a .zvfield..)r"r r%r#r$r)r)r*vfield*s r2c Osd}t|s|gd}}ttt|}t||}g}|D]}||q8t||\}}|jdurt dd|Dg}t ||d\|_} t |j |j|j } g} tdt|dD]"} | | t|| | dq|r| | dfS| | fSdS) aConstruct a field deriving generators and domain from options and input expressions. Parameters ========== exprs : py:class:`~.Expr` or sequence of :py:class:`~.Expr` (sympifiable) symbols : sequence of :py:class:`~.Symbol`/:py:class:`~.Expr` options : keyword arguments understood by :py:class:`~.Options` Examples ======== >>> from sympy import exp, log, symbols, sfield >>> x = symbols("x") >>> K, f = sfield((x*log(x) + 4*x**2)*exp(1/x + log(x)/3)/x**2) >>> K Rational function field in x, exp(1/x), log(x), x**(1/3) over ZZ with lex order >>> f (4*x**2*(exp(1/x)) + x*(exp(1/x))*(log(x)))/((x**(1/3))**5) FTNcSsg|]}t|qSr))listvalues)r.repr)r)r*r0Yr1zsfield..)optr)rr3maprrextendZas_numer_denomrr&sumrr"r#r'rangelenappendtuple) exprsr%optionssingler6ZnumdensexprZrepsZcoeffs_r(Zfracsir)r)r*sfield1s&     rEc@seZdZdZefddZddZddZdd Zd d Z d d Z ddZ d#ddZ d$ddZ ddZddZddZeZddZddZdd Zd!d"ZdS)%r"z2Multivariate distributed rational function field. c Csddlm}||||}|j}|j}|j}|j}|j||||f}t|}|dur t |}||_ t ||_ ||_tdtfd|i|_||_||_||_||_||j|_||j|_||_t|j|jD].\} } t| tr| j} t|| st|| | q|t|<|S)NrPolyRing FracElementr+)sympy.polys.ringsrGr%ngensr&r'__name__ _field_cachegetobject__new__ _hash_tuplehash_hashringtyperHdtypezeroone_gensr#zip isinstancerr-hasattrsetattr) clsr%r&r'rGrSrJrPobjsymbol generatorr-r)r)r*rOks8         zFracField.__new__cstfddjjDS)z(Return a list of polynomial generators. csg|]}|qSr)rUr.genselfr)r*r0r1z#FracField._gens..)r>rSr#rdr)rdr*rXszFracField._genscCs|j|j|jfSN)r%r&r'rdr)r)r*__getnewargs__szFracField.__getnewargs__cCs|jSrf)rRrdr)r)r*__hash__szFracField.__hash__cCs2t||jr|j|Std|j|fdS)Nzexpected a %s, got %s instead)rZrUrSindexto_poly ValueError)rercr)r)r*ris zFracField.indexcCs2t|to0|j|j|j|jf|j|j|j|jfkSrf)rZr"r%rJr&r'reotherr)r)r*__eq__s  zFracField.__eq__cCs ||k Srfr)rlr)r)r*__ne__szFracField.__ne__NcCs |||Srfrarenumerdenomr)r)r*raw_newszFracField.raw_newcCs*|dur|jj}||\}}|||Srf)rSrWcancelrsrpr)r)r*newsz FracField.newcCs |j|Srf)r&convert)reelementr)r)r* domain_newszFracField.domain_newcCsz||j|WSty|j}|js||jr||j}|}||}|| |}|| |}| ||YSYn0dSrf) rurS ground_newrr&is_Fieldhas_assoc_Field get_fieldrvrqrrrs)rerwr&rS ground_fieldrqrrr)r)r*rys   zFracField.ground_newcCsvt|trp||jkr|St|jtr<|jj|jkr<||St|jtrd|jj|jkrd||St dnt|t r| \}}t|jtr|j|jjkr|j|}n8t|jtr|j|jj kr|j|}n | |j}|j|}|||St|trr<r3r8Zring_newrustrr from_expr)rerwrrrqr)r)r* field_newsB                     zFracField.field_newcs:|jtddDfddt|S)Ncss*|]"}|jst|tr||fVqdSrf)is_PowrZr as_base_exprbr)r)r* sz*FracField._rebuild_expr..cs|}|dur|S|jr2tttt|jS|jrNtttt|jS|j