o 8VaUF@sdZddlmZddlmZddlmZddlmZddl m Z ddl m Z ddl mZdd lmZdd lmZGd d d eZGd ddeZGdddeZGdddZGdddeeZeZe_GdddeeZeZe_dS)zDomains of Gaussian type.)I)CoercionFailed)ZZ)QQ)AlgebraicField)Domain) DomainElement)Field)RingcseZdZdZdZdZdZd5ddZefddZ d d Z d d Z d dZ ddZ ddZddZddZddZeddZddZeZddZdd Zd!d"ZeZd#d$Zd%d&Zd'd(Zd)d*Zd+d,Zd-d.Zd/d0Zd1d2Z d3d4Z!Z"S)6GaussianElementz1Base class for elements of Gaussian type domains.N)xyrcCs|jj}|||||SN)baseconvertnew)clsr r convrE/usr/lib/python3/dist-packages/sympy/polys/domains/gaussiandomains.py__new__szGaussianElement.__new__cst|}||_||_|S)z0Create a new GaussianElement of the same domain.)superrr r )rr r obj __class__rrrs zGaussianElement.newcCs|jS)z4The domain that this is an element of (ZZ_I or QQ_I))_parentselfrrrparent!szGaussianElement.parentcCst|j|jfSr)hashr r rrrr__hash__%zGaussianElement.__hash__cCs(t||jr|j|jko|j|jkStSr) isinstancerr r NotImplementedrotherrrr__eq__(s zGaussianElement.__eq__cCs&t|tstS|j|jg|j|jgkSr)r"r r#r r r$rrr__lt__.s zGaussianElement.__lt__cC|Srrrrrr__pos__3szGaussianElement.__pos__cCs||j |j Srrr r rrrr__neg__6zGaussianElement.__neg__cCsd|jj|j|jfS)Nz %s(%s, %s))rrepr r rrrr__repr__9szGaussianElement.__repr__cCst|j|Sr)strrto_sympyrrrr__str__<r!zGaussianElement.__str__cCs<t||sz|j|}Wn tyYdSw|j|jfS)N)NN)r"rrrr r )rr%rrr_get_xy?s   zGaussianElement._get_xycCs2||\}}|dur||j||j|StSrr2rr r r#rr%r r rrr__add__HzGaussianElement.__add__cCs2||\}}|dur||j||j|StSrr3r4rrr__sub__Qr6zGaussianElement.__sub__cCs2||\}}|dur|||j||jStSrr3r4rrr__rsub__Xr6zGaussianElement.__rsub__cCsF||\}}|dur!||j||j||j||j|StSrr3r4rrr__mul___s,zGaussianElement.__mul__cCs|dkr |ddS|dkrd|| }}|dkr|S|}|dr$|n|jj}|d}|r@||9}|dr:||9}|d}|s.|S)Nr)rrone)rZexpZpow2Zprodrrr__pow__hs  zGaussianElement.__pow__cCst|jp t|jSr)boolr r rrrr__bool__yr,zGaussianElement.__bool__cCsJ|jdkr|jdkr dSdS|jdkr|jdkrdSdS|jdkr#dSdS)zIReturn quadrant index 0-3. 0 is included in quadrant 0. rr:r;)r r rrrrquadrant|s  zGaussianElement.quadrantcCs2z|j|}Wn tytYSw||Sr)rrrr# __divmod__r$rrr __rdivmod__s   zGaussianElement.__rdivmod__cCs0zt|}Wn tytYSw||Sr)QQ_Irrr# __truediv__r$rrr __rtruediv__s   zGaussianElement.__rtruediv__cC||}|tur |S|dSNrrBr#rr%Zqrrrr __floordiv__ zGaussianElement.__floordiv__cCrGrHrCr#rJrrr __rfloordiv__rLzGaussianElement.__rfloordiv__cCrGNr:rIrJrrr__mod__rLzGaussianElement.__mod__cCrGrOrMrJrrr__rmod__rLzGaussianElement.__rmod__)r)#__name__ __module__ __qualname____doc__rr __slots__r classmethodrrr r&r'r)r+r.r1r2r5__radd__r7r8r9__rmul__r=r?rArCrFrKrNrPrQ __classcell__rrrrr sB   r c@$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.)rDrr$rrrrEszGaussianInteger.__truediv__c Cs|s td|||\}}|durtS|j||j||j ||j|}}||||}d||d|}d||d|}t||} | || |fS)Nz divmod({}, 0)r;)ZeroDivisionErrorformatr2r#r r r\) rr%r r abcZqxZqyqrrrrBs, zGaussianInteger.__divmod__N)rRrSrTrUrrrErBrrrrr\s   r\c@r[)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)r]z{} / 0N)r^r_r2r#rdr r )rr%r r rbrrrrEszGaussianRational.__truediv__cCsHz|j|}Wn tytYSw|std|||tjfS)Nz{} % 0)rrrr#r^r_rDzeror$rrrrBs zGaussianRational.