~9\c@ s}dZddlmZmZddlmZmZmZmZm Z m Z m Z m Z m Z mZddlmZmZddlmZddlmZddlmZddlmZmZdd lmZdd lmZdd lm Z m!Z!dd l"m#Z#dd l$m%Z%m&Z&m'Z'm(Z(ddl)m*Z*m+Z+m,Z,m-Z-m.Z.m/Z/m0Z0m1Z1m2Z2m3Z3ddl4m5Z5m6Z6ddl7m8Z8ddl9m:Z:ddl;m<Z<ddl=m>Z>ddl?m@Z@ddlAmBZBddlCmDZDddlEmFZFmGZGmHZHmIZImJZJddlKmLZLmMZMe!dddZNdZOdZPd6d6dZRdZSd ZTd6d!ZUd"ZVd#ZWd$ZXd%ZYd&ZZd'Z[d(Z\eId6e]e^d6d)Z_d*Z`e_Zaebjcd+d,ZdeId6d-Zed.Zfd/Zgd0ZheId1ZieId6d2Zjd3e@fd4YZkeId6e^d5Zld6S(7s&Computational algebraic field theory. i(tprint_functiontdivision( tStRationaltAlgebraicNumbertAddtMultsympifytDummyt expand_multItpi(treducetrange(tFactors(t_mexpand(texp(tcostsin(tsieve(tdivisors(tZZtQQ(tdup_chebyshevt(tIsomorphismFailedtCoercionFailedt NotAlgebraictGeneratorsError( tPolytPurePolytinvertt factor_listtgroebnert resultanttdegreetpoly_from_exprtparallel_poly_from_exprtlcm(tdict_from_exprtexpr_from_dict(trs_compose_add(tring(tCRootOf(tcyclotomic_poly(t LambdaPrinter(t _split_gcd(t _is_sum_surds(tnumbered_symbolst variationstlambdifytpublictsift(tpslqtmpiicC st|dtr3g|D]}|d^q}nt|dkrM|dSi||6}t|drr|jng}tdd} xt|t|D]} d} | } x&|D]} | ||| <| |} qWxtrg}| | d}xB|D]:}t|j j | ||kr|j |qqW|rJ|}nt|dkrd|dS| |krtPn| d9} qWqWt d|dS(se Return a factor having root ``v`` It is assumed that one of the factors has root ``v``. iitsymbolsi is4multiple candidates for the minimal polynomial of %sN( t isinstancettupletlenthasattrR6RR tTruetabstas_exprtevalftappendtNotImplementedError(tfactorstxtvtdomtprectboundtftpointsR6tttntprec1tn_temptst candidatesteps((s7lib/python2.7/site-packages/sympy/polys/numberfields.pyt_choose_factor,s6     $  cC s5ddlm}d}g}x|jD]}|js||ra|jtj|dfq)|jr|j|tjfq)|jr|j j r|j|tjfq)t q)n||j|dt \}}|jt |t |dfq)W|jdd|ddtjkr0|Sg|D]\}}|^q7}x.tt|D]} || dkrbPqbqbWt|| \} } } g} g}xS|D]K\}}|| kr| j||tjq|j||tjqWt| }t|}t|dt|d}|S( s7 helper function for ``_minimal_polynomial_sq`` It selects a rational ``g`` such that the polynomial ``p`` consists of a sum of terms whose surds squared have gcd equal to ``g`` and a sum of terms with surds squared prime with ``g``; then it takes the field norm to eliminate ``sqrt(g)`` See simplify.simplify.split_surds and polytools.sqf_norm. Examples ======== >>> from sympy import sqrt >>> from sympy.abc import x >>> from sympy.polys.numberfields import _separate_sq >>> p= -x + sqrt(2) + sqrt(3) + sqrt(7) >>> p = _separate_sq(p); p -x**2 + 2*sqrt(3)*x + 2*sqrt(7)*x - 2*sqrt(21) - 8 >>> p = _separate_sq(p); p -x**4 + 4*sqrt(7)*x**3 - 32*x**2 + 8*sqrt(7)*x + 20 >>> p = _separate_sq(p); p -x**8 + 48*x**6 - 536*x**4 + 1728*x**2 - 400 i(R3cS