sbt |t t fr| \}}D]<\}\}}||krrt||dkrr|t||Sqr|jr|tjurЈ|t|Sz |WStyjsjr|YSYn0dS)Nr)rMZis_Addrrr3r8argsZis_MulrrrZrr rr intZ is_IntegerrZOnervrrzr{r|)rBr`bercbgZeg_rebuildr&mappingZpowersr)r*rs(   z)FracField._rebuild_expr.._rebuild)r&r>keysr)rerBrr)rr* _rebuild_exprszFracField._rebuild_exprcCsXttt|j|j}z|||}Wn"tyHtd||fYn 0||SdS)NzGexpected an expression convertible to a rational function in %s, got %s) dictr3rYr%r#rrrkr)rerBrfracr)r)r*r s  zFracField.from_exprcCst|Srfrrdr)r)r* to_domainszFracField.to_domaincCsddlm}||j|j|jS)NrrF)rIrGr%r&r')rerGr)r)r*rs zFracField.to_ring)N)N)rK __module__ __qualname____doc__rrOrXrgrhrirnrorsrurxryr__call__rrrrr)r)r)r*r"hs$ &  %! r"c@s<eZdZdZdKddZddZddZd d Zd d Zd dZ dZ ddZ ddZ ddZ ddZddZddZddZddZdd Zd!d"Zd#d$Zd%d&Zd'd(Zd)d*Zd+d,Zd-d.Zd/d0Zd1d2Zd3d4Zd5d6Zd7d8Zd9d:Z d;d<Z!d=d>Z"d?d@Z#dAdBZ$dCdDZ%dLdEdFZ&dMdGdHZ'dNdIdJZ(dS)OrHz=Element of multivariate distributed rational function field. NcCs0|dur|jjj}n |s td||_||_dS)Nzzero denominator)r+rSrWZeroDivisionErrorrqrrrpr)r)r*__init__s  zFracElement.__init__cCs |||Srf) __class__frqrrr)r)r*rs(szFracElement.raw_newcCs|j||Srf)rsrtrr)r)r*ru*szFracElement.newcCs|jdkrtd|jS)Nzf.denom should be 1)rrrkrqrr)r)r*rj-s zFracElement.to_polycCs |jSrf)r+rrdr)r)r*parent2szFracElement.parentcCs|j|j|jfSrf)r+rqrrrdr)r)r*rg5szFracElement.__getnewargs__cCs,|j}|dur(t|j|j|jf|_}|Srf)rRrQr+rqrr)rerRr)r)r*rh:szFracElement.__hash__cCs||j|jSrf)rsrqcopyrrrdr)r)r*r@szFracElement.copycCs<|j|kr|S|j}|j|}|j|}|||SdSrf)r+rSrqrrrru)reZ new_fieldZnew_ringrqrrr)r)r* set_fieldCs    zFracElement.set_fieldcGs|jj||jj|Srf)rqas_exprrr)rer%r)r)r*rLszFracElement.as_exprcCsLt|tr.|j|jkr.|j|jko,|j|jkS|j|koF|j|jjjkSdSrf)rZrHr+rqrrrSrWrgr)r)r*rnOszFracElement.__eq__cCs ||k Srfr)rr)r)r*roUszFracElement.__ne__cCs t|jSrf)boolrqrr)r)r*__bool__XszFracElement.__bool__cCs|j|jfSrf)rrsort_keyrqrdr)r)r*r[szFracElement.sort_keycCs(t||jjr |||StSdSrf)rZr+rUrNotImplemented)f1f2opr)r)r*_cmp^szFracElement._cmpcCs ||tSrf)rrrrr)r)r*__lt__dszFracElement.__lt__cCs ||tSrf)rrrr)r)r*__le__fszFracElement.__le__cCs ||tSrf)rr rr)r)r*__gt__hszFracElement.__gt__cCs ||tSrf)rr rr)r)r*__ge__jszFracElement.__ge__cCs||j|jSz"Negate all coefficients in ``f``. rsrqrrrr)r)r*__pos__mszFracElement.__pos__cCs||j |jSrrrr)r)r*__neg__qszFracElement.__neg__c Cs|jj}z||}Wndtyz|jst|jrt|}z||}WntyXYn0d||||fYSYdS0d|dfSdS)N)rNNr) r+r&rvrrzr{r|rqrr)rerwr&r}r)r)r*_extract_groundus   zFracElement._extract_groundcCs(|j}|s|S|s|St||jrn|j|jkrD||j|j|jS||j|j|j|j|j|jSnt||jjr||j|j||jSt|trt|jt r|jj|jkrn*t|jjt r|jjj|kr| |St Sn6t|t rt|jt r|jj|jkrn | |S| |S)z(Add rational functions ``f`` and ``g``. )r+rZrUrrrurqrSrHr&r__radd__rrrrrr+r)r)r*__add__s,  *    zFracElement.