gc!dZddlmZmZmZmZddlmZmZm Z m Z ddl m Z m Z ddlmZmZddlmZmZddlmZddlmZmZmZed Zed Zed Zeed fd ZeddZdS)z/High-level polynomials manipulation functions. )SBasicsymbolsDummy)PolificationFailedComputationFailedMultivariatePolynomialError OptionError) allowed_flags build_options)poly_from_exprPoly)symmetric_polyinterpolating_poly)sring)numbered_symbolstakepubliccpt|ddgd}t|dsd}|g}t|g|Ri|\}}|j}t ||}|jfdt t |Dg}|D]A}|\}} } ||j | j |fBdt| D} |j s2t|D]"\} \} }| | |f|| <#|s|\}|j s|S|r|| fS|| fzS)a Rewrite a polynomial in terms of elementary symmetric polynomials. A symmetric polynomial is a multivariate polynomial that remains invariant under any variable permutation, i.e., if `f = f(x_1, x_2, \dots, x_n)`, then `f = f(x_{i_1}, x_{i_2}, \dots, x_{i_n})`, where `(i_1, i_2, \dots, i_n)` is a permutation of `(1, 2, \dots, n)` (an element of the group `S_n`). Returns a tuple of symmetric polynomials ``(f1, f2, ..., fn)`` such that ``f = f1 + f2 + ... + fn``. Examples ======== >>> 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)]) formalrT__iter__Fc.g|]}tS)next).0irs Q/var/www/html/ai-engine/env/lib/python3.11/site-packages/sympy/polys/polyfuncs.py zsymmetrize..=s777tG}}777cFg|]\}\}}||fSr)as_expr)rs_gs rrzsymmetrize..Es- ? ? ?)!Vaa  ? ? ?r)r hasattrrrr rangelen symmetrizeappendr!zipr enumeratesubs)FgensargsiterableRoptresultfprmpolysrsymnon_symrs @rr(r(sB$9-...H 1j ! ! C  "T " " "T " "DAq 9D d # #CkG7777eCII&6&6777G F ??,,..1a yqy'*IAIt,<=>>>> ? ?s7A ? ? ?E :3!*6!2!2 3 3 A~W%'2F1II  :%  %5= UH$ $rct|g t|g|Ri|\}}n#t$r}|jcYd}~Sd}~wwxYwtj|j}}|jr |D] }||z|z} nGt|||dd}}|D]}||zt|g|Ri|z}|S)a 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 N) r r rexprrZerogen is_univariate all_coeffsrhorner) r4r.r/r-r2excformr?coeffs rrBrBWsB$1D111D1133 x#D;\\^^ $ $E8e#DD $q#,,QRR4\\^^ ; ;E8fU:T:::T:::DD Ks& A<AAcDt|}t|trE||vrt||St t |\}}nt|dtrFt t |\}}||vr(t|||Sn\|td|dzvrt||dz St |}t td|dz} t|||| S#t$rIt}t|||| ||cYSwxYw)a) Construct an interpolating polynomial for the data points evaluated at point x (which can be symbolic or numeric). Examples ======== >>> from sympy.polys.polyfuncs import interpolate >>> from sympy.abc import a, b, 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 If the interpolation is going to be used only once then the value of interest can be passed instead of passing a symbol: >>> interpolate([1, 4, 9], 5) 25 Symbolic coordinates are also supported: >>> [(i,interpolate((a, b), i)) for i in range(1, 4)] [(1, a), (2, b), (3, -a + 2*b)] rr<)r' isinstancedictrlistr*itemstupleindexr&rexpand ValueErrorrr,)dataxnXYds r interpolaterUsT D A$ & 99T!W:: C&''11 d1gu % % &T ##DAqAvv1771::'''E!QUOO##a!e~~%T AU1a!e__%%AB!!Q1--44666 BBB GG!!Q1--4466;;AqAAAAABs(#E AFFrPc ddlm}tt|\}}t |z dz }|dkrt d||zdz|zdz}t t|D]5}t |zdzD]} || |f|| z|| |dzf<6t |dzD]?}t |zdzD]'} || ||z f || z|| |zdz|z f<(@|d t fdt dzDt fdt |dzDz S)a Returns a rational interpolation, where the data points are element of any integral domain. The first argument contains the data (as a list of coordinates). The ``degnum`` argument is the degree in the numerator of the rational function. Setting it too high will decrease the maximal degree in the denominator for the same amount of data. Examples ======== >>> 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 r)onesr<z'Too few values for the required degree.c34K|]}||zzVdSNr)rrrRr6s r z'rational_interpolate..s/77!q!t 777777rc3@K|]}|zdz|zzVdS)r<Nr)rrrRdegnumr6s rr[z'rational_interpolate..s9AAq!AJN#ad*AAAAAAr) sympy.matrices.denserWrIr*r'r r&max nullspacesum) rOr]rRrWxdataydatakcjrr6s `` @rrational_interpolatergsR*)))))T ##LE5 E VaA1uuCDDD VaZ!^VaZ!^,,A 3vq>> " "++vzA~&& + +AAqD'%(*AaQhKK + 1q5\\==vzA~&& = =A()!QU( |E!H'>> 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)] Nvieter<z(multivariate polynomials are not allowedz8Cannot derive Viete's formulas for a constant polynomialr6)startz required z roots, got )r rGrr rris_multivariater degreerNrrr'LCrAr+rr))r4rootsr.r/r2rCrQlccoeffsr3signrrEpolys rriri s"$%,hote11D111D1133 111C0001 8) 688 8  A1uu FHH H } A... NNECJJj3u:::FGGGBrDFfQRRj))5a!eU++eBh tUm$$$u MsA A# AA#rZ)__doc__ sympy.corerrrrsympy.polys.polyerrorsrrr r sympy.polys.polyoptionsr r sympy.polys.polytoolsr rsympy.polys.specialpolysrrsympy.polys.ringsrsympy.utilitiesrrrr(rBrUrgrirrrr|s550///////////............A@@@@@@@66666666((((((((######::::::::::D%D%D%N222j>B>B>BB)08C8C8C8Cv555555r