s|jo|jtjkS(N(tis_PowRRtHalf(texpr((s7lib/python2.7/site-packages/sympy/polys/numberfields.pytis_sqrtmsitbinarytkeycS s|dS(Ni((tz((s7lib/python2.7/site-packages/sympy/polys/numberfields.pyt~ti(tsympy.utilities.iterablesR3targstis_MulR?RtOnetis_AtomRQRt is_integerR@R;RtsortR R9R-RRRR(tpR3RTtatytTtFRWtsurdstitgtb1tb2ta1ta2tp1tp2((s7lib/python2.7/site-packages/sympy/polys/numberfields.pyt _separate_sqRsB    '   cC s.ddlm}t|}t|}|j sL|dk sL|| rPdS|td|}||8}x@t|}||kr|ji|||6}Pqp|}qpW|dkrt|}|j ||j |dkr| }n|j d}|St |d}t |||}|S(s Returns the minimal polynomial for the ``nth-root`` of a sum of surds or ``None`` if it fails. Parameters ========== p : sum of surds n : positive integer x : variable of the returned polynomial Examples ======== >>> from sympy.polys.numberfields import _minimal_polynomial_sq >>> from sympy import sqrt >>> from sympy.abc import x >>> q = 1 + sqrt(2) + sqrt(3) >>> _minimal_polynomial_sq(q, 3, x) x**12 - 4*x**9 - 4*x**6 + 16*x**3 - 8 i(R.iiN(tsympy.simplify.simplifyR.Rt is_IntegertNoneRRotsubsRtcoeffR"t primitiveRRP(RaRJRBR.tpnRmRAtresult((s7lib/python2.7/site-packages/sympy/polys/numberfields.pyt_minimal_polynomial_sqs,  $      " cC s(tt|}|dkr3t|||}n|dkrTt|||}n|ji||6}|tkr |tkrtdt\}} |t|d} |t|d} q9t |||f||\\} } } | j | } | j }n-|t kr-t |||}n td|t ksQ|tkrot||d||g} n$t| | } t| j|} t||}t||}|t kr|dks|dkr| St| |d|} | j\} }t||||||}|j S(s return the minimal polynomial for ``op(ex1, ex2)`` Parameters ========== op : operation ``Add`` or ``Mul`` ex1, ex2 : expressions for the algebraic elements x : indeterminate of the polynomials dom: ground domain mp1, mp2 : minimal polynomials for ``ex1`` and ``ex2`` or None Examples ======== >>> from sympy import sqrt, Add, Mul, QQ >>> from sympy.polys.numberfields import _minpoly_op_algebraic_element >>> from sympy.abc import x, y >>> p1 = sqrt(sqrt(2) + 1) >>> p2 = sqrt(sqrt(2) - 1) >>> _minpoly_op_algebraic_element(Mul, p1, p2, x, QQ) x - 1 >>> q1 = sqrt(y) >>> q2 = 1 / y >>> _minpoly_op_algebraic_element(Add, q1, q2, x, QQ.frac_field(y)) x**2*y**2 - 2*x*y - y**3 + 1 References ========== .. [1] https://en.wikipedia.org/wiki/Resultant .. [2] I.M. Isaacs, Proc. Amer. Math. Soc. 25 (1970), 638 "Degrees of sums in a separable field extension". tXisoption not availabletgensitdomainN(RtstrRrt_minpoly_composeRsRRR)R&R$tcomposeR=Rt_mulyR@R!R(R't as_expr_dictR"RRRP(toptex1tex2RBRDtmp1tmp2RctRRyRmRnt_trtmp1atdeg1tdeg2RAtres((s7lib/python2.7/site-packages/sympy/polys/numberfields.pyt_minpoly_op_algebraic_elements:$    (  $cC s]t||d}t|}g|jD]!