U Zʹjc!@sdZddlmZmZmZmZddlmZmZm Z m Z ddl m Z m Z ddlmZmZddlmZmZddlmZddlmZmZmZed d Zed d Zed dZeedfddZedddZdS)z/High-level polynomials manipulation functions. )SBasicsymbolsDummy)PolificationFailedComputationFailedMultivariatePolynomialError OptionError) allowed_flags build_options)poly_from_exprPoly)symmetric_polyinterpolating_poly)sring)numbered_symbolstakepubliccst|ddgd}t|ds&d}|g}t|f||\}}|j}t||}|jfddtt|Dg}|D],}|\}} } ||j | j |fqrddt | D} |j st |D]\} \} }| | |f|| <q|s|\}|j s|S|r|| fS|| fSd S) 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__Fcsg|] }tqS)next.0irrT/var/www/html/bwave/python/venv/lib/python3.8/site-packages/sympy/polys/polyfuncs.py =szsymmetrize..cSs g|]\}\}}||fqSr)as_expr)rs_grrrrEs N)r hasattrrrr rangelen symmetrizeappendrzipr enumeratesubs)FgensargsiterableRoptresultfprmZpolysrsymZnon_symrrrr%s2!  r%c Ost|gzt|f||\}}Wn,tk rN}z|jWYSd}~XYnXtj|j}}|jr|D]}|||}qln>t |||dd}}|D]}||t |f||}q|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 rexprrZZerogenZ is_univariate all_coeffsr horner) r1r+r,r*r/excformr8coeffrrrr:Ws!   r:cCst|}t|tr<||kr&t||Stt|\}}nvt|dtrvtt|\}}||krt|||Sn<|t d|dkrt||dSt|}tt d|d}zt |||| WSt k rt }t |||| ||YSXdS)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)] rr6N)r$ isinstancedictrlistr'itemstupleindexr#rexpand ValueErrorrr))dataxnXYdrrr interpolates$*