~9\c@ sdZddlmZmZddlmZmZmZmZm Z m Z m Z m Z m Z mZmZddlmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZm Z m!Z!m"Z"m#Z#m$Z$m%Z%m&Z&m'Z'm(Z(ddl)m*Z*m+Z+m,Z,m-Z-m.Z.m/Z/m0Z0m1Z1m2Z2m3Z3m4Z4m5Z5m6Z6m7Z7m8Z8m9Z9m:Z:m;Z;m<Z<m=Z=m>Z>m?Z?m@Z@mAZAmBZBmCZCddlDmEZEmFZFmGZGmHZHmIZImJZJmKZKmLZLmMZMmNZNmOZOmPZPmQZQmRZRmSZSddlTmUZUmVZVmWZWddlXmYZYmZZZm[Z[m\Z\m]Z]dd l^m_Z_dd l`maZadd lbmcZcmdZdmeZemfZfdd lgmhZhmiZimjZjdd lkmlZlddlmmnZompZqddlrmsZsdZtdZudZvdZwdZxdZydZzdZ{dZ|e}dZ~dZdZdZdZdZdZd Zd!Zd"Zd#Zd$Zd1d1d%Zd&Zd'Zd(Zd)Zd*Zd+Zd,Zd-Zd.Zd/Zd0Zd1S(2s:Polynomial factorization routines in characteristic zero. i(tprint_functiontdivision( tgf_from_int_polytgf_to_int_polyt gf_lshiftt gf_add_multgf_multgf_divtgf_remtgf_gcdextgf_sqf_pt gf_factor_sqft gf_factor(tdup_LCtdmp_LCt dmp_ground_LCtdup_TCt dup_convertt dmp_convertt dup_degreet dmp_degreet dmp_degree_intdmp_degree_listt dmp_from_dictt dmp_zero_ptdmp_onetdmp_nestt dmp_raiset dup_stript dmp_groundt dup_inflatet dmp_excludet dmp_includet dmp_injectt dmp_ejectt dup_terms_gcdt dmp_terms_gcd(tdup_negtdmp_negtdup_addtdmp_addtdup_subtdmp_subtdup_multdmp_multdup_sqrtdmp_powtdup_divtdmp_divtdup_quotdmp_quot dmp_expandt dmp_add_mult dup_sub_mult dmp_sub_mult dup_lshiftt dup_max_normt dmp_max_normt dup_l1_normtdup_mul_groundtdmp_mul_groundtdup_quo_groundtdmp_quo_ground(tdup_clear_denomstdmp_clear_denomst dup_trunctdmp_ground_trunct dup_contentt dup_monictdmp_ground_monict dup_primitivetdmp_ground_primitivet dmp_eval_tailt dmp_eval_intdmp_diff_eval_int dmp_composet dup_shiftt dup_mirror(t dmp_primitivet dup_inner_gcdt dmp_inner_gcd(t dup_sqf_pt dup_sqf_normt dmp_sqf_normt dup_sqf_partt dmp_sqf_part(t _sort_factors(tquery(tExtraneousFactorst DomainErrortCoercionFailedtEvaluationFailed(t nextprimetisprimet factorint(tsubsets(tceiltlog(trangecC szg}xg|D]_}d}x=trXt|||\}}|sT||d}}qPqW|j||fq Wt|S(s:Determine multiplicities of factors using trial division. ii(tTrueR/tappendRV(tftfactorstKtresulttfactortktqtr((s6lib/python2.7/site-packages/sympy/polys/factortools.pytdup_trial_divisionRs  c C sg}xs|D]k}d}xItrdt||||\}}t||r`||d}}qPqW|j||fq Wt|S(s:Determine multiplicities of factors using trial division. ii(RcR0RRdRV( ReRftuRgRhRiRjRkRl((s6lib/python2.7/site-packages/sympy/polys/factortools.pytdmp_trial_divisionfs  cC sWt||}tt||}t|}|j||dd|||S(s5Mignotte bound for univariate polynomials in `K[x]`. ii(R8tabsR Rtsqrt(ReRgtatbtn((s6lib/python2.7/site-packages/sympy/polys/factortools.pytdup_zz_mignotte_boundzs cC sft|||}tt|||}tt||}|j||dd|||S(s7Mignotte bound for multivariate polynomials in `K[X]`. ii(R9RpRtsumRRq(ReRnRgRrRsRt((s6lib/python2.7/site-packages/sympy/polys/factortools.pytdmp_zz_mignotte_boundscC s|d}t||||}t|||}tt|||||\} } t| ||} t| ||} tt|||t| |||} tt|| |||} tt|| |||} tt|| |t|| ||} tt| |jg|||}tt|||| |\}}t|||}t|||}tt|||t|| ||} tt|||||}tt|| |||}| | ||fS(s One step in Hensel lifting in `Z[x]`. Given positive integer `m` and `Z[x]` polynomials `f`, `g`, `h`, `s` and `t` such that:: f == g*h (mod m) s*g + t*h == 1 (mod m) lc(f) is not a zero divisor (mod m) lc(h) == 1 deg(f) == deg(g) + deg(h) deg(s) < deg(h) deg(t) < deg(g) returns polynomials `G`, `H`, `S` and `T`, such that:: f == G*H (mod m**2) S*G + T**H == 1 (mod m**2) References ========== .. [1] [Gathen99]_ i(R5RAR/R+R'R)tone(tmRetgthtsttRgtMteRkRlRntGtHRstctdtStT((s6lib/python2.7/site-packages/sympy/polys/factortools.pytdup_zz_hensel_steps$ $**$$*c C st|}t||}|dkrdt||j|||d|}t||||gS|}|d} ttt|d} t|g|} x0|| D]$} t | t| |||} qWt|| |} x4|| dD]$} t | t| |||} qWt | | ||\}}}t | |} t | |} t ||}t ||}xPt d| dD];}t ||| | ||||d\} } }}}qWt|| || ||t|| || ||S(s Multifactor Hensel lifting in `Z[x]`. Given a prime `p`, polynomial `f` over `Z[x]` such that `lc(f)` is a unit modulo `p`, monic pair-wise coprime polynomials `f_i` over `Z[x]` satisfying:: f = lc(f) f_1 ... f_r (mod p) and a positive integer `l`, returns a list of monic polynomials `F_1`, `F_2`, ..., `F_r` satisfying:: f = lc(f) F_1 ... F_r (mod p**l) F_i = f_i (mod p), i = 1..r References ========== .. [1] [Gathen99]_ iii(tlenR R;tgcdexRAtintt_ceilt_logRRR RRbRtdup_zz_hensel_lift(tpRetf_listtlRgRltlctFRyRjRRztf_iR{R|R}t_((s6lib/python2.7/site-packages/sympy/polys/factortools.pyRs.  & ""9cC s5||dkr||}n|s'tS||dkS(Nii(Rc(tfcRktpl((s6lib/python2.7/site-packages/sympy/polys/factortools.pyt_test_pls  c!C st|}|dkr|gS|d}t||}t||}tt|j||dd|||}t|dd||d|d}ttdt|d}td|t|} g} xtd| dD]} t |  s|| dkr qn|j | } t || } t | | |sVqnt | | |d} | j| | ft| dkst| dkrPqqWt| dd \}}tttd|d|}g|D]}t||^q}t|||||}tt|}t|}gd}}||}x[d|t|krx>t||D]#}|dkrd}x |D]}|||d}qW||}t|||s\qq\n{|g}x$|D]}t||||}qWt|||}t||d}|d}|r\||dkr\qn|g}t|}||}|dkr|g}x$|D]}t||||}qWt|||}nx$|D]}t||||}qWt|||}t||}t||} || |kr|}g|D]}||kr@|^q@}t||d}t||d}|j|t||}PqqW|d7}q[W||gS( s4Factor primitive square-free polynomials in `Z[x]`. iiiiiiitkeycS st|dS(Ni(R(tx((s6lib/python2.7/site-packages/sympy/polys/factortools.pyt%t(RR8R RRpRqRRRbR]tconvertRR R RdRtminRRtsetR_RR+RARFR:(!ReRgRtRtARstBtCtgammatboundRrtpxRtfsqfxRtfsqfRtfftmodularRztsorted_TRRfR|RRRktiRRtT_StG_normtH_norm((s6lib/python2.7/site-packages/sympy/polys/factortools.pytdup_zz_zassenhauss   5($#"                 % cC st||}t||}t|d|}|rtt|}x3|jD]"}||rV||drVtSqVWndS(s2Test