o 3IhL@sdZddlmZddlmZddlmZddlmZddl m Z ddl m Z ddl mZdd lmZdd lmZdd lmZdd lmZGd ddeZGdddeZGdddeZGdddZGdddeeZeZe_GdddeeZeZe_dS)zDomains of Gaussian type.) annotations)I)DMP)CoercionFailed)ZZ)QQ)AlgebraicField)Domain) DomainElement)Field)RingcseZdZUdZded<ded<dZd7ddZefd d Zd d Z d dZ ddZ ddZ ddZ ddZddZddZeddZddZeZdd Zd!d"Zd#d$ZeZd%d&Zd'd(Zd)d*Zd+d,Zd-d.Zd/d0Zd1d2Zd3d4Zd5d6Z Z!S)8GaussianElementz1Base class for elements of Gaussian type domains.r base_parent)xyrcCs|jj}|||||SN)rconvertnew)clsrrconvrc/home/air/sanwanet/gpt-api/venv/lib/python3.10/site-packages/sympy/polys/domains/gaussiandomains.py__new__szGaussianElement.__new__cst|}||_||_|S)z0Create a new GaussianElement of the same domain.)superrrr)rrrobj __class__rrrs zGaussianElement.newcCs|jS)z4The domain that this is an element of (ZZ_I or QQ_I))rselfrrrparent#szGaussianElement.parentcCst|j|jfSr)hashrrrrrr__hash__'zGaussianElement.__hash__cCs(t||jr|j|jko|j|jkStSr) isinstancerrrNotImplementedrotherrrr__eq__*s zGaussianElement.__eq__cCs&t|tstS|j|jg|j|jgkSr)r$r r%rrr&rrr__lt__0s zGaussianElement.__lt__cC|Srrrrrr__pos__5zGaussianElement.__pos__cCs||j |j Srrrrrrrr__neg__8zGaussianElement.__neg__cCsd|jj|j|jfS)Nz %s(%s, %s))rreprrrrrr__repr__;szGaussianElement.__repr__cCst|j|Sr)strrto_sympyrrrr__str__>r#zGaussianElement.__str__cCs<t||sz|j|}Wn tyYdSw|j|jfS)N)NN)r$rrrrr)rr'rrr_get_xyAs   zGaussianElement._get_xycCs2||\}}|dur||j||j|StSrr5rrrr%rr'rrrrr__add__JzGaussianElement.__add__cCs2||\}}|dur||j||j|StSrr6r7rrr__sub__Sr9zGaussianElement.__sub__cCs2||\}}|dur|||j||jStSrr6r7rrr__rsub__Zr9zGaussianElement.__rsub__cCsF||\}}|dur!||j||j||j||j|StSrr6r7rrr__mul__as,zGaussianElement.__mul__cCs|dkr |ddS|dkrd|| }}|dkr|S|}|dr$|n|jj}|d}|r@||9}|dr:||9}|d}|s.|S)Nr)rrone)rexppow2prodrrr__pow__js  zGaussianElement.__pow__cCst|jp t|jSr)boolrrrrrr__bool__{r/zGaussianElement.__bool__cCsJ|jdkr|jdkr dSdS|jdkr|jdkrdSdS|jdkr#dSdS)zIReturn quadrant index 0-3. 0 is included in quadrant 0. rr=r>)rrrrrrquadrant~s  zGaussianElement.quadrantcCs2z|j|}Wn tytYSw||Sr)rrrr% __divmod__r&rrr __rdivmod__s   zGaussianElement.__rdivmod__cCs0zt|}Wn tytYSw||Sr)QQ_Irrr% __truediv__r&rrr __rtruediv__s   zGaussianElement.__rtruediv__cC||}|tur |S|dSNrrHr%rr'qrrrr __floordiv__ zGaussianElement.__floordiv__cCrMrNrIr%rPrrr __rfloordiv__rSzGaussianElement.__rfloordiv__cCrMNr=rOrPrrr__mod__rSzGaussianElement.__mod__cCrMrVrTrPrrr__rmod__rSzGaussianElement.__rmod__)r)"__name__ __module__ __qualname____doc____annotations__ __slots__r classmethodrr r"r(r)r+r.r1r4r5r8__radd__r:r;r<__rmul__rCrErGrIrLrRrUrWrX __classcell__rrrrr sB    r c@$eZdZdZeZddZddZdS)GaussianIntegerzGaussian integer: domain element for :ref:`ZZ_I` >>> from sympy import ZZ_I >>> z = ZZ_I(2, 3) >>> z (2 + 3*I) >>> type(z) cCst||S)Return a Gaussian rational.)rJrr&rrrrKszGaussianInteger.