ó ~9­\c@ s?dZddlmZmZddlmZmZmZmZm Z ddl m Z ddl m Z ddlmZmZmZmZddlmZddlmZmZmZdd lmZmZdd lmZmZmZed „ƒZ ed „ƒZ!ed „ƒZ"ee dƒd„ƒZ#edd„ƒZ%dS(s/High-level polynomials manipulation functions. iÿÿÿÿ(tprint_functiontdivision(tStBasictAddtMultsymbols(trange(t factorial(tPolificationFailedtComputationFailedtMultivariatePolynomialErrort OptionError(t allowed_flags(tpoly_from_exprtparallel_poly_from_exprtPoly(tsymmetric_polytinterpolating_poly(tnumbered_symbolsttaketpublicc" sžt|ddgƒt}t|dƒs:t}|g}nyt|||Ž\}}Wn¤tk rü}g}xŒ|jD]@}|jr |j|t j fƒqxt dt |ƒ|ƒ‚qxW|sÎ|\}n|j jsÞ|S|rî|gfS|gfSnXg|j}} |j|j}} xUtt |ƒƒD]A} t| d|dtƒ} |jt| ƒ| j| ƒfƒq3Wttt |ƒdƒƒ} ttt |ƒddƒƒ}g}x2|D]*}g}|jsú|j|jƒƒ||jƒ8}nxÍ|rÉd \}}}xšt|jƒƒD]†\} \‰}t‡fd †| Dƒƒr%tgt|ˆƒD]\}}||^qfƒ}||kr«|ˆ|}}}q«q%q%W|dkrË||‰}nPg}x6tˆˆdd ƒD]\}}|j||ƒqêWgt||ƒD]\\}}}||^q}gt||ƒD]\\}}}||^qM} |jt||Œƒ| dj |ƒ}x!| dD]}|j |ƒ}q£W||8}qýW|jt!|Œ|j"ƒfƒq¿Wg|D]\}}||j"ƒf^qô}|js`x<t|ƒD]+\} \} }!| j#|ƒ|!f|| >> from sympy.polys.polyfuncs import symmetrize >>> from sympy.abc import x, y >>> symmetrize(x**2 + y**2) (-2*x*y + (x + y)**2, 0) >>> symmetrize(x**2 + y**2, formal=True) (s1**2 - 2*s2, 0, [(s1, x + y), (s2, x*y)]) >>> symmetrize(x**2 - y**2) (-2*x*y + (x + y)**2, -2*y**2) >>> symmetrize(x**2 - y**2, formal=True) (s1**2 - 2*s2, -2*y**2, [(s1, x + y), (s2, x*y)]) tformalRt__iter__t symmetrizeitpolysiiÿÿÿÿc3 s'|]}ˆ|ˆ|dkVqdS(iN((t.0ti(tmonom(s4lib/python2.7/site-packages/sympy/polys/polyfuncs.pys hsN(iÿÿÿÿNN(i($R tTruethasattrtFalseRR texprst is_NumbertappendRtZeroR tlentoptRRtgenstdomainRRtnextt set_domaintlisttis_homogeneoustTCtNonet enumeratettermstalltmaxtzipRtmulRtas_exprtsubs("tFR&targstiterableR%texctresulttexprRRtdomRtpolytindicestweightstft symmetrict_heightt_monomt_coefftcoefftntmtheightt exponentstm1tm2tst_ttermtptproducttsymtnon_sym((Rs4lib/python2.7/site-packages/sympy/polys/polyfuncs.pyRs‚!     &   %2  $22#+     c O sÖt|gƒyt|||Ž\}}Wntk rB}|jSXtj|j}}|jr‡xp|jƒD]}|||}qlWnKt ||ƒ|d}}x.|jƒD] }||t |||Ž}q®W|S(sê Rewrite a polynomial in Horner form. Among other applications, evaluation of a polynomial at a point is optimal when it is applied using the Horner scheme ([1]). Examples ======== >>> from sympy.polys.polyfuncs import horner >>> from sympy.abc import x, y, a, b, c, d, e >>> horner(9*x**4 + 8*x**3 + 7*x**2 + 6*x + 5) x*(x*(x*(9*x + 8) + 7) + 6) + 5 >>> horner(a*x**4 + b*x**3 + c*x**2 + d*x + e) e + x*(d + x*(c + x*(a*x + b))) >>> f = 4*x**2*y**2 + 2*x**2*y + 2*x*y**2 + x*y >>> horner(f, wrt=x) x*(x*y*(4*y + 2) + y*(2*y + 1)) >>> horner(f, wrt=y) y*(x*y*(4*x + 2) + x*(2*x + 1)) References ========== [1] - https://en.wikipedia.org/wiki/Horner_scheme i( R RR R;RR#tgent is_univariatet all_coeffsRthorner( R@R&R7R6R%R9tformRSRE((s4lib/python2.7/site-packages/sympy/polys/polyfuncs.pyRV—s!  c C s’t|ƒ}d}t|tƒrWtt|jƒŒƒ\}}t||||ƒ}n1t|dtƒrštt|Œƒ\}}t||||ƒ}nît|ƒ}t gt d|dƒD]}||^q½Œ}|ddkr÷t |dƒ n t |dƒ}g} xFt d|dƒD]1}| j ||||ƒ||||}q!Wt gt| |ƒD]\} } | | ^qiŒ}|jƒS(s¯ Construct an interpolating polynomial