irreducibility using Eisenstein's criterion. iiN(R RRCR^RtkeysRc(ReRgRttcte_fcte_ffR((s6lib/python2.7/site-packages/sympy/polys/factortools.pytdup_zz_irreducible_plscC sw|jrJy)||j}}t|||}WqWtk rFtSXn |jsWtSt||}t||}|dks|dkr|dkrtS|st||\}}||j ks||dfgkrtSnt |}gg} } x.t |ddD]} | j d|| qWx2t |dddD]} | j d|| qDWt t| |} t t| |} t| t| d||} |jt| |rt| |} n| |krtSt||} |jt| |rt| |} n| | kr<t| |r<tSt| |} t | || krst| |rstStS(sh Efficiently test if ``f`` is a cyclotomic polnomial. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> f = x**16 + x**14 - x**10 + x**8 - x**6 + x**2 + 1 >>> R.dup_cyclotomic_p(f) False >>> g = x**16 + x**14 - x**10 - x**8 - x**6 + x**2 + 1 >>> R.dup_cyclotomic_p(g) True iiii(tis_QQtget_ringRRZtFalsetis_ZZR Rtdup_factor_listRxRRbtinsertR-RR)R7t is_negativeR%RcRMtdup_cyclotomic_pRT(ReRgt irreducibletK0RRtcoeffRfRtRzR{RRR((s6lib/python2.7/site-packages/sympy/polys/factortools.pyR{sL   $$   $cC sr|j|j g}xXt|jD]D\}}tt|||||}t|||d|}q&W|S(s0Efficiently generate n-th cyclotomic polnomial. i(RxR^titemsR1R(RtRgR{RRj((s6lib/python2.7/site-packages/sympy/polys/factortools.pytdup_zz_cyclotomic_polys c C s|j|j gg}xt|jD]\}}g|D]$}tt|||||^q<}|j|xItd|D]8}g|D]}t|||^q}|j|qWq)W|S(Ni(RxR^RR1RtextendRb( RtRgRRRjR{tQRRk((s6lib/python2.7/site-packages/sympy/polys/factortools.pyt_dup_cyclotomic_decomposes1 %cC st||t||}}t|dkr5dS|dksM|dkrQdStd|dd!DrrdSt|}t||}|j|s|Sg}x7td||D]"}||kr|j|qqW|SdS(s Efficiently factor polynomials `x**n - 1` and `x**n + 1` in `Z[x]`. Given a univariate polynomial `f` in `Z[x]` returns a list of factors of `f`, provided that `f` is in the form `x**n - 1` or `x**n + 1` for `n >= 1`. Otherwise returns None. Factorization is performed using using cyclotomic decomposition of `f`, which makes this method much faster that any other direct factorization approach (e.g. Zassenhaus's). References ========== .. [1] [Weisstein09]_ iiics s|]}t|VqdS(N(tbool(t.0tcf((s6lib/python2.7/site-packages/sympy/polys/factortools.pys siN(ii(R RRtNonetanyRtis_oneRd(ReRgtlc_fttc_fRtRRR{((s6lib/python2.7/site-packages/sympy/polys/factortools.pytdup_zz_cyclotomic_factors   cC st||\}}t|}t||dkrP| t||}}n|dkrf|gfS|dkr||gfStdrt||r||gfSnd}tdrt||}n|dkrt||}n|t |dt fS(s9Factor square-free (non-primitive) polyomials in `Z[x]`. iitUSE_IRREDUCIBLE_IN_FACTORtUSE_CYCLOTOMIC_FACTORtmultipleN( RFRR R%RWRRRRRVR(ReRgtcontRzRtRf((s6lib/python2.7/site-packages/sympy/polys/factortools.pytdup_zz_factor_sqf s"        cC s#t||\}}t|}t||dkrP| t||}}n|dkrf|gfS|dkr||dfgfStdrt||r||dfgfSnt||}d}tdrt||}n|dkrt ||}nt |||}||fS(s Factor (non square-free) polynomials