__truediv__c Cs|s td|||\}}|durtS|j||j||j ||j|}}||||}d||d|}d||d|}t||} | || |fS)Nz divmod({}, 0)r>)ZeroDivisionErrorformatr5r%rrrd) rr'rrabcqxqyqrrrrHs, zGaussianInteger.__divmod__N)rYrZr[r\rrrKrHrrrrrds   rdc@rc)GaussianRationalaGaussian rational: domain element for :ref:`QQ_I` >>> from sympy import QQ_I, QQ >>> z = QQ_I(QQ(2, 3), QQ(4, 5)) >>> z (2/3 + 4/5*I) >>> type(z) cCsp|s td|||\}}|durtS||||}t|j||j|||j ||j||S)rez{} / 0N)rfrgr5r%rnrr)rr'rrrjrrrrKszGaussianRational.__truediv__cCsHz|j|}Wn tytYSw|std|||tjfS)Nz{} % 0)rrrr%rfrgrJzeror&rrrrHs zGaussianRational.__divmod__N)rYrZr[r\rrrKrHrrrrrns   rnc@seZdZUdZded<dZdZdZdZddZ ddZ d d Z d d Z d dZ ddZddZddZddZddZddZddZddZdd Zd!d"Zd#S)$GaussianDomainz Base class for Gaussian domains.r domTcCs |jj}||jt||jS)z!Convert ``a`` to a SymPy object. )rqr3rrr)rrhrrrrr3szGaussianDomain.to_sympycCsb|\}}|j|}|s||dS|\}}|j|}|tur*|||Std|)z)Convert a SymPy object to ``self.dtype``.rz{} is not Gaussian) as_coeff_Addrq from_sympyr as_coeff_Mulrrrg)rrhrrirrrrrrss      zGaussianDomain.from_sympycGs |j|S)z$Inject generators into this domain. ) poly_ring)rgensrrrinjects zGaussianDomain.injectcCs|j| }|Sr)unitsrG)rdunitrrrcanonical_unitszGaussianDomain.canonical_unitcCdSz/Returns ``False`` for any ``GaussianElement``. Frrelementrrr is_negativezGaussianDomain.is_negativecCr}r~rrrrr is_positiverzGaussianDomain.is_positivecCr}r~rrrrris_nonnegative"rzGaussianDomain.is_nonnegativecCr}r~rrrrris_nonpositive&rzGaussianDomain.is_nonpositivecC||S)z%Convert a GMPY mpz to ``self.dtype``.rK1rhK0rrr from_ZZ_gmpy*zGaussianDomain.from_ZZ_gmpycCrz.Convert a ZZ_python element to ``self.dtype``.rrrrrfrom_ZZ.rzGaussianDomain.from_ZZcCrrrrrrrfrom_ZZ_python2rzGaussianDomain.from_ZZ_pythoncCrz%Convert a GMPY mpq to ``self.dtype``.rrrrrfrom_QQ6rzGaussianDomain.from_QQcCrrrrrrr from_QQ_gmpy:rzGaussianDomain.from_QQ_gmpycCr)z.Convert a QQ_python element to ``self.dtype``.rrrrrfrom_QQ_python>rzGaussianDomain.from_QQ_pythoncCs$|jjdtkr|||SdS)z9Convert an element from ZZ or QQ to ``self.dtype``.rN)extargsrrsr3rrrrfrom_AlgebraicFieldBsz"GaussianDomain.from_AlgebraicFieldN)rYrZr[r\r] is_Numericalis_Exacthas_assoc_Ringhas_assoc_Fieldr3rsrxr|rrrrrrrrrrrrrrrrps,   rpc@seZdZdZeZeejejejgeZ e Z e ededZe ededZe ededZ ee e e fZ dZdZdZdZddZdd Zd d Zed d ZddZddZddZddZddZddZddZddZddZd S)!GaussianIntegerRinga{ Ring of Gaussian integers ``ZZ_I`` The :ref:`ZZ_I` domain represents the `Gaussian integers`_ `\mathbb{Z}[i]` as a :py:class:`~.Domain` in the domain system (see :ref:`polys-domainsintro`). By default a :py:class:`~.Poly` created from an expression with