\\}}||||^q,}t|S(s@ Returns ``expand_mul(x**degree(p, x)*p.subs(x, 1/x))`` i(R#R"ttermsR(RaRBRmRJRgtcRb((s7lib/python2.7/site-packages/sympy/polys/numberfields.pyt_invertxs 4cC set||d}t|}g|jD])\\}}||||||^q,}t|S(s8 Returns ``_mexpand(y**deg*p.subs({x:x / y}))`` i(R#R"RR(RaRBRcRmRJRgRRb((s7lib/python2.7/site-packages/sympy/polys/numberfields.pyR%s <c C sCt|}|s't|||}n|jsCtd|n|dkr||krntd|nt||}|dkr|S| }d|}ntt|}|ji||6}|j \}}t t |||||d|g|d|}|j \} } t | ||||}|jS(s Returns ``minpoly(ex**pw, x)`` Parameters ========== ex : algebraic element pw : rational number x : indeterminate of the polynomial dom: ground domain mp : minimal polynomial of ``p`` Examples ======== >>> from sympy import sqrt, QQ, Rational >>> from sympy.polys.numberfields import _minpoly_pow, minpoly >>> from sympy.abc import x, y >>> p = sqrt(1 + sqrt(2)) >>> _minpoly_pow(p, 2, x, QQ) x**2 - 2*x - 1 >>> minpoly(p**2, x) x**2 - 2*x - 1 >>> _minpoly_pow(y, Rational(1, 3), x, QQ.frac_field(y)) x**3 - y >>> minpoly(y**Rational(1, 3), x) x**3 - y s*%s doesn't seem to be an algebraic elementis %s is zeroiiRzR{(RR}t is_rationalRtZeroDivisionErrorRRR|Rstas_numer_denomRR!RRPR=( textpwRBRDR5RcRJtdRRRA((s7lib/python2.7/site-packages/sympy/polys/numberfields.pyt _minpoly_pow0s(      3c G sstt|d|d||}|d|d}x:|dD].}tt||||d|}||}q=W|S(s. returns ``minpoly(Add(*a), dom, x)`` iiiR(RR(RBRDRbR5Ratpx((s7lib/python2.7/site-packages/sympy/polys/numberfields.pyt _minpoly_addes  c G sstt|d|d||}|d|d}x:|dD].}tt||||d|}||}q=W|S(s. returns ``minpoly(Mul(*a), dom, x)`` iiiR(RR(RBRDRbR5RaR((s7lib/python2.7/site-packages/sympy/polys/numberfields.pyt _minpoly_mulqs  c C s|jdj\}}|tkr|jr|j}t|}|jrt|t}t gt |D] }|||d||^qkS|j dkr|dkrd|dd|dd|d d Sn|d dkr^t|t}gt |dD]}|||||^q}t |}t |\}} t | ||} | Sdtd |td tj} t| |t} | Sntd |d S( st Returns the minimal polynomial of ``sin(ex)`` see http://mathworld.wolfram.com/TrigonometryAngles.html iii i@ii`ii$iis*%s doesn't seem to be an algebraic elementN(R[t as_coeff_MulR RtqRtis_primeRRRR RaRRPRRRRR}RR( RRBRRbRJRRgRRRARRS((s7lib/python2.7/site-packages/sympy/polys/numberfields.pyt _minpoly_sin}s,     7 +3 #c C sddlm}|jdj\}}|tkr|jr|jdkr|jdkrd|dd|d d|dS|jd kr d|dd |dSna|jd kr t|j}|j r t ||}t |j i|d|d |6Snt |j}t|t}gt|dD]}|||||^q:}t|d|j} t| \} } t| ||} | Sntd |d S(st Returns the minimal polynomial of ``cos(ex)`` see http://mathworld.wolfram.com/TrigonometryAngles.html i(tsqrtiiiiiiii is*%s doesn't seem to be an algebraic elementN(tsympyRR[RR RRaRRRRRRstintRRR RRRPR( RRBRRRbRRMRJRgRRRAR((s7lib/python2.7/site-packages/sympy/polys/numberfields.pyt _minpoly_coss,  $ +3c C s|jdj\}}t|j}|ttkr|jr|jdks_|jdkrW|dkr{|d|dS|dkr|ddS|dkr|d|ddS|dkr|ddS|d kr|d|ddS|d kr|d|d|d|ddS|jrWd}x#t |D]}|| |7}q7W|Sngt d|D]}t ||^qh}t |||}|St d |nt d |d S( s7 Returns the minimal polynomial of ``exp(ex)`` iiiiiiiii i s*%s doesn't seem to be an algebraic elementN(R[RRRR R RRaRR RR+RPR( RRBRRbRRMRgRAR5((s7lib/python2.7/site-packages/sympy/polys/numberfields.pyt _minpoly_exps6         $ ,cC sT|j}|ji||jjd6}t||\}}t|||}|S(sA Returns the minimal polynomial of a ``CRootOf`` object. i(RSRstpolyRzRRP(RRBRaRRARw((s7lib/python2.7/site-packages/sympy/polys/numberfields.pyt_minpoly_rootofs   c C sn|jr|j||jS|tkrpt|dd|d|\}}t|dkrh|ddS|tSt|dr||jkr||S|jrt |r||8}x)t |}||kr|S|}qWn|j rt |||j }nh|jrt|j}t|jd}|tr|tkrtg|t|d D]\} }| |^q^}t|t} g| jD]} | j^q} tt| d} tj}| j|tj}g| jD]$\}} || j| | j^q}t|}t ||}|j|| |j||| }|||t!d| }t"t||||d|d|}qjt#|||j }n|j$rt%|j&|j'||}n|j(t)krt*||}ns|j(t+krt,||}nR|j(t'kr9t-||}n1|j(t.krZt/||}nt0d||S( s Computes the minimal polynomial of an algebraic element using operations on minimal polynomials Examples ======== >>> from sympy import minimal_polynomial, sqrt, Rational >>> from sympy.abc import x, y >>> minimal_polynomial(sqrt(2) + 3*Rational(1, 3), x, compose=True) x**2 - 2*x - 1 >>> minimal_polynomial(sqrt(y) + 1/y, x, compose=True) x**2*y**2 - 2*x*y - y**3 + 1 iiR{R6cS s|djo|djS(Nii(t is_Rational(titx((s7lib/python2.7/site-packages/sympy/polys/numberfields.pyRXRYRRs*%s doesn't seem to be an algebraic elementN(1RRRaR RR9R:R6tis_QQR.Rotis_AddRR[R\RRAR3titemsR;RRtFalseRrtdicttvaluesR R%Rt NegativeOnetpoptZerotminimal_polynomialRRRRQRtbaseRt __class__RRRRRR*RR(RRBRDRRARRRGRtbxtr1Rctdenstlcmdenstneg1texpn1RtnumsRRR((s7lib/python2.7/site-packages/sympy/polys/numberfields.pyR}sZ  #&      5" 7 $' c C sddlm}ddlm}ddlm}t|}|jrZt|dt }nx'||D]}|j rgt }PqgqgW|d k rt|t }} ntdt}} |s|jr|tt|j}qt}nt|dr#||jkr#td||fn|rt|||} | jd } | j||| |} | jrt| } n|r| | |d t S| j|S|jstd nt||| } |r| | |d t S| j|S( s- Computes the minimal polynomial of an algebraic element. Parameters ========== ex : Expr Element or expression whose minimal polynomial is to be calculated. x : Symbol, optional Independent variable of the minimal polynomial compose : boolean, optional (default=True) Method to use for computing minimal polynomial. If ``compose=True`` (default) then ``_minpoly_compose`` is used, if ``compose=False`` then groebner bases are used. polys : boolean, optional (default=False) If ``True`` returns a ``Poly`` object else an ``Expr`` object. domain : Domain, optional Ground domain Notes ===== By default ``compose=True``, the minimal polynomial of the subexpressions of ``ex`` are computed, then the arithmetic operations on them are performed using the resultant and factorization. If ``compose=False``, a