a RG5dUF@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$Zd%d&Zd'd(Zd)d*Zd+d,Zd-d.Zd/d0Zd1d2Z d3d4Z!Z"S)6GaussianElementz1Base class for elements of Gaussian type domains.N)xyrcCs|jj}|||||SN)baseconvertnew)clsr r convr_/var/www/html/django/DPS/env/lib/python3.9/site-packages/sympy/polys/domains/gaussiandomains.py__new__szGaussianElement.__new__cst|}||_||_|S)z0Create a new GaussianElement of the same domain.)superrr r )rr r obj __class__rrrs zGaussianElement.newcCs|jS)z4The domain that this is an element of (ZZ_I or QQ_I))_parentselfrrrparent!szGaussianElement.parentcCst|j|jfSr)hashr r rrrr__hash__%szGaussianElement.__hash__cCs,t||jr$|j|jko"|j|jkStSdSr) 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__cCs|Srrrrrr__pos__3szGaussianElement.__pos__cCs||j |j Srrr r rrrr__neg__6szGaussianElement.__neg__cCsd|jj|j|jfS)Nz %s(%s, %s))rrepr r rrrr__repr__9szGaussianElement.__repr__cCst|j|Sr)strrto_sympyrrrr__str__<szGaussianElement.__str__cCs<t||s0z|j|}Wnty.YdS0|j|jfS)N)NN)r!rrrr r )rr$rrr_get_xy?s   zGaussianElement._get_xycCs6||\}}|dur.||j||j|StSdSrr/rr r r"rr$r r rrr__add__HszGaussianElement.__add__cCs6||\}}|dur.||j||j|StSdSrr0r1rrr__sub__QszGaussianElement.__sub__cCs6||\}}|dur.|||j||jStSdSrr0r1rrr__rsub__XszGaussianElement.__rsub__cCsJ||\}}|durB||j||j||j||j|StSdSrr0r1rrr__mul___s,zGaussianElement.__mul__cCs|dkr|ddS|dkr,d|| }}|dkr8|S|}|drH|n|jj}|d}|r~||9}|drt||9}|d}qX|S)Nr)rrone)rexpZpow2prodrrr__pow__hs  zGaussianElement.__pow__cCst|jpt|jSr)boolr r rrrr__bool__yszGaussianElement.__bool__cCsN|jdkr|jdkrdSdS|jdkr8|jdkr4dSdS|jdkrFdSdSdS)zIReturn quadrant index 0-3. 0 is included in quadrant 0. rr6r7N)r r rrrrquadrant|s   zGaussianElement.quadrantcCs6z|j|}Wnty&tYS0||SdSr)rrrr" __divmod__r#rrr __rdivmod__s   zGaussianElement.__rdivmod__cCs4zt|}Wnty$tYS0||SdSr)QQ_Irrr" __truediv__r#rrr __rtruediv__s   zGaussianElement.__rtruediv__cCs||}|tur|S|dSNrr@r"rr$qrrrr __floordiv__s zGaussianElement.__floordiv__cCs||}|tur|S|dSrErAr"rGrrr __rfloordiv__s zGaussianElement.__rfloordiv__cCs||}|tur|S|dSNr6rFrGrrr__mod__s zGaussianElement.__mod__cCs||}|tur|S|dSrLrJrGrrr__rmod__s zGaussianElement.__rmod__)r)#__name__ __module__ __qualname____doc__rr __slots__r classmethodrrr r%r&r'r)r+r.r/r2__radd__r3r4r5__rmul__r;r=r?rArDrIrKrMrN __classcell__rrrrr s@   r c@s$eZdZdZeZddZddZdS)GaussianIntegerzGaussian integer: domain element for :ref:`ZZ_I` >>> from sympy import ZZ_I >>> z = ZZ_I(2, 3) >>> z (2 + 3*I) >>> type(z) cCst||S)Return a Gaussian rational.)rBrr#rrrrCszGaussianInteger.__truediv__c Cs|std|||\}}|dur,tS|j||j||j ||j|}}||||}d||d|}d||d|}t||} | || |fS)Nz divmod({}, 0)r7)ZeroDivisionErrorformatr/r"r r rX) rr$r r abcqxqyqrrrr@s, zGaussianInteger.