coefficients that are combinations of integers and ``I`` (`\sqrt{-1}`) will have the domain :ref:`ZZ_I`. >>> from sympy import Poly, Symbol, I >>> x = Symbol('x') >>> p = Poly(x**2 + I) >>> p Poly(x**2 + I, x, domain='ZZ_I') >>> p.domain ZZ_I The :ref:`ZZ_I` domain can be used to factorise polynomials that are reducible over the Gaussian integers. >>> from sympy import factor >>> factor(x**2 + 1) x**2 + 1 >>> factor(x**2 + 1, domain='ZZ_I') (x - I)*(x + I) The corresponding `field of fractions`_ is the domain of the Gaussian rationals :ref:`QQ_I`. Conversely :ref:`ZZ_I` is the `ring of integers`_ of :ref:`QQ_I`. >>> from sympy import ZZ_I, QQ_I >>> ZZ_I.get_field() QQ_I >>> QQ_I.get_ring() ZZ_I When using the domain directly :ref:`ZZ_I` can be used as a constructor. >>> ZZ_I(3, 4) (3 + 4*I) >>> ZZ_I(5) (5 + 0*I) The domain elements of :ref:`ZZ_I` are instances of :py:class:`~.GaussianInteger` which support the rings operations ``+,-,*,**``. >>> z1 = ZZ_I(5, 1) >>> z2 = ZZ_I(2, 3) >>> z1 (5 + 1*I) >>> z2 (2 + 3*I) >>> z1 + z2 (7 + 4*I) >>> z1 * z2 (7 + 17*I) >>> z1 ** 2 (24 + 10*I) Both floor (``//``) and modulo (``%``) division work with :py:class:`~.GaussianInteger` (see the :py:meth:`~.Domain.div` method). >>> z3, z4 = ZZ_I(5), ZZ_I(1, 3) >>> z3 // z4 # floor division (1 + -1*I) >>> z3 % z4 # modulo division (remainder) (1 + -2*I) >>> (z3//z4)*z4 + z3%z4 == z3 True True division (``/``) in :ref:`ZZ_I` gives an element of :ref:`QQ_I`. The :py:meth:`~.Domain.exquo` method can be used to divide in :ref:`ZZ_I` when exact division is possible. >>> z1 / z2 (1 + -1*I) >>> ZZ_I.exquo(z1, z2) (1 + -1*I) >>> z3 / z4 (1/2 + -3/2*I) >>> ZZ_I.exquo(z3, z4) Traceback (most recent call last): ... ExactQuotientFailed: (1 + 3*I) does not divide (5 + 0*I) in ZZ_I The :py:meth:`~.Domain.gcd` method can be used to compute the `gcd`_ of any two elements. >>> ZZ_I.gcd(ZZ_I(10), ZZ_I(2)) (2 + 0*I) >>> ZZ_I.gcd(ZZ_I(5), ZZ_I(2, 1)) (2 + 1*I) .. _Gaussian integers: https://en.wikipedia.org/wiki/Gaussian_integer .. _gcd: https://en.wikipedia.org/wiki/Greatest_common_divisor rr=ZZ_ITcCr})zFor constructing ZZ_I.Nrrrrr__init__zGaussianIntegerRing.__init__cCt|trdStSz0Returns ``True`` if two domains are equivalent. T)r$rr%r&rrrr( zGaussianIntegerRing.__eq__cCtdS)Compute hash code of ``self``. rr!rrrrr"rzGaussianIntegerRing.