bottom-up algorithm is used with ``groebner``. The default algorithm stalls less frequently. If no ground domain is given, it will be generated automatically from the expression. Examples ======== >>> from sympy import minimal_polynomial, sqrt, solve, QQ >>> from sympy.abc import x, y >>> minimal_polynomial(sqrt(2), x) x**2 - 2 >>> minimal_polynomial(sqrt(2), x, domain=QQ.algebraic_field(sqrt(2))) x - sqrt(2) >>> minimal_polynomial(sqrt(2) + sqrt(3), x) x**4 - 10*x**2 + 1 >>> minimal_polynomial(solve(x**3 + x + 3)[0], x) x**3 + x + 3 >>> minimal_polynomial(sqrt(y), x) x**2 - y i(R"(t FractionField(tpreorder_traversalt recursiveRBR6s5the variable %s is an element of the ground domain %sitfields!groebner method only works for QQN(tsympy.polys.polytoolsR"tsympy.polys.domainsRtsympy.core.basicRRt is_numberRR;tis_AlgebraicNumberRRrRRRt free_symbolsRtlistR:R6RR}RuRtt is_negativeR tcollectRR@t_minpoly_groebner( RRBR~tpolysR{R"RRRStclsRwR((s7lib/python2.7/site-packages/sympy/polys/numberfields.pyR<s>6       & c s5ddlm}ddlm}tddtiid fdfdd}t}||}|jr|j j |S|j r|j ||j }n$||}|r|d}nd }|jr&d |jjr&d |j} t|j| |}n$t|rJt|tj|}n|d k r_|}n|d kr|} || gtj} t| tj|gd d } t| d\} }t|||}n|r1t||}|j||||d kr1t| }q1n|S(s/ Computes the minimal polynomial of an algebraic number using Groebner bases Examples ======== >>> from sympy import minimal_polynomial, sqrt, Rational >>> from sympy.abc import x >>> minimal_polynomial(sqrt(2) + 3*Rational(1, 3), x, compose=False) x**2 - 2*x - 1 i(R"(texpand_multinomialRbRc sNt}||<|dk r7||||d|jjr>|jdkr>|jjr;tSq>n|jrt}xI|jD]>}|jrjtS|jrW|jjr|jdkrtSqWqWW|rtSntS(s Returns True if it is more likely that the minimal polynomial algorithm works better with the inverse ii( RQRR_RRR;R\R[R(RthitRa((s7lib/python2.7/site-packages/sympy/polys/numberfields.pytsimpler_inverses       itordertlexiN(RR"tsympy.core.functionRR/RRrRRRR=RRRaRQRRqRxRR.RR]RRR RRPRRtR (RRBRR"RRtinvertedRwRRJtbusRetGRRA((RRRR6Rs7lib/python2.7/site-packages/sympy/polys/numberfields.pyRsF  1            ("Rcc sTxMtddddddg|dtD]$}|ddkr(t|Vq(q(Wd S( s1Generate coefficients for `primitive_element()`. iiiiiit repetitioniN(R0R;R(RJtcoeffs((s7lib/python2.7/site-packages/sympy/polys/numberfields.pyt_coeffs_generator*s.cK sX|stdn|dk r7t|t}}ntdt}}|jdtsM|ddg}}t|t r|j j |}nt ||dt }xl|dD]`}t|d|\}} t| ||}|j\} }}|| |7}|j| qW|jdts:|j|fS|||fSntdd t} gg} } xz|D]r}t| }|jr|jr|j|}qtd |nt ||}| j|| j|qsW|jd t}x|t| D]}|tgt|| D]\}}||^q*}t| |g| |gd d dt }|d ||d|dd}}xftt|| D]N\}\}}y't|||ddj||R5tdpsR4RRRRtreversedRR~tremtis_zeroR9R(RbRRGRhRJRtprevRgtAtBtbasisRRRRRtapproxRt((s7lib/python2.7/site-packages/sympy/polys/numberfields.pytfield_isomorphism_pslqsB 