) 1 - \zeta_{7} r)cyclotomic_poly)rr&)sympy.polys.specialpolysr,r)r))r nssrgenr&r,s r!cyclotomic_fieldDomain.cyclotomic_fields<H = SVOE''(?u3=(? ?r$c[e)z$Inject generators into this domain. rrs r!inject Domain.inject!!r$c4UR(aU$[e)z"Drop generators from this domain. ) is_Simplerrs r!r Domain.drops >>K!!r$cU(+$)zReturns True if ``a`` is zero. rzrks r!is_zeroDomain.is_zeros u r$cXR:H$)zReturns True if ``a`` is one. )rrks r!is_one Domain.is_onesHH}r$c US:$)z#Returns True if ``a`` is positive. rrzrks r! is_positiveDomain.is_positive 1u r$c US:$)z#Returns True if ``a`` is negative. rrzrks r! is_negativeDomain.is_negativerCr$c US:*$)z'Returns True if ``a`` is non-positive. rrzrks r!is_nonpositiveDomain.is_nonpositive Av r$c US:$)z'Returns True if ``a`` is non-negative. rrzrks r!is_nonnegativeDomain.is_nonnegativerJr$c`URU5(a UR*$UR$r)rErrks r!canonical_unitDomain.canonical_units(   A  HH9 88Or$c[U5$)z.Absolute value of ``a``, implies ``__abs__``. )absrks r!rR Domain.abs s 1v r$cU*$)z,Returns ``a`` negated, implies ``__neg__``. rzrks r!neg Domain.neg r r$cU7$)z-Returns ``a`` positive, implies ``__pos__``. rzrks r!pos Domain.posrWr$c X-$)z.Sum of ``a`` and ``b``, implies ``__add__``. rzr rlbs r!add Domain.add u r$c X- $)z5Difference of ``a`` and ``b``, implies ``__sub__``. rzr\s r!sub Domain.subr`r$c X-$)z2Product of ``a`` and ``b``, implies ``__mul__``. rzr\s r!mul Domain.mulr`r$c X-$)z2Raise ``a`` to power ``b``, implies ``__pow__``. rzr\s r!pow Domain.pow" v r$c[e)a Exact quotient of *a* and *b*. Analogue of ``a / b``. Explanation =========== This is essentially the same as ``a / b`` except that an error will be raised if the division is inexact (if there is any remainder) and the result will always be a domain element. When working in a :py:class:`~.Domain` that is not a :py:class:`~.Field` (e.g. :ref:`ZZ` or :ref:`K[x]`) ``exquo`` should be used instead of ``/``. The key invariant is that if ``q = K.exquo(a, b)`` (and ``exquo`` does not raise an exception) then ``a == b*q``. Examples ======== We can use ``K.exquo`` instead of ``/`` for exact division. >>> from sympy import ZZ >>> ZZ.exquo(ZZ(4), ZZ(2)) 2 >>> ZZ.exquo(ZZ(5), ZZ(2)) Traceback (most recent call last): ... ExactQuotientFailed: 2 does not divide 5 in ZZ Over a :py:class:`~.Field` such as :ref:`QQ`, division (with nonzero divisor) is always exact so in that case ``/`` can be used instead of :py:meth:`~.Domain.exquo`. >>> from sympy import QQ >>> QQ.exquo(QQ(5), QQ(2)) 5/2 >>> QQ(5) / QQ(2) 5/2 Parameters ========== a: domain element The dividend b: domain element The divisor Returns ======= q: domain element The exact quotient Raises ====== ExactQuotientFailed: if exact division is not possible. ZeroDivisionError: when the divisor is zero. See also ======== quo: Analogue of ``a // b`` rem: Analogue of ``a % b`` div: Analogue of ``divmod(a, b)`` Notes ===== Since the default :py:attr:`~.Domain.dtype` for :ref:`ZZ` is ``int`` (or ``mpz``) division as ``a / b`` should not be used as it would give a ``float`` which is not a domain element. >>> ZZ(4) / ZZ(2) # doctest: +SKIP 