__hash__cCr}NTrrrrrhas_CharacteristicZerorz*GaussianIntegerRing.has_CharacteristicZerocCr}rNrrrrrcharacteristicr,z"GaussianIntegerRing.characteristiccCr*z)Returns a ring associated with ``self``. rrrrrget_ringrzGaussianIntegerRing.get_ringcCtSz*Returns a field associated with ``self``. )rJrrrr get_fieldrzGaussianIntegerRing.get_fieldcs:|||9}tfdd|D}|r|f|S|S)zReturn first quadrant element associated with ``d``. Also multiply the other arguments by the same power of i. c3s|]}|VqdSrr).0rhr{rr sz0GaussianIntegerRing.normalize..)r|tuple)rrzrrrr normalizes zGaussianIntegerRing.normalizecCs |r |||}}|s||S)z-Greatest common divisor of a and b over ZZ_I.)rrrhrirrrgcds zGaussianIntegerRing.gcdcCs||j}|j}|j}|j}|r/||}||||}}||||}}||||}}|s||||\}}}|||fS)z6Return x, y, g such that x * a + y * b = g = gcd(a, b))r?ror)rrhrix_ax_by_ay_brmrrrgcdexs zGaussianIntegerRing.gcdexcCs|||||S)z+Least common multiple of a and b over ZZ_I.)rrrrrlcmszGaussianIntegerRing.lcmcC|S)zConvert a ZZ_I element to ZZ_I.rrrrrfrom_GaussianIntegerRingrz,GaussianIntegerRing.from_GaussianIntegerRingcC|t|jt|jS)zConvert a QQ_I element to ZZ_I.)rrrrrrrrrfrom_GaussianRationalFieldz.GaussianIntegerRing.from_GaussianRationalFieldN) rYrZr[r\rrqrr?romodrddtype imag_unitryr0is_GaussianRingis_ZZ_Iis_PIDrr(r"propertyrrrrrrrrrrrrrrrHs6c   rc@seZdZdZeZeejejejgeZ e Z e ededZe ededZe ededZ ee e e fZ dZdZdZddZdd Zd d Zed d ZddZddZddZddZddZddZddZddZddZd S)!GaussianRationalFielda Field of Gaussian rationals ``QQ_I`` The :ref:`QQ_I` domain represents the `Gaussian rationals`_ `\mathbb{Q}(i)` as a :py:class:`~.Domain` in the domain system (see :ref:`polys-domainsintro`). By default a :py:class:`~.Poly` created from an expression with coefficients that are combinations of rationals and ``I`` (`\sqrt{-1}`) will have the domain :ref:`QQ_I`. >>> from sympy import Poly, Symbol, I >>> x = Symbol('x') >>> p = Poly(x**2 + I/2) >>> p Poly(x**2 + I/2, x, domain='QQ_I') >>> p.domain QQ_I The polys option ``gaussian=True`` can be used to specify that the domain should be :ref:`QQ_I` even if the coefficients do not contain ``I`` or are all integers. >>> Poly(x**2) Poly(x**2, x, domain='ZZ') >>> Poly(x**2 + I) Poly(x**2 + I, x, domain='ZZ_I') >>> Poly(x**2/2) Poly(1/2*x**2, x, domain='QQ') >>> Poly(x**2, gaussian=True) Poly(x**2, x, domain='QQ_I') >>> Poly(x**2 + I, gaussian=True) Poly(x**2 + I, x, domain='QQ_I') >>> Poly(x**2/2, gaussian=True) Poly(1/2*x**2, x, domain='QQ_I') The :ref:`QQ_I` domain can be used to factorise polynomials that are reducible over the Gaussian rationals. >>> from sympy import factor, QQ_I >>> factor(x**2/4 + 1) (x**2 + 4)/4 >>> factor(x**2/4 + 1, domain='QQ_I') (x - 2*I)*(x + 2*I)/4 >>> factor(x**2/4 + 1, domain=QQ_I) (x - 2*I)*(x + 2*I)/4 It is also possible to specify the :ref:`QQ_I` domain explicitly with polys functions like :py:func:`~.apart`. >>> from sympy import apart >>> apart(1/(1 + x**2)) 1/(x**2 + 1) >>> apart(1/(1 + x**2), domain=QQ_I) I/(2*(x + I)) - I/(2*(x - I)) The corresponding `ring of integers`_ is the domain of the Gaussian