\hc!SrSSKJrJrJrJr SSKJrJrJ r J r SSK J r J r SSKJrJr SSKJrJr SSKJr SSKJrJrJr \S 5r\S 5r\S 5r\\"S 54S j5r\SSj5rg)z/High-level polynomials manipulation functions. )SBasicsymbolsDummy)PolificationFailedComputationFailedMultivariatePolynomialError OptionError) allowed_flags build_options)poly_from_exprPoly)symmetric_polyinterpolating_poly)sring)numbered_symbolstakepublicc[USS/5 Sn[US5(dSnU/n[U/UQ70UD6up@URn[ X5nURn[ [ U55Vs/sHn[U5PM nn/nUHCn U R5upn URU R"U6U R"U645 ME [UW 5V VVs/sHun upXR54PM nnn nUR(d-[U5HununnURU5U4X'M U(dUunUR(dU$U(aUU4$UU4-$s snfs snnn f)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__F)r hasattrrrr rangelennext symmetrizeappendas_exprzipr enumeratesubs)FgensargsiterableRoptriresultfprms_gpolyssymnon_syms M/var/www/auris/envauris/lib/python3.13/site-packages/sympy/polys/polyfuncs.pyrrscB$9-.H 1j ! ! C  "T "T "DA 99D  #CkkG&+CI&67&6tG}&6G7 F ,,.a qyy'*AIIt,<=>037A ?)!Vaa E ? ::!*6!2 A~W%'2FI"3  :: 5= UH$ $/8 @s 1E4# E9c[U/5 [U/UQ70UD6up4[R UR pvUR(a UR5H nXg-U-nM U$[X75USSpUR5HnXg-[U/UQ70UD6-nM U$![anURsSnA$SnAff=f)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) r*r#r$r"r'excformr9coeffs r4r<r<WsB$1D1D1#\\^E8e#D$ K q,QR4\\^E8fU:T:T::D$ K xxsB'' C1 C<CCc[U5n[U[5(a4X;a [X5$[ [ UR 565up4O[US[5(a5[ [ U65up4X;a[XCRU55$OFU[SUS-5;a[XS- 5$[ U5n[ [SUS-55n[X!X45R5$![a6 [5n[X%X45R5RXQ5s$f=f)a Construct an interpolating polynomial for the data points evaluated at point x (which can be symbolic or numeric). Examples ======== >>> from sympy.polys.polyfuncs import interpolate >>> from sympy.abc import a, b, x A list is interpreted as though it were paired with a range starting from 1: >>> interpolate([1, 4, 9, 16], x) x**2 This can be made explicit by giving a list of coordinates: >>> interpolate([(1, 1), (2, 4), (3, 9)], x) x**2 The (x, y) coordinates can also be given as keys and values of a dictionary (and the points need not be equispaced): >>> interpolate([(-1, 2), (1, 2), (2, 5)], x) x**2 + 1 >>> interpolate({-1: 2, 1: 2, 2: 5}, x) x**2 + 1 If the interpolation is going to be used only once then the value of interest can be passed instead of passing a symbol: >>> interpolate([1, 4, 9], 5) 25 Symbolic coordinates are also supported: >>> [(i,interpolate((a, b), i)) for i in range(1, 4)] [(1, a), (2, b), (3, -a + 2*b)] rr6)r isinstancedictrlistritemstupleindexrrexpand ValueErrorrr!)dataxnXYds r4 interpolaterOsT D A$ 9TW: C&'1 d1gu % %T #DAv771:''E!QUO#!e~%T AU1a!e_%AB!!-4466 B G!!-446;;AAABs)D=EErJcv^^^ SSKJn [[U65upE[ U5T- S- nUS:a [ S5eU"TU-S-TU-S-5n[ [TU55H-n[ TU-S-5Hn XyU4XI-XyUS-4'M M/ [ US-5H6n[ TU-S-5Hn XyXh- 4*XY-XyTU-S-U- 4'M M8 UR5Sm [UU 4Sj[ TS-555[UUU 4Sj[ US-555- $)a Returns a rational interpolation, where the data points are element of any integral domain. The first argument contains the data (as a list of coordinates). The ``degnum`` argument is the degree in the numerator of the rational function. Setting it too high will decrease the maximal degree in the denominator for the same amount of data. Examples ======== >>> from sympy.polys.polyfuncs import rational_interpolate >>> data = [(1, -210), (2, -35), (3, 105), (4, 231), (5, 350), (6, 465)] >>> rational_interpolate(data, 2) (105*x**2 - 525)/(x + 1) Values do not need to be integers: >>> from sympy import sympify >>> x = [1, 2, 3, 4, 5, 6] >>> y = sympify("[-1, 0, 2, 22/5, 7, 68/7]") >>> rational_interpolate(zip(x, y), 2) (3*x**2 - 7*x + 2)/(x + 1) The symbol for the variable can be changed if needed: >>> from sympy import symbols >>> z = symbols('z') >>> rational_interpolate(data, 2, X=z) (105*z**2 - 525)/(z + 1) References ========== .. [1] Algorithm is adapted from: http://axiom-wiki.newsynthesis.org/RationalInterpolation r)onesr6z'Too few values for the required degree.c3:># UHnTUTU--v M g7fN).0r(rLr,s r4 'rational_interpolate..s7%6!q!t %6sc3F># UHnTUT-S-TU--v M g7f)r6NrU)rVr(rLdegnumr,s r4rWrXs'ALq!AJN#ad*Ls!) sympy.matrices.denserQrCrrr rmax nullspacesum) rIrZrLrQxdataydatakcjr(r,s `` @r4rational_interpolaterdsDR*T #LE E VaA1uCDD VaZ!^VaZ!^,A 3vq> "vzA~&AqD'%(*AQhK'#1q5\vzA~&A()QU( |EH'>> from sympy.polys.polyfuncs import viete >>> from sympy import symbols >>> x, a, b, c, r1, r2 = symbols('x,a:c,r1:3') >>> viete(a*x**2 + b*x + c, [r1, r2], x) [(r1 + r2, -b/a), (r1*r2, c/a)] Nvieter6z(multivariate polynomials are not allowedz8Cannot derive Viete's formulas for a constant polynomialr,)startz required z roots, got )r rArr rris_multivariater degreerHrrrLCr;r rr)r*rootsr#r$r'r=rKlccoeffsr)signr(r?polys r4rgrg sH"$%hote11D1D1 ) 68 8  A1u FH H } A. NECJ3u:FGGrDfQRj)a!eU+Bh tm$u * M= 1C001sD!! D=+ D88D=rT)__doc__ sympy.corerrrrsympy.polys.polyerrorsrrr r sympy.polys.polyoptionsr r sympy.polys.polytoolsr rsympy.polys.specialpolysrrsympy.polys.ringsrsympy.utilitiesrrrrr<rOrdrgrUrer4rzs50/..A6(#::D%D%N22j>B>BB)08C8Cv55re