in `Z[x]`. Given a univariate polynomial `f` in `Z[x]` computes its complete factorization `f_1, ..., f_n` into irreducibles over integers:: f = content(f) f_1**k_1 ... f_n**k_n The factorization is computed by reducing the input polynomial into a primitive square-free polynomial and factoring it using Zassenhaus algorithm. Trial division is used to recover the multiplicities of factors. The result is returned as a tuple consisting of:: (content(f), [(f_1, k_1), ..., (f_n, k_n)) Examples ======== Consider the polynomial `f = 2*x**4 - 2`:: >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_zz_factor(2*x**4 - 2) (2, [(x - 1, 1), (x + 1, 1), (x**2 + 1, 1)]) In result we got the following factorization:: f = 2 (x - 1) (x + 1) (x**2 + 1) Note that this is a complete factorization over integers, however over Gaussian integers we can factor the last term. By default, polynomials `x**n - 1` and `x**n + 1` are factored using cyclotomic decomposition to speedup computations. To disable this behaviour set cyclotomic=False. References ========== .. [1] [Gathen99]_ iiRRN( RFRR R%RWRRTRRRRm(ReRgRRzRtRRf((s6lib/python2.7/site-packages/sympy/polys/factortools.pyt dup_zz_factor)s&.       cC s||g}x|D]x}t|}xVt|D]H}x,|dkrg|j||}||}q<W|j|r3dSq3W|j|qW|dS(s,Wang/EEZ: Compute a set of valid divisors. iN(RptreversedtgcdRRRd(tEtcstctRgRhRkRl((s6lib/python2.7/site-packages/sympy/polys/factortools.pytdmp_zz_wang_non_divisorsts   cC s!tt||||d|s1tdnt||||}t||sdtdnt||\}}|jt||r| t||}}n|d} g|D]!\} } t| || |^q} t| |||} | dk r||| fStddS(s2Wang/EEZ: Test evaluation points for suitability. isno luckN( RHRR[RQRFRR R%RR(ReRRRRnRgRzRR{tvR}RRtD((s6lib/python2.7/site-packages/sympy/polys/factortools.pytdmp_zz_wang_test_pointss" .  c C sgdgt||d}} } x|D]} t| |} t| ||} xttt|D]}d||||}}\}}x#| |s| ||d} }qW|dkrmt| t||| || |d} | |s(RRR RRbR,R.RdRRXtzipRHRRR;R<(ReRRRRRRnRgRtJRR{RRRRjRR}RtCCtHHRtccRztCCCtHHH((s6lib/python2.7/site-packages/sympy/polys/factortools.pytdmp_zz_wang_lead_coeffssB% "  6       #c C s;t|dkr|\}}t||}t||}t||||\}} } t|||}t| ||} t||||\} }t| | |||} t||}t| |} || g} nY|dg} x;t|dd!D]&}| jdt || d|qWgdgg} }xet || D]T\}}t ||g|dgd|d|\} }|j | | j |qLWg| |dg} } xxt | |D]g\}}t||}t||}t t||||||}t||}| j |qW| S(s2Wang/EEZ: Solve univariate Diophantine equations. iiii(RRR RRRRRRR+Rtdmp_zz_diophantineRdR(RRyRRgRrRsReRzR|R}RRkRhRRRl((s6lib/python2.7/site-packages/sympy/polys/factortools.pytdup_zz_diophantines8  $. !c C s|sg|D] }g^q }t|} xt|D]\} } | sPq8nt|| | ||} x]tt|| D]F\} \}}t|| |}tt||||||| t factorialR((RRRRRRnRgRRRtRRRRR|R}RRrRRReRRRsRyR~Rj((s6lib/python2.7/site-packages/sympy/polys/factortools.pyRsV (-   # (",((c C s|gt||d}}} t|}xktt|dD]S\} } t|d| || || |} |jdt| || | |qDWtt||d} xt t d|d||D]\}} } t||d}}||d ||d}}xytt ||D]b\} \}}tt ||| |||d|}|gt |dd|d||| RRR4RX(ReRtLCRRRnRgRRtRRRrR|RRRtwtIRR{RRyR~RtdjRjRRR}((s6lib/python2.7/site-packages/sympy/polys/factortools.pytdmp_zz_wang_hensel_liftingLs@! #$'/((,!