2.0 >>> ZZ(5) / ZZ(2) # doctest: +SKIP 2.5 On the other hand with `SYMPY_GROUND_TYPES=flint` elements of :ref:`ZZ` are ``flint.fmpz`` and division would raise an exception: >>> ZZ(4) / ZZ(2) # doctest: +SKIP Traceback (most recent call last): ... TypeError: unsupported operand type(s) for /: 'fmpz' and 'fmpz' Using ``/`` with :ref:`ZZ` will lead to incorrect results so :py:meth:`~.Domain.exquo` should be used instead. rr\s r!exquo Domain.exquo&s r"!r$c[e)aQuotient of *a* and *b*. Analogue of ``a // b``. ``K.quo(a, b)`` is equivalent to ``K.div(a, b)[0]``. See :py:meth:`~.Domain.div` for more explanation. See also ======== rem: Analogue of ``a % b`` div: Analogue of ``divmod(a, b)`` exquo: Analogue of ``a / b`` rr\s r!quo Domain.quo "!r$c[e)aModulo division of *a* and *b*. Analogue of ``a % b``. ``K.rem(a, b)`` is equivalent to ``K.div(a, b)[1]``. See :py:meth:`~.Domain.div` for more explanation. See also ======== quo: Analogue of ``a // b`` div: Analogue of ``divmod(a, b)`` exquo: Analogue of ``a / b`` rr\s r!rem Domain.remrqr$c[e)a{Quotient and remainder for *a* and *b*. Analogue of ``divmod(a, b)`` Explanation =========== This is essentially the same as ``divmod(a, b)`` except that is more consistent when working over some :py:class:`~.Field` domains such as :ref:`QQ`. When working over an arbitrary :py:class:`~.Domain` the :py:meth:`~.Domain.div` method should be used instead of ``divmod``. The key invariant is that if ``q, r = K.div(a, b)`` then ``a == b*q + r``. The result of ``K.div(a, b)`` is the same as the tuple ``(K.quo(a, b), K.rem(a, b))`` except that if both quotient and remainder are needed then it is more efficient to use :py:meth:`~.Domain.div`. Examples ======== We can use ``K.div`` instead of ``divmod`` for floor division and remainder. >>> from sympy import ZZ, QQ >>> ZZ.div(ZZ(5), ZZ(2)) (2, 1) If ``K`` is a :py:class:`~.Field` then the division is always exact with a remainder of :py:attr:`~.Domain.zero`. >>> QQ.div(QQ(5), QQ(2)) (5/2, 0) Parameters ========== a: domain element The dividend b: domain element The divisor Returns ======= (q, r): tuple of domain elements The quotient and remainder Raises ====== ZeroDivisionError: when the divisor is zero. See also ======== quo: Analogue of ``a // b`` rem: Analogue of ``a % b`` exquo: Analogue of ``a / b`` Notes ===== If ``gmpy`` is installed then the ``gmpy.mpq`` type will be used as the :py:attr:`~.Domain.dtype` for :ref:`QQ`. The ``gmpy.mpq`` type defines ``divmod`` in a way that is undesirable so :py:meth:`~.Domain.div` should be used instead of ``divmod``. >>> a = QQ(1) >>> b = QQ(3, 2) >>> a # doctest: +SKIP mpq(1,1) >>> b # doctest: +SKIP mpq(3,2) >>> divmod(a, b) # doctest: +SKIP (mpz(0), mpq(1,1)) >>> QQ.div(a, b) # doctest: +SKIP (mpq(2,3), mpq(0,1)) Using ``//`` or ``%`` with :ref:`QQ` will lead to incorrect results so :py:meth:`~.Domain.div` should be used instead. rr\s r!div Domain.divs h"!r$c[e)z5Returns inversion of ``a mod b``, implies something. rr\s r!invert Domain.invertr6r$c[e)z!Returns ``a**(-1)`` if possible. rrks r!revert