for the data points. Examples ======== >>> from sympy.polys.polyfuncs import interpolate >>> from sympy.abc import x A list is interpreted as though it were paired with a range starting from 1: >>> interpolate([1, 4, 9, 16], x) x**2 This can be made explicit by giving a list of coordinates: >>> interpolate([(1, 1), (2, 4), (3, 9)], x) x**2 The (x, y) coordinates can also be given as keys and values of a dictionary (and the points need not be equispaced): >>> interpolate([(-1, 2), (1, 2), (2, 5)], x) x**2 + 1 >>> interpolate({-1: 2, 1: 2, 2: 5}, x) x**2 + 1 iiiN(R$R-t isinstancetdictR*R2titemsRttupleRRRR"Rtexpand( tdatatxRFR=tXtYRtnumerttdenomtcoeffsREty((s4lib/python2.7/site-packages/sympy/polys/polyfuncs.pyt interpolateÍs"  012R^c  s©ddlm}tt|Œƒ\}}t|ƒˆd}|dkrWtdƒ‚n|ˆ|dˆ|dƒ}xbttˆ|ƒƒD]K}xBtˆ|dƒD],} || |f|| || |df>> from sympy.polys.polyfuncs import rational_interpolate >>> data = [(1, -210), (2, -35), (3, 105), (4, 231), (5, 350), (6, 465)] >>> rational_interpolate(data, 2) (105*x**2 - 525)/(x + 1) Values do not need to be integers: >>> from sympy import sympify >>> x = [1, 2, 3, 4, 5, 6] >>> y = sympify("[-1, 0, 2, 22/5, 7, 68/7]") >>> rational_interpolate(zip(x, y), 2) (3*x**2 - 7*x + 2)/(x + 1) The symbol for the variable can be changed if needed: >>> from sympy import symbols >>> z = symbols('z') >>> rational_interpolate(data, 2, X=z) (105*z**2 - 525)/(z + 1) References ========== .. [1] Algorithm is adapted from: http://axiom-wiki.newsynthesis.org/RationalInterpolation iÿÿÿÿ(tonesiis'Too few values for the required degree.ic3 s!|]}ˆ|ˆ|VqdS(N((RR(R_tr(s4lib/python2.7/site-packages/sympy/polys/polyfuncs.pys =sc3 s)|]}ˆ|ˆdˆ|VqdS(iN((RR(R_tdegnumRg(s4lib/python2.7/site-packages/sympy/polys/polyfuncs.pys >s( tsympy.matrices.denseRfR*R2R$R RR1t nullspacetsum( R]RhR_RftxdatatydatatktctjR((R_RhRgs4lib/python2.7/site-packages/sympy/polys/polyfuncs.pytrational_interpolates) .;&cO st|gƒt|tƒr3|f|d }}nyt|||Ž\}}Wn%tk rv}tdd|ƒ‚nX|jrtdƒ‚n|j ƒ}|dkr¶t dƒ‚n|d kr×t dddƒ}nt ||ƒ}|t |ƒkrt d|t |ƒfƒ‚n|jƒ|jƒ}}gd} } xYt|dƒD]G\} } t| d|ƒ} | | |} | j| | fƒ| } qNW| S( s# Generate Viete's formulas for ``f``. Examples ======== >>> from sympy.polys.polyfuncs import viete >>> from sympy import symbols >>> x, a, b, c, r1, r2 = symbols('x,a:c,r1:3') >>> viete(a*x**2 + b*x + c, [r1, r2], x) [(r1 + r2, -b/a), (r1*r2, c/a)] tvieteis(multivariate polynomials are not alloweds7can't derive Viete's formulas for a constant polynomialRgtstartsrequired %s roots, got %siÿÿÿÿN(R RXRR-RR R tis_multivariateR tdegreet ValueErrorRRR$tLCRUR.RR"(R@trootsR&R7R%R9RFtlcRcR:tsignRRER=((s4lib/python2.7/site-packages/sympy/polys/polyfuncs.pyRrAs6         N(&t__doc__t __future__RRt sympy.coreRRRRRtsympy.core.compatibilityRt(sympy.functions.combinatorial.factorialsRtsympy.polys.polyerrorsR R R R tsympy.polys.polyoptionsR tsympy.polys.polytoolsRRRtsympy.polys.specialpolysRRtsympy.utilitiesRRRRRVReRqR-Rr(((s4lib/python2.7/site-packages/sympy/polys/polyfuncs.pyts ("…68;