&"(+!  c C s{ddlm}||}tt|||d|\}}t|||} |t| } |d kr|dkrd}qd}ntgg|jg|d f\} } } }ytt |||| ||\}}}t ||\}}t |}|dkr|gS||||| fg} Wnt k rDnXt d}t d}t d}xyt | |krx`t|D]H}gt|D]}||| |^q} t| | kr| jt| nqy(t |||| ||\}}}Wnt k r)qnXt ||\}}t |}|d k r||kr||krg|} }qqqn|}|dkr|gS| j||||| ft | |krPqqW||7}qlWd \}}}xf| D]^\}}}}}t||}|d k rI||krO|}|}qOn|}|d7}qW| |\}}}}} |}yLt|||||| ||\}}}t|||| | ||}Wn@tk r t d rt||||dStd nXg}xc|D][}t|||\}}|jt|||rft|||}n|j|qW|S( s^ Factor primitive square-free polynomials in `Z[X]`. Given a multivariate polynomial `f` in `Z[x_1,...,x_n]`, which is primitive and square-free in `x_1`, computes factorization of `f` into irreducibles over integers. The procedure is based on Wang's Enhanced Extended Zassenhaus algorithm. The algorithm works by viewing `f` as a univariate polynomial in `Z[x_2,...,x_n][x_1]`, for which an evaluation mapping is computed:: x_2 -> a_2, ..., x_n -> a_n where `a_i`, for `i = 2, ..., n`, are carefully chosen integers. The mapping is used to transform `f` into a univariate polynomial in `Z[x_1]`, which can be factored efficiently using Zassenhaus algorithm. The last step is to lift univariate factors to obtain true multivariate factors. For this purpose a parallel Hensel lifting procedure is used. The parameter ``seed`` is passed to _randint and can be used to seed randint (when an integer) or (for testing purposes) can be a sequence of numbers. References ========== .. [1] [Wang78]_ .. [2] [Geddes92]_ i(t_randintiitEEZ_NUMBER_OF_CONFIGStEEZ_NUMBER_OF_TRIEStEEZ_MODULUS_STEPitEEZ_RESTART_IF_NEEDEDs3we need to restart algorithm with better parametersN(Nii(tsympy.utilities.randtestRt dmp_zz_factorRRwR\RRtzeroRRRR[RWRbttupletaddRdR8RRRXt dmp_zz_wangRGRRR&( ReRnRgtmodtseedRtrandintRRRsRthistorytconfigsRRlRR|RRRteez_num_configst eez_num_triest eez_mod_steptrrts_normts_argRt_s_normtorig_fRRfRh((s6lib/python2.7/site-packages/sympy/polys/factortools.pyRs %    .