7    + c C st|jd|\}}x|D]\}}|jdkr"|jjj}t|dg}}x6t|D](\}} |j| |j ||qyWt |} |j | j dt dkr|S|j | j dt dkrg|D] } | ^qSq"q"WdSdS(s/Construct field isomorphism via factorization. RitchopiN(RRR"treptTCt to_sympy_listR9RR?RRR>R;Rr( RbRRRARGRRRRgRtRR((s7lib/python2.7/site-packages/sympy/polys/numberfields.pytfield_isomorphism_factors  cK st|t|}}|js1t|}n|jsIt|}n||kr_|jS|jj}|jj}|dkr|jgS||dkrdS|jdt ry#t ||}|dk r|SWqt k rqXnt ||S(s4Construct an isomorphism between two number fields. iitfastN( RRRRRR"RRrRR;R R@R(RbRR[RJRRw((s7lib/python2.7/site-packages/sympy/polys/numberfields.pytfield_isomorphisms*        c K sD|jd}t|dr-t|}n |g}t|dkrlt|dtkrlt|dSt||dt\}}t gt ||D]\}}||^q}|dkrt||fSt |}|j st|d|}nt||}|dk r't||Std||jfdS(s7Express `extension` in the field generated by `theta`. Rt__iter__iiRs%s is not in a subfield of %sN(RR:RR9ttypeR8RRR;RRRrRRRRR( RtthetaR[RRRRtRR((s7lib/python2.7/site-packages/sympy/polys/numberfields.pytto_number_fields$ (2     tIntervalPrintercB s)eZdZdZdZdZRS(s?Use ``lambda`` printer but print numbers as ``mpi`` intervals. cC sdtt|j|S(Ns mpi('%s')(tsuperRt_print_Integer(tselfRS((s7lib/python2.7/site-packages/sympy/polys/numberfields.pyR2scC sdtt|j|S(Ns mpi('%s')(RRt_print_Rational(RRS((s7lib/python2.7/site-packages/sympy/polys/numberfields.pyR5scC stt|j|dtS(Ntrational(RRt _print_PowR;(RRS((s7lib/python2.7/site-packages/sympy/polys/numberfields.pyR8s(t__name__t __module__t__doc__RRR(((s7lib/python2.7/site-packages/sympy/polys/numberfields.pyR/s  c C s;t|}|jr||fS|js7tdntd |dddt}t|dt}|jdt}t j t }}zhxa|s|}xK|D]4\}} ||j kr|j | krt}PqqWt j d9_ qWWd|t _ X|dk r1|j|| d |d |\}} n|| fS( s<Give a rational isolating interval for an algebraic number. s+complex algebraic numbers are not supportedtmodulestmpmathtprinterRtsqfiNROR((RRRR@R1RRR;t intervalsR5RRRbRRrt refine_root( RRORtfuncRR'RtdoneRbR((s7lib/python2.7/site-packages/sympy/polys/numberfields.pytisolate<s,         'N(mR"t __future__RRRRRRRRRRR R R tsympy.core.compatibilityR R tsympy.core.exprtoolsRRRt&sympy.functions.elementary.exponentialRt(sympy.functions.elementary.trigonometricRRt sympy.ntheoryRtsympy.ntheory.factor_RRRRtsympy.polys.orthopolysRtsympy.polys.polyerrorsRRRRRRRRRR R!R"R#R$R%tsympy.polys.polyutilsR&R'tsympy.polys.ring_seriesR(tsympy.polys.ringsR)tsympy.polys.rootoftoolsR*tsympy.polys.specialpolysR+tsympy.printing.lambdareprR,tsympy.simplify.radsimpR-RpR.tsympy.utilitiesR/R0R1R2R3R$R4R5RPRoRxRrRRRRRRRRRRR}R;RRRRt__all__R?RRRR RRRRR+(((s7lib/python2.7/site-packages/sympy/polys/numberfields.pytslF"F(& A 8O 5 & $ J_   K  6 #