o 8Va(@sdZddlmZmZmZmZmZmZddlm Z m Z m Z m Z ddl mZddlmZmZmZddlmZmZddlmZmZmZedd Zed d Zed d ZeedfddZedddZdS)z/High-level polynomials manipulation functions. )SBasicAddMulsymbolsDummy)PolificationFailedComputationFailedMultivariatePolynomialError OptionError) allowed_flags)poly_from_exprparallel_poly_from_exprPoly)symmetric_polyinterpolating_poly)numbered_symbolstakepublicc sXt|ddgd}t|dsd}|g}zt|g|Ri|\}}WnOtyq}zCg}|jD]}|jr=||tjfq/t dt |||sJ|\}|j j sV|WYd}~S|rb|gfWYd}~S|gfWYd}~Sd}~wwg|j }} |j|j}} tt |D]} t| d|dd } |t| | | fqttt |d} ttt |d d }g}|D]}g}|js|||||j8}|rhd \}}}t|D]+\} \}tfd d| Drtddt|D}||kr||}}}q|d kr||}nnUg}tdddD] \}}|||q ddt||D}ddt||D} |t|g|R| d |}| ddD]}||}qY||8}|s|t|| fqdd|D}|j st|D]\} \}}|!||f|| <q|s|\}|j s|S|r||fS||fS)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, ..., x_n)``, then ``f = f(x_{i_1}, x_{i_2}, ..., x_{i_n})``, where ``(i_1, i_2, ..., i_n)`` is a permutation of ``(1, 2, ..., 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 symmetrizeN)polysr)rNNc3s$|] }||dkVqdSrN.0iZmonomr7/usr/lib/python3/dist-packages/sympy/polys/polyfuncs.py es"zsymmetrize..cSsg|]\}}||qSrr)rnmrrr! fszsymmetrize..)rcSsg|] \\}}}||qSrr)rs_r#rrr!r%ucSsg|] \\}}}||qSrr)rr'pr#rrr!r%vr(cSsg|] \}}||fqSr)as_expr)rr&r)rrr!r%r()"r hasattrrrZexprsZ is_NumberappendrZeror lenoptrrgensdomainrangernextZ set_domainlistZis_homogeneousZTCZas_poly enumerateZtermsallmaxziprmulrr*subs)Fr0argsiterabler/excresultexprrrZdomrpolyindicesZweightsfZ symmetricZ_heightZ_monomZ_coeffcoeffZheightZ exponentsZm1Zm2Ztermproductr)ZsymZnon_symrr r!rs!        rc Ost|gzt|g|Ri|\}}Wnty)}z |jWYd}~Sd}~wwtj|j}}|jrC|D]}|||}q8|St |||dd}}|D]}||t |g|Ri|}qS|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 Nr) r r rr@rr-genZ is_univariate all_coeffsrhorner) rCr0r<r;r/r>ZformrFrDrrr!rHs !   rHcCst|}t|tr||vrt||Stt|\}}n;t|dtr;tt|\}}||vr:t|||Sn|t d|dvrLt||dSt|}tt d|d}z t |||| WSt y|t }t |||| ||YSw)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. isinstancedictrr4r8itemstupleindexr2rexpand ValueErrorrr:)dataxr#XYdrrr! interpolates(*   rUrQc sDddlm}tt|\}}t|d}|dkrtd||d|d}tt|D]}t|dD]} || |f|| || |df<q>> 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)onesrz'Too few values for the required degree.c3s |] }||VqdSNrr)rRrrr!r"Dsz'rational_interpolate..c3s(|]}|d|VqdSrrrrRdegnumrYrr!r"Es&) Zsympy.matrices.denserVr4r8r.r r2r7Z nullspacesum) rPr[rRrVZxdataZydatakcjrrrZr!rational_interpolate s$ )"0  r`Nc Os4t|gt|tr|f|d}}zt|g|Ri|\}}Wnty3}ztdd|d}~ww|jr;td|}|dkrGt d|durQt ddd}t ||}|t |krft d|t |f| |}}gd } } t|ddD]\} } t| d|} | | |} | | | f| } q|| S) a# 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)] Nvieterz(multivariate polynomials are not allowedz7can't derive Viete's formulas for a constant polynomialrY)startzrequired %s roots, got %sr)r rIrr rr Zis_multivariater ZdegreerOrrr.ZLCrGr5rr,)rCrootsr0r<r/r>r#ZlcZcoeffsr?signrrDrArrr!raHs>        rarX)__doc__Z sympy.corerrrrrrZsympy.polys.polyerrorsrr r r Zsympy.polys.polyoptionsr Zsympy.polys.polytoolsr rrZsympy.polys.specialpolysrrZsympy.utilitiesrrrrrHrUr`rarrrr!s$    5 A;