$      /(          *"    c C sh|st||St||r/|jgfSt|||\}}t|||dkr|| t|||}}ntdt||Dr|gfSt|||\}}g}t ||dkrt |||}t |||}t ||||}nxAt ||d|dD]%\}}|jd|g|fq/W|t|fS(s Factor (non square-free) polynomials in `Z[X]`. Given a multivariate polynomial `f` in `Z[x]` computes its complete factorization `f_1, ..., f_n` into irreducibles over integers:: f = content(f) f_1**k_1 ... f_n**k_n The factorization is computed by reducing the input polynomial into a primitive square-free polynomial and factoring it using Enhanced Extended Zassenhaus (EEZ) algorithm. Trial division is used to recover the multiplicities of factors. The result is returned as a tuple consisting of:: (content(f), [(f_1, k_1), ..., (f_n, k_n)) Consider polynomial `f = 2*(x**2 - y**2)`:: >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_zz_factor(2*x**2 - 2*y**2) (2, [(x - y, 1), (x + y, 1)]) In result we got the following factorization:: f = 2 (x - y) (x + y) References ========== .. [1] [Gathen99]_ ics s|]}|dkVqdS(iN((RR((s6lib/python2.7/site-packages/sympy/polys/factortools.pys =si(RRRRGRR&tallRRNRRURRoRRRV( ReRnRgRRzRRfRRj((s6lib/python2.7/site-packages/sympy/polys/factortools.pyRs$$   'cC sgt|t||}}t||}|dkrA|gfS|dkr`||dfgfSt|||}}t||\}}}t||j}t|dkr|||t|fgfS||j} xlt |D]^\} \} } t | |j|} t | ||\} } }t | | |} | || Factor multivariate polynomials over algebraic number fields. cs s|]}|dkVqdS(iN((RR((s6lib/python2.7/site-packages/sympy/polys/factortools.pys vsii(R RRERRRURStdmp_factor_list_includeR RRRxR RRRPRKRo(ReRnRgRRR|RzRlRfRRRiRR{((s6lib/python2.7/site-packages/sympy/polys/factortools.pytdmp_ext_factorns$   %cC st|||j}t||j|j\}}x?t|D]1\}\}}t||j||f||RRRRxRRRV(ReRnRRRRRfRRgRtlevelsRRRjRRtterm((s6lib/python2.7/site-packages/sympy/polys/factortools.pyRsb       &   #" $&cC s|st||St|||\}}|sGt||dfgSt|dd|||}||ddfg|dSdS(s0Factor polynomials into irreducibles in `K[X]`. iiN(R RRR<(ReRnRgRRfRz((s6lib/python2.7/site-packages/sympy/polys/factortools.pyR /s cC st|d|S(s:Returns ``True`` if ``f`` has no factors over its domain. i(tdmp_irreducible_p(ReRg((s6lib/python2.7/site-packages/sympy/polys/factortools.pytdup_irreducible_p=scC sVt|||\}}|s"tSt|dkr8tS|d\}}|dkSdS(s:Returns ``True`` if ``f`` has no factors over its domain. iiN(RRcRR(ReRnRgRRfRj((s6lib/python2.7/site-packages/sympy/polys/factortools.pyR!BsN(t__doc__t __future__RRtsympy.polys.galoistoolsRRRRRRRR R R R tsympy.polys.densebasicR RRRRRRRRRRRRRRRRRRR R!R"R#R$tsympy.polys.densearithR%R&R'R(R)R*R+R,R-R.R/R0R1R2R3R4R5R6R7R8R9R:R;R<R=R>tsympy.polys.densetoolsR?R@RARBRCRDRERFRGRHRIRJRKRLRMtsympy.polys.euclidtoolsRNRORPtsympy.polys.sqfreetoolsRQRRRSRTRUtsympy.polys.polyutilsRVtsympy.polys.polyconfigRWtsympy.polys.polyerrorsRXRYRZR[t sympy.ntheoryR\R]R^tsympy.utilitiesR_tmathR`RRaRtsympy.core.compatibilityRbRmRoRuRwRRRRRRRRRRRRRRRRRRRRRR RRRRR RR R"R!(((s6lib/python2.7/site-packages/sympy/polys/factortools.pyts^Ld ("   9 9  g  L  ,  K   6 0 D 4 A    > I