+ ic!Rt^RIHtHtHtHt^RIHtHtH t H t ^RI H t H t ^RIHtHt^RIHtHt^RIHt^RIHtHtHt]R4t]R 4t]R 4t]]!R 43R l4t]RRl4tR #)z/High-level polynomials manipulation functions. )SBasicsymbolsDummy)PolificationFailedComputationFailedMultivariatePolynomialError OptionError) allowed_flags build_options)poly_from_exprPoly)symmetric_polyinterpolating_poly)sring)numbered_symbolstakepublicc\VRR.4Rp\VR4'gRpV.p\V.VO5/VBwr@VPp\ W4pVPp\ \ V44Uu.uFp\V4NK pp.pVFEp V P4wrp VPV P!V!V P!V!34KG \VX 4U UUu.uFwp wrWP43NK ppp pVP'g.\V4FwpwppVPV4V3W&K V'gVwpVP'gV#V'dVV3#VV3,#uupiuuppp i)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&*, u/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/polys/polyfuncs.pyrrsgB$9-.H 1j ! ! C  "T "T "DA 99D  #CkkG&+CI&67&6tG}&6G7 F ,,.a qyy'*AIIt,<=>037A ?)!Vaa E ? :::!*6!2 A~W%'2FI"3  ::: 5= UH$ $/8 @s 2E=%Fc\V.4\V.VO5/VBwr4\P TP rvTP'd)TP4FpYg,T,pK T#\Y74TR,rTP4FpYg,\T.TO5/TB,pK! T# \dpTPuRp?#Rp?ii;i)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 NNN) r r rexprrZerogen is_univariate all_coeffsr horner) r)r"r#r!r&excformr9coeffs &*, r3r<r<WsB$1D1D1#\\^E8e#D$ K q,R4\\^E8fU:T:T::D$ K xxsB<< C CCCc\V4p\V\4'd;W9d\W,4#\ \ VP 4!4wr4M\V^,\4'd<\ \ V!4wr4W9d!\WCPV4,4#MWV\^V^,49d\W^, ,4#\ V4p\ \^V^,44p\W!W44P4# \d7\4p\Y%Y44P4PYQ4u#i;i)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)] )r isinstancedictrlistritemstupleindexrrexpand ValueErrorrr )dataxnXYds&& r3 interpolaterOsT D A$ 9TW: C&'1 d1gu % %T #DAv771:''E!QUO#!e~%T AU1a!e_%AB!!-4466 B G!!-446;;AAABs D&&>E'&E'rJcJaaa ^RIHp\\V!4wrE\ V4S, ^, pV^8d \ R4hV!SV,^,SV,^,4p\ \SV44FEp\ SV,^,4F%p WyV3,WI,,WyV^,3&K' KG \ V^,4FZp\ SV,^,4F:p WyWh, 3,)WY,,WySV,^,V, 3&K< K\ VP4^,o \VV 3Rl\ S^,444\VVV 3Rl\ V^,444, #)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 )onesz'Too few values for the required degree.c3R<"TFpSV,SV,,xK R#5iN).0r'rLr+s& r3 'rational_interpolate..s!7%6!q!t %6s$'c3n<"TF*pSVS,^,,SV,,xK, R#5i)r6NrT)rUr'rLdegnumr+s& r3rVrWs*ALq!AJN#ad**Ls25) sympy.matrices.denserQrCrrr rmax nullspacesum) rIrYrLrQxdataydatakcjr'r+s &ff @r3rational_interpolatercsDR*T #LE E VaA1uCDD VaZ!^VaZ!^,A 3vq> "vzA~&AqD'%(*AQhK'#1q5\vzA~&A()QU( |EH' T # \dp\ R^T4hRp?ii;i) z Generate Viete's formulas for ``f``. Examples ======== >>> from sympy.polys.polyfuncs import viete >>> from sympy import symbols >>> x, a, b, c, r1, r2 = symbols('x,a:c,r1:3') >>> viete(a*x**2 + b*x + c, [r1, r2], x) [(r1 + r2, -b/a), (r1*r2, c/a)] Nvietez(multivariate polynomials are not allowedz8Cannot derive Viete's formulas for a constant polynomialr+)startz required z roots, got r5)r rArr rris_multivariaterdegreerHrrrLCr;rrr)r)rootsr"r#r&r=rKlccoeffsr(signr'r?polys&&*, r3rfrf sI"$%hote11D1D1 ) 68 8  A1u FH H } A. NECJ3u:FGGrDfRj)a!eU+Bh tm$u * M= 1C001sD== E EErS)__doc__ sympy.corerrrrsympy.polys.polyerrorsrrrr sympy.polys.polyoptionsr r sympy.polys.polytoolsr r sympy.polys.specialpolysrrsympy.polys.ringsrsympy.utilitiesrrrrr<rOrcrfrTrdr3rys50/..A6(#::D%D%N22j>B>BB)08C8Cv55rd