Domain.revertr6r$c[e)zReturns numerator of ``a``. rrks r!numer Domain.numerr6r$c[e)zReturns denominator of ``a``. rrks r!denom Domain.denomr6r$c0URX5up4nX54$)z&Half extended GCD of ``a`` and ``b``. )gcdex)r rlr]sths r! half_gcdexDomain.half_gcdexs**Q"at r$c[e)z!Extended GCD of ``a`` and ``b``. rr\s r!r Domain.gcdex r6r$cpURX5nURX5nURX#5nX4U4$)z.Returns GCD and cofactors of ``a`` and ``b``. )gcdro)r rlr]rcfacfbs r! cofactorsDomain.cofactorss5hhqnhhqhhq}r$c[e)z Returns GCD of ``a`` and ``b``. rr\s r!r Domain.gcdr6r$c[e)z Returns LCM of ``a`` and ``b``. rr\s r!lcm Domain.lcmr6r$c[e)z#Returns b-base logarithm of ``a``. rr\s r!log Domain.logr6r$c[e)aReturns a (possibly inexact) square root of ``a``. Explanation =========== There is no universal definition of "inexact square root" for all domains. It is not recommended to implement this method for domains other then :ref:`ZZ`. See also ======== exsqrt rrks r!sqrt Domain.sqrt!rqr$c[e)a>Returns whether ``a`` is a square in the domain. Explanation =========== Returns ``True`` if there is an element ``b`` in the domain such that ``b * b == a``, otherwise returns ``False``. For inexact domains like :ref:`RR` and :ref:`CC`, a tiny difference in this equality can be tolerated. See also ======== exsqrt rrks r! is_squareDomain.is_square0s "!r$c[e)aPrincipal square root of a within the domain if ``a`` is square. Explanation =========== The implementation of this method should return an element ``b`` in the domain such that ``b * b == a``, or ``None`` if there is no such ``b``. For inexact domains like :ref:`RR` and :ref:`CC`, a tiny difference in this equality can be tolerated. The choice of a "principal" square root should follow a consistent rule whenever possible. See also ======== sqrt, is_square rrks r!exsqrt Domain.exsqrt@s "!r$c FURU5R"U40UD6$)z*Returns numerical approximation of ``a``. )rpevalf)r rlroptionss r!r Domain.evalfQs!}}Q%%d6g66r$cU$rrzrks r!real Domain.realWsr$cUR$r)rrks r!imag Domain.imagZs yyr$c X:H$)z+Check if ``a`` and ``b`` are almost equal. rz)r rlr] tolerances r!almosteqDomain.almosteq]rjr$c[S5e)z*Return the characteristic of this domain. zcharacteristic()rrs r!rDomain.characteristicas!"455r$rzr)N)FzetaNr)r/ __module__ __qualname____firstlineno____doc__r__annotations__rris_Ringrrhas_assoc_Fieldis_FiniteFieldis_FFris_ZZris_QQris_ZZ_Iris_QQ_Iris_RRris_CCr is_Algebraicris_Polyris_Fracis_SymbolicDomainris_SymbolicRawDomainrris_Exactrar8ris_PIDrrrr"r&r*r0r6propertyr:r=r@rNrbrZrmrprdrvr~rrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrr r)r1r4rr;r>rArErHrLrOrRrUrYr^rbrerhrlrorsrvryr|rrrrrrrrrrrrr.rrrr__static_attributes__rzr$r!rrshTE;,D# CO G$H"N O #"NU""NU$$u %%Og!&&w  L5##Oe',, "''!&&w %%&++8HLIL F"#CE:";!!r"AdF, ["z"8*, /# . , +BBG ! HI+),4 *-3 8 7 15J(8T(?T""  Y"v " "T"l"""" """" "" ""7 A6r$rN)r __future__rtypingrsympy.core.numbersr sympy.corerrsympy.core.sortingrsympy.external.gmpyr !sympy.polys.domains.domainelementr sympy.polys.orderingsr sympy.polys.polyerrorsr r rsympy.polys.polyutilsrrsympy.utilitiesrsympy.utilities.iterablesrr__all__rzr$r!rsU/".%&,;%QQ;"1P6P6P6f* *r$