o o h@sdZddlmZddlmZddlmZddlmZm Z ddl m Z ddl m Z ddlmZdd lmZdd lmZmZmZdd lmZmZdd lmZdd lmZeGdddZdgZdS)z)Implementation of :class:`Domain` class. ) annotations)Any)AlgebraicNumber)Basicsympify)ordered) GROUND_TYPES) DomainElement)lex)UnificationFailedCoercionFailed DomainError) _unify_gens _not_a_coeff)public) is_sequencec@seZdZUdZdZded< dZded< dZded< dZ dZ dZ dZ dZ Z dZZdZZdZZdZZdZZdZZdZZdZZdZZdZ Z!dZ"Z#dZ$d Z%dZ&dZ'dZ(dZ) dZ*dZ+d ed <dZ,d ed <d dZ-ddZ.ddZ/ddZ0ddZ1e2ddZ3ddZ4ddZ5ddZ6ddd Z7d!d"Z8d#d$Z9d%d&Z:d'd(Z;d)d*Zd/d0Z?d1d2Z@d3d4ZAd5d6ZBd7d8ZCd9d:ZDd;d<ZEd=d>ZFd?d@ZGdAdBZHdCdDZIdEdFZJdGdHZKdIdJZLdKdLZMdMdNZNdOdPZOddQdRZPdSdTZQdUdVZRdWdXZSdYdZZTd[d\ZUd]d^ZVd_d`ZWeXdadbdcZYeXdadddeZZdfdgZ[dhdiZ\ddjdkdlZ]ddndoZ^ddqdrZ_dsdtZ`dudvZadwdxZbdydzZcd{d|Zdd}d~ZeddZfddZgddZhddZiddZjddZkddZlddZmddZnddZoddZpddZqddZrddZsddZtddZuddZvddZwddZxddZyddZzddZ{ddZ|ddZ}ddZ~ddZddZdddZeZddZddZdddZddZdS)DomainaySuperclass 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 # class of the elements >>> 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 | NonerepaliascCtNNotImplementedErrorselfrn/var/www/html/construction_image-detection-poc/venv/lib/python3.10/site-packages/sympy/polys/domains/domain.py__init__gzDomain.__init__cC|jSr)rrrrr__str__jzDomain.__str__cCst|Sr)strrrrr__repr__mszDomain.__repr__cCst|jj|jfSr)hash __class____name__rrrrr__hash__pszDomain.__hash__cG |j|Srrrargsrrrnews z Domain.newcCr")z#Alias for :py:attr:`~.Domain.dtype`r,rrrrtpvsz Domain.tpcGr+)z7Construct an element of ``self`` domain from ``args``. )r/r-rrr__call__{ zDomain.__call__cGr+rr,r-rrrnormalr0z Domain.normalcCsb|jdur d|j}nd|jj}t||}|dur%|||}|dur%|Std|t|||f)z=Convert ``element`` to ``self.dtype`` given the base domain. Nfrom_z*Cannot convert %s of type %s from %s to %s)rr(r)getattrr type)relementbasemethod_convertresultrrr convert_froms     zDomain.convert_fromc Cs|durt|rtd||||S||r|St|r%td|ddlm}m}m}m}||r<|||St |t rI||||St dkret ||j rY|||St ||j re|||St |t rw|dd}||||St |tr|dd}||||St |tr|||S|jrt|ddr||St |trz||WSttfyYn$wt|szt|d d }t |tr||WSWn ttfyYnwtd |t||f) z'Convert ``element`` to ``self.dtype``. Nz%s is not in any domainr)ZZQQ RealField ComplexFieldpythonF)tol is_groundT)strictz"Cannot convert %s of type %s to %s)rr r=of_typesympy.polys.domainsr>r?r@rA isinstanceintrr1floatcomplexr parent is_Numericalr6convertLCr from_sympy TypeError ValueErrorrrr7)rr8r9r>r?r@rArLrrrrNsX                     zDomain.convertcCs t||jS)z%Check if ``a`` is of type ``dtype``. )rHr1)rr8rrrrF zDomain.of_typecCs2zt|rt||WdStyYdSw)z'Check if ``a`` belongs to this domain. FT)rr rNrarrr __contains__s  zDomain.