__add__cCst||jjjr*||j|j||jS||\}}}|dkr\||j|j||jS|sdtS||j||j||j|SdSNr rZr+rSrUrurqrrrrrcrg_numerg_denomr)r)r*rszFracElement.__radd__cCs|j}|s|S|s| St||jrp|j|jkrF||j|j|jS||j|j|j|j|j|jSnt||jjr||j|j||jSt|trt|jt r|jj|jkrn*t|jjt r|jjj|kr| |St Sn6t|t r t|jt r|jj|jkrn | |S||\}}}|dkrT||j|j||jS|s^t S||j||j||j|SdS)z-Subtract rational functions ``f`` and ``g``. rN)r+rZrUrrrurqrSrHr&r__rsub__rrrrrrr+rrrr)r)r*__sub__s6  *     zFracElement.__sub__cCst||jjjr,||j |j||jS||\}}}|dkr`||j |j||jS|shtS||j ||j||j|SdSrrrr)r)r*rszFracElement.__rsub__cCs|j}|r|s|jSt||jr<||j|j|j|jSt||jjr^||j||jSt|trt|j t r|j j|jkrqt|jj t r|jj j|kr| |St Sn0t|t rt|j tr|j j|jkrn | |S| |S)z-Multiply rational functions ``f`` and ``g``. )r+rVrZrUrurqrrrSrHr&r__rmul__rrrrr)r)r*__mul__s$     zFracElement.__mul__cCstt||jjjr$||j||jS||\}}}|dkrP||j||jS|sXtS||j||j|SdSrrrr)r)r*rszFracElement.__rmul__cCs0|j}|stnt||jr8||j|j|j|jSt||jjrZ||j|j|St|trt|j t r|j j|jkrqt|jj t r|jj j|kr| |St Sn0t|t rt|j tr|j j|jkrn | |S||\}}}|dkr ||j|j|S|st S||j||j|SdS)z0Computes quotient of fractions ``f`` and ``g``. rN)r+rrZrUrurqrrrSrHr&r __rtruediv__rrrrrr)r)r* __truediv__s.      zFracElement.__truediv__cCs~|s tn$t||jjjr.||j||jS||\}}}|dkrZ||j||jS|sbt S||j||j|SdSr) rrZr+rSrUrurrrqrrrr)r)r*r0szFracElement.__rtruediv__cCsJ|dkr ||j||j|S|s*tn||j| |j| SdS)z+Raise ``f`` to a non-negative power ``n``. rN)rsrqrrr)rnr)r)r*__pow__?s zFracElement.__pow__cCs:|}||j||j|j|j||jdS)aComputes partial derivative in ``x``. Examples ======== >>> from sympy.polys.fields import field >>> from sympy.polys.domains import ZZ >>> _, x, y, z = field("x,y,z", ZZ) >>> ((x**2 + y)/(z + 1)).diff(x) 2*x/(z + 1) r7)rjrurqdiffrr)rxr)r)r*rHszFracElement.diffcGsTdt|kr|jjkr8nn|tt|jj|Std|jjt|fdS)Nrz1expected at least 1 and at most %s values, got %s)r<r+rJevaluater3rYr#rk)rr4r)r)r*rYs zFracElement.__call__cCsxt|tr<|dur.) rZr3rqrrrrjrSrru)rrrrqrrr+r)r)r*r_s zFracElement.evaluatecCsnt|tr<|dur.)rZr3rqsubsrrrjru)rrrrqrrr)r)r*rjs zFracElement.subscCstdSrf)r)rrrr)r)r*composetszFracElement.compose)N)N)N)N))rKrrrrrsrurjrrgrRrhrrrrnrorrrrrrrrrrrrrrrrrrrrrrrrr)r)r)r*rHsL   &  !  rHN)>rtypingrrZtDict functoolsroperatorrrrrr r Zsympy.core.exprr Zsympy.core.modr Zsympy.core.numbersr Zsympy.core.singletonrZsympy.core.symbolrZsympy.core.sympifyrrZ&sympy.functions.elementary.exponentialrZ!sympy.polys.domains.domainelementrZ!sympy.polys.domains.fractionfieldrZ"sympy.polys.domains.polynomialringrZsympy.polys.constructorrZsympy.polys.orderingsrZsympy.polys.polyerrorsrZsympy.polys.polyoptionsrZsympy.polys.polyutilsrrIrZsympy.printing.defaultsrZsympy.utilitiesrZsympy.utilities.iterablesrZsympy.utilities.magicr r+r,r2rErLr"rHr)r)r)r*sF                      45