integers :ref:`ZZ_I`. Conversely :ref:`QQ_I` is the `field of fractions`_ of :ref:`ZZ_I`. >>> from sympy import ZZ_I, QQ_I, QQ >>> ZZ_I.get_field() QQ_I >>> QQ_I.get_ring() ZZ_I When using the domain directly :ref:`QQ_I` can be used as a constructor. >>> QQ_I(3, 4) (3 + 4*I) >>> QQ_I(5) (5 + 0*I) >>> QQ_I(QQ(2, 3), QQ(4, 5)) (2/3 + 4/5*I) The domain elements of :ref:`QQ_I` are instances of :py:class:`~.GaussianRational` which support the field operations ``+,-,*,**,/``. >>> z1 = QQ_I(5, 1) >>> z2 = QQ_I(2, QQ(1, 2)) >>> z1 (5 + 1*I) >>> z2 (2 + 1/2*I) >>> z1 + z2 (7 + 3/2*I) >>> z1 * z2 (19/2 + 9/2*I) >>> z2 ** 2 (15/4 + 2*I) True division (``/``) in :ref:`QQ_I` gives an element of :ref:`QQ_I` and is always exact. >>> z1 / z2 (42/17 + -2/17*I) >>> QQ_I.exquo(z1, z2) (42/17 + -2/17*I) >>> z1 == (z1/z2)*z2 True Both floor (``//``) and modulo (``%``) division can be used with :py:class:`~.GaussianRational` (see :py:meth:`~.Domain.div`) but division is always exact so there is no remainder. >>> z1 // z2 (42/17 + -2/17*I) >>> z1 % z2 (0 + 0*I) >>> QQ_I.div(z1, z2) ((42/17 + -2/17*I), (0 + 0*I)) >>> (z1//z2)*z2 + z1%z2 == z1 True .. _Gaussian rationals: https://en.wikipedia.org/wiki/Gaussian_rational rr=rJTcCr})zFor constructing QQ_I.NrrrrrrrzGaussianRationalField.__init__cCrr)r$rr%r&rrrr(rzGaussianRationalField.__eq__cCr)rrJrrrrrr"rzGaussianRationalField.__hash__cCr}rrrrrrrrz,GaussianRationalField.has_CharacteristicZerocCr}rNrrrrrrr,z$GaussianRationalField.characteristiccCrr)rrrrrrrzGaussianRationalField.get_ringcCr*rrrrrrrrzGaussianRationalField.get_fieldcCs t|jtS)z0Get equivalent domain as an ``AlgebraicField``. )rrqrrrrras_AlgebraicFields z'GaussianRationalField.as_AlgebraicFieldcCs|}||||S)zGet the numerator of ``a``.)rrdenom)rrhrrrrnumerszGaussianRationalField.numercCs@|j}|j}|}|||j||j}|||jS)zGet the denominator of ``a``.)rqrrrrrro)rrhrrrdenom_ZZrrrrs  zGaussianRationalField.denomcCs||j|jS)zConvert a ZZ_I element to QQ_I.r-rrrrrsz.GaussianRationalField.from_GaussianIntegerRingcCr)zConvert a QQ_I element to QQ_I.rrrrrrrz0GaussianRationalField.from_GaussianRationalFieldcCr)z'Convert a ComplexField element to QQ_I.)rrrrealimagrrrrfrom_ComplexFieldrz'GaussianRationalField.from_ComplexFieldN)rYrZr[r\rrqrr?rorrnrrryr0is_GaussianFieldis_QQ_Irr(r"rrrrrrrrrrrrrrrrs4t  rN) r\ __future__rsympy.core.numbersrsympy.polys.polyclassesrsympy.polys.polyerrorsrsympy.polys.domains.integerringr!sympy.polys.domains.rationalfieldr"sympy.polys.domains.algebraicfieldrsympy.polys.domains.domainr !sympy.polys.domains.domainelementr sympy.polys.domains.fieldr sympy.polys.domains.ringr r rdrnrprrrrrJrrrrs.           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