__contains__cCr)a Convert 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 rrTrrrto_sympys[zDomain.to_sympycCr)aConvert 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 rrTrrrrP:szDomain.from_sympycCst||jdS)N)start)sumrr-rrrrYVsz Domain.sumcCdSz.Convert ``ModularInteger(int)`` to ``dtype``. NrK1rUK0rrrfrom_FFYzDomain.from_FFcCrZr[rr\rrrfrom_FF_python]r`zDomain.from_FF_pythoncCrZ)z.Convert a Python ``int`` object to ``dtype``. Nrr\rrrfrom_ZZ_pythonar`zDomain.from_ZZ_pythoncCrZ)z3Convert a Python ``Fraction`` object to ``dtype``. Nrr\rrrfrom_QQ_pythoner`zDomain.from_QQ_pythoncCrZ)z.Convert ``ModularInteger(mpz)`` to ``dtype``. Nrr\rrr from_FF_gmpyir`zDomain.from_FF_gmpycCrZ)z,Convert a GMPY ``mpz`` object to ``dtype``. Nrr\rrr from_ZZ_gmpymr`zDomain.from_ZZ_gmpycCrZ)z,Convert a GMPY ``mpq`` object to ``dtype``. Nrr\rrr from_QQ_gmpyqr`zDomain.from_QQ_gmpycCrZ)z,Convert a real element object to ``dtype``. Nrr\rrrfrom_RealFieldur`zDomain.from_RealFieldcCrZ)z(Convert a complex element to ``dtype``. Nrr\rrrfrom_ComplexFieldyr`zDomain.from_ComplexFieldcCrZ)z*Convert an algebraic number to ``dtype``. Nrr\rrrfrom_AlgebraicField}r`zDomain.from_AlgebraicFieldcCs|jr ||j|jSdS)#Convert a polynomial to ``dtype``. N)rDrNrOdomr\rrrfrom_PolynomialRingszDomain.from_PolynomialRingcCrZ)z*Convert a rational function to ``dtype``. Nrr\rrrfrom_FractionFieldr`zDomain.from_FractionFieldcCs||j|jS)z.Convert an ``ExtensionElement`` to ``dtype``. )r=rringr\rrrfrom_MonogenicFiniteExtensionsz$Domain.from_MonogenicFiniteExtensioncCs ||jSz&Convert a ``EX`` object to ``dtype``. )rPexr\rrrfrom_ExpressionDomainrSzDomain.from_ExpressionDomaincCs ||Srp)rPr\rrrfrom_ExpressionRawDomainr3zDomain.from_ExpressionRawDomaincCs"|dkr|||jSdS)rjrN)degreerNrOrkr\rrrfrom_GlobalPolynomialRings z Domain.from_GlobalPolynomialRingcCs |||Sr)rmr\rrrfrom_GeneralizedPolynomialRings z%Domain.from_GeneralizedPolynomialRingcCsP|jr t|jt|@s|jr#t|jt|@r#td||t|f||S)Nz,Cannot unify %s with %s, given %s generators) is_Compositesetsymbolsr tupleunify)r^r]ryrrrunify_with_symbolss0 zDomain.unify_with_symbolsc Cs|jr|jn|}|jr|jn|}|jr|jnd}|jr|jnd}||}t||}|jr0|jn|j}|jr9|js?|jrO|jrO|jrE|jsO|jrO|j rO| }|jr_|jr[|js[|jr_|j } n|j } ddl m } | | krq| ||S| |||S)z2Unify two domains where at least one is composite.rr)GlobalPolynomialRing)rwrkryr{rorderis_FractionFieldis_PolynomialRingis_Fieldhas_assoc_Ringget_ringr(&sympy.polys.domains.old_polynomialringr}) r^r] K0_ground K1_ground K0_symbols K1_symbolsdomainryr~clsr}rrrunify_composites8      zDomain.unify_compositecCs|dur |||S||kr|S|jr|js+||kr&td||f||S|jr0|S|jr5|S|jr:|S|jr?|S|jsE|jrz|jrM||}}|jritt |j |j gd|j krd||}}| |S| |j }|j|}| |S|js|jr||Sdd}|jr||}}|jr|js|jr||j||S|S|jr||}}|jr|jr||j||S|js|jrddlm}||j|jdS|S|jr||}}|jr|jr|}|jr|}|jr|j|j|jgt|j|jRS|S|jr|S|jr |S|jr|j r|}|S|jr'|j r%|}|S|j r-|S|j r3|S|j!r9|S|j!r?