__divmod__N)rRrSrTrUrrrErBrrrrrds   rdc@seZdZdZdZdZdZdZdZddZ ddZ dd Z d d Z d d Z ddZddZddZddZddZddZddZddZddZd d!ZdS)"GaussianDomainz Base class for Gaussian domains.NTcCs |jj}||jt||jS)z!Convert ``a`` to a SymPy object. )domr0r rr )rr`rrrrr0szGaussianDomain.to_sympycCsb|\}}|j|}|s||dS|\}}|j|}|tur*|||Std|)z)Convert a SymPy object to ``self.dtype``.rz{} is not Gaussian)Z as_coeff_Addrg from_sympyrZ as_coeff_Mulrrr_)rr`rrar r rrrrhs      zGaussianDomain.from_sympycGs |j|S)z$Inject generators into this domain. )Z poly_ring)rZgensrrrinjects zGaussianDomain.injectcCs|j| }|Sr)unitsrA)rdunitrrrcanonical_unitszGaussianDomain.canonical_unitcCdSz/Returns ``False`` for any ``GaussianElement``. Frrelementrrr is_negativezGaussianDomain.is_negativecCrorprrqrrr is_positivertzGaussianDomain.is_positivecCrorprrqrrris_nonnegative rtzGaussianDomain.is_nonnegativecCrorprrqrrris_nonpositive$rtzGaussianDomain.is_nonpositivecC||S)z%Convert a GMPY mpz to ``self.dtype``.rZK1r`ZK0rrr from_ZZ_gmpy(zGaussianDomain.from_ZZ_gmpycCrxz.Convert a ZZ_python element to ``self.dtype``.rryrrrfrom_ZZ,r{zGaussianDomain.from_ZZcCrxr|rryrrrfrom_ZZ_python0r{zGaussianDomain.from_ZZ_pythoncCrxz%Convert a GMPY mpq to ``self.dtype``.rryrrrfrom_QQ4r{zGaussianDomain.from_QQcCrxrrryrrr from_QQ_gmpy8r{zGaussianDomain.from_QQ_gmpycCrx)z.Convert a QQ_python element to ``self.dtype``.rryrrrfrom_QQ_python<r{zGaussianDomain.from_QQ_pythoncCs$|jjdtkr|||SdS)z9Convert an element from ZZ or QQ to ``self.dtype``.rN)extargsrrhr0ryrrrfrom_AlgebraicField@sz"GaussianDomain.from_AlgebraicField)rRrSrTrUrgZ is_NumericalZis_ExactZhas_assoc_RingZhas_assoc_Fieldr0rhrjrnrsrurvrwrzr}r~rrrrrrrrrfs,  rfc@eZdZdZeZeZeededZeededZ eededZ e e e e fZ dZ dZ dZddZdd Zd d Zd d ZddZddZddZddZdS)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_ITcCro)zFor constructing ZZ_I.Nrrrrr__init__zGaussianIntegerRing.__init__cCr(z)Returns a ring associated with ``self``. rrrrrget_ringrtzGaussianIntegerRing.get_ringcCtSz*Returns a field associated with ``self``. )rDrrrr get_fieldrtzGaussianIntegerRing.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).0r`rmrr sz0GaussianIntegerRing.normalize..)rntuple)rrlrrrr normalizes zGaussianIntegerRing.normalizecCs |r |||}}|s||S)z-Greatest common divisor of a and b over ZZ_I.)rrr`rarrrgcds zGaussianIntegerRing.gcdcCs|||||S)z+Least common multiple of a and b over ZZ_I.)rrrrrlcmszGaussianIntegerRing.lcmcC|S)zConvert a ZZ_I element to ZZ_I.rryrrrfrom_GaussianIntegerRingrtz,GaussianIntegerRing.from_GaussianIntegerRingcCs|t|jt|jS)zConvert a QQ_I element to ZZ_I.)rrrr r ryrrrfrom_GaussianRationalFieldsz.GaussianIntegerRing.from_GaussianRationalFieldN)rRrSrTrUrrgr\dtyperer< imag_unitrkr-Zis_GaussianRingZis_ZZ_IrrrrrrrrrrrrrFs&c  rc@r)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:rDTcCro)zFor constructing QQ_I.NrrrrrrarzGaussianRationalField.__init__cCrr)rrrrrrdrtzGaussianRationalField.get_ringcCr(rrrrrrrhrtzGaussianRationalField.get_fieldcCs t|jtS)z0Get equivalent domain as an ``AlgebraicField``. )rrgrrrrras_AlgebraicFieldls z'GaussianRationalField.as_AlgebraicFieldcCs|}||||S)zGet the numerator of ``a``.)rrdenom)rr`rrrrnumerpszGaussianRationalField.numercCs@|j}|j}|}|||j||j}|||jS)zGet the denominator of ``a``.)rgrrrr r re)rr`rrrZdenom_ZZrrrrus  zGaussianRationalField.denomcCs||j|jS)zConvert a ZZ_I element to QQ_I.r*ryrrrr}sz.GaussianRationalField.from_GaussianIntegerRingcCr)zConvert a QQ_I element to QQ_I.rryrrrrrtz0GaussianRationalField.from_GaussianRationalFieldN)rRrSrTrUrrgrdrrer<rrkr-Zis_GaussianFieldZis_QQ_Irrrrrrrrrrrrrs&t rN)rUZsympy.core.numbersrZsympy.polys.polyerrorsrZsympy.polys.domains.integerringrZ!sympy.polys.domains.rationalfieldrZ"sympy.polys.domains.algebraicfieldrZsympy.polys.domains.domainrZ!sympy.polys.domains.domainelementrZsympy.polys.domains.fieldr Zsympy.polys.domains.ringr r r\rdrfrrrrrDrrrrs*         (#R &