__divmod__N)rOrPrQrRrrrCr@rrrrrXs rXc@s$eZdZdZeZddZddZdS)GaussianRationalaGaussian rational: domain element for :ref:`QQ_I` >>> from sympy import QQ_I, QQ >>> z = QQ_I(QQ(2, 3), QQ(4, 5)) >>> z (2/3 + 4/5*I) >>> type(z) cCsp|std|||\}}|dur,tS||||}t|j||j|||j ||j||S)rYz{} / 0N)rZr[r/r"rbr r )rr$r r r^rrrrCszGaussianRational.__truediv__cCsNz|j|}Wnty&tYS0|s or QQ to ``self.dtype``.rN)extargsrrgr-rxrrrfrom_AlgebraicField@sz"GaussianDomain.from_AlgebraicField)rOrPrQrRre is_Numericalis_Exacthas_assoc_Ringhas_assoc_Fieldr-rgrlrprtrurvrwr{r}r~rrrrrrrrrds* rdc@seZdZdZeZeZeededZeededZ 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 rr6ZZ_ITcCsdS)zFor constructing ZZ_I.Nrrrrr__init__szGaussianIntegerRing.__init__cCs|Sz)Returns a ring associated with ``self``. rrrrrget_ringszGaussianIntegerRing.get_ringcCstSz*Returns a field associated with ``self``. )rBrrrr get_fieldszGaussianIntegerRing.get_fieldcs:|||9}tfdd|D}|r6|f|S|S)zReturn first quadrant element associated with ``d``. Also multiply the other arguments by the same power of i. c3s|]}|VqdSrr).0r\rorr z0GaussianIntegerRing.normalize..)rptuple)rrnrrrr normalizes zGaussianIntegerRing.normalizecCs|r|||}}q||S)z-Greatest common divisor of a and b over ZZ_I.)rrr\r]rrrgcdszGaussianIntegerRing.gcdcCs|||||S)z+Least common multiple of a and b over ZZ_I.)rrrrrlcmszGaussianIntegerRing.lcmcCs|S)zConvert a ZZ_I element to ZZ_I.rrxrrrfrom_GaussianIntegerRingsz,GaussianIntegerRing.from_GaussianIntegerRingcCs|t|jt|jS)zConvert a QQ_I element to ZZ_I.)rrrr r rxrrrfrom_GaussianRationalFieldsz.GaussianIntegerRing.from_GaussianRationalFieldN)rOrPrQrRrrerXdtypercr8 imag_unitrmr*is_GaussianRingis_ZZ_IrrrrrrrrrrrrrFs$c rc@seZdZdZeZeZeededZeededZ eededZ e e e e fZ dZ dZ dZddZdd Zd d Zd d ZddZddZddZddZdS)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 rr6rBTcCsdS)zFor constructing QQ_I.NrrrrrraszGaussianRationalField.__init__cCstSr)rrrrrrdszGaussianRationalField.get_ringcCs|SrrrrrrrhszGaussianRationalField.get_fieldcCs t|jtS)z0Get equivalent domain as an ``AlgebraicField``. )rrerrrrras_AlgebraicFieldlsz'GaussianRationalField.as_AlgebraicFieldcCs|}||||S)zGet the numerator of ``a``.)rrdenom)rr\rrrrnumerpszGaussianRationalField.numercCs@|j}|j}|}|||j||j}|||jS)zGet the denominator of ``a``.)rerrrr r rc)rr\rrrZdenom_ZZrrrrus  zGaussianRationalField.denomcCs||j|jS)zConvert a ZZ_I element to QQ_I.r(rxrrrr}sz.GaussianRationalField.from_GaussianIntegerRingcCs|S)zConvert a QQ_I element to QQ_I.rrxrrrrsz0GaussianRationalField.from_GaussianRationalFieldN)rOrPrQrRrrerbrrcr8rrmr*is_GaussianFieldis_QQ_Irrrrrrrrrrrrrs$trN)rRsympy.core.numbersrsympy.polys.polyerrorsrZsympy.polys.domains.integerringrZ!sympy.polys.domains.rationalfieldrZ"sympy.polys.domains.algebraicfieldrZsympy.polys.domains.domainr!sympy.polys.domains.domainelementrsympy.polys.domains.fieldr sympy.polys.domains.ringr r rXrbrdrrrrrBrrrrs(         (#R &