|Sdd l"m#}|S) aZ 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`` NzCannot unify %s with %scSs(t|j|j}t|j|j}|||dS)NprecrC)max precision tolerance)rr^r]rrCrrr mkinexacts zDomain.unify..mkinexactr)rAr)EX)$r|has_CharacteristicZerocharacteristicr ris_EXRAWis_EXis_FiniteExtensionlistrmodulus set_domaindropsymbolrr{rwis_ComplexField is_RealFieldr(is_GaussianRingis_GaussianField sympy.polys.domains.complexfieldrArris_AlgebraicField get_fieldas_AlgebraicFieldrkrorig_extis_RationalFieldis_IntegerRingrGr)r^r]ryrrArrrrr{s                  & z Domain.unifycCst|to |j|jkS)z0Returns ``True`` if two domains are equivalent. )rHrrrotherrrr__eq__Dsz Domain.__eq__cCs ||k S)z1Returns ``False`` if two domains are equivalent. rrrrr__ne__Ir3z Domain.__ne__cCs<g}|D]}t|tr|||q|||q|S)z5Rersively apply ``self`` to all elements of ``seq``. )rHrappendmap)rseqr<eltrrrrMs  z Domain.mapcC td|)z)Returns a ring associated with ``self``. z#there is no ring associated with %sr rrrrrYrSzDomain.get_ringcCr)z*Returns a field associated with ``self``. z$there is no field associated with %srrrrrr]rSzDomain.get_fieldcCs|S)z2Returns an exact domain associated with ``self``. rrrrr get_exactar`zDomain.get_exactcCst|dr |j|S||S)z0The mathematical way to make a polynomial ring. __iter__)hasattr poly_ringrryrrr __getitem__es   zDomain.__getitem__)r~cGddlm}||||Sz(Returns a polynomial ring, i.e. `K[X]`. r)PolynomialRing)"sympy.polys.domains.polynomialringr)rr~ryrrrrrl  zDomain.poly_ringcGrz'Returns a fraction field, i.e. `K(X)`. r) FractionField)!sympy.polys.domains.fractionfieldr)rr~ryrrrr frac_fieldqrzDomain.frac_fieldcO"ddlm}||g|Ri|Sr)rr)rrykwargsrrrr old_poly_ringv zDomain.old_poly_ringcOrr)%sympy.polys.domains.old_fractionfieldr)rryrrrrrold_frac_field{rzDomain.old_frac_fieldrcGr)z6Returns an algebraic field, i.e. `K(\alpha, \ldots)`. z%Cannot create algebraic field over %sr)rr extensionrrralgebraic_fieldrSzDomain.algebraic_fieldcCs0ddlm}|||}t||d}|j||dS)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.rootoftoolsrrr)rpolyr root_indexrrootalpharrralg_field_from_polys %  zDomain.alg_field_from_polyzetacCs2ddlm}|r|t|7}|j|||||dS)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.specialpolysrr%r)rnssrgenrrrrrcyclotomic_fields $ zDomain.cyclotomic_fieldcGr)z$Inject generators into this domain. rrrrrinjectr`z Domain.injectcGs|jr|St)z"Drop generators from this domain. ) is_Simplerrrrrrsz Domain.dropcCs| S)zReturns True if ``a`` is zero. rrTrrris_zerozDomain.is_zerocCs ||jkS)zReturns True if ``a`` is one. )rrTrrris_oner3z Domain.is_onecCs|dkS)z#Returns True if ``a`` is positive. rrrTrrr is_positivezDomain.is_positivecCs|dkS)z#Returns True if ``a`` is negative. rrrTrrr is_negativerzDomain.is_negativecCs|dkS)z'Returns True if ``a`` is non-positive. rrrTrrris_nonpositiverzDomain.is_nonpositivecCs|dkS)z'Returns True if ``a`` is non-negative. rrrTrrris_nonnegativerzDomain.is_nonnegativecCs||r |j S|jSr)rrrTrrrcanonical_units zDomain.canonical_unitcCst|S)z.Absolute value of ``a``, implies ``__abs__``. )absrTrrrrrz Domain.abscCs| S)z,Returns ``a`` negated, implies ``__neg__``. rrTrrrnegrz Domain.negcCs| S)z-Returns ``a`` positive, implies ``__pos__``. rrTrrrposrz Domain.poscCs||S)z.Sum of ``a`` and ``b``, implies ``__add__``. rrrUbrrradd rz Domain.addcCs||S)z5Difference of ``a`` and ``b``, implies ``__sub__``. rrrrrsubrz Domain.subcCs||S)z2Product of ``a`` and ``b``, implies ``__mul__``. rrrrrmulrz Domain.mulcCs||S)z2Raise ``a`` to power ``b``, implies ``__pow__``. rrrrrpowrz Domain.powcCr)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. rrrrrexquosYz Domain.exquocCr)aGQuotient 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`` rrrrrquow z Domain.quocCr)aNModulo 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`` rrrrrremrz Domain.remcCr)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. rrrrrdivsTz Domain.divcCr)z5Returns inversion of ``a mod b``, implies something. rrrrrinvertr`z Domain.invertcCr)z!Returns ``a**(-1)`` if possible. rrTrrrrevertr`z Domain.revertcCr)zReturns numerator of ``a``. rrTrrrnumerr`z Domain.numercCr)zReturns denominator of ``a``. rrTrrrdenomr`z Domain.denomcCs|||\}}}||fS)z&Half extended GCD of ``a`` and ``b``. )gcdex)rrUrsthrrr half_gcdexszDomain.half_gcdexcCr)z!Extended GCD of ``a`` and ``b``. rrrrrrr`z Domain.gcdexcCs.|||}|||}|||}|||fS)z.Returns GCD and cofactors of ``a`` and ``b``. )gcdr)rrUrrcfacfbrrr cofactorss    zDomain.cofactorscCr)z Returns GCD of ``a`` and ``b``. rrrrrr r`z Domain.gcdcCr)z Returns LCM of ``a`` and ``b``. rrrrrlcmr`z Domain.lcmcCr)z#Returns b-base logarithm of ``a``. rrrrrlogr`z Domain.logcCr)aJReturns 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 rrTrrrsqrtrz Domain.sqrtcCr)aReturns 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 rrTrrr is_square&szDomain.is_squarecCr)a'Principal 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 rrTrrrexsqrt6sz Domain.exsqrtcKs||j|fi|S)z*Returns numerical approximation of ``a``. )rWevalf)rrUroptionsrrrrGsz Domain.evalfcCs|SrrrTrrrrealMr!z Domain.realcCr"r)rrTrrrimagPr$z Domain.imagcCs||kS)z+Check if ``a`` and ``b`` are almost equal. r)rrUrrrrralmosteqSrzDomain.almosteqcCstd)z*Return the characteristic of this domain. zcharacteristic()rrrrrrWrzDomain.characteristicr)Nr)FrNr)r) __module__ __qualname____doc__r__annotations__rris_Ringrrhas_assoc_Fieldis_FiniteFieldis_FFris_ZZris_QQris_ZZ_Iris_QQ_Iris_RRris_CCr is_Algebraicris_Polyris_Fracis_SymbolicDomainris_SymbolicRawDomainrris_ExactrMrrwis_PIDrrrr r#r&r*r/propertyr1r2r4r=rNrFrVrWrPrYr_rarbrcrdrerfrgrhrirlrmrorrrsrurvr|rr{rrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrs  +      ; ] !   **[V   rN)r  __future__rtypingrsympy.core.numbersr sympy.corerrsympy.core.sortingrsympy.external.gmpyr!sympy.polys.domains.domainelementr sympy.polys.orderingsr sympy.polys.polyerrorsr r r sympy.polys.polyutilsrrsympy.utilitiesrsympy.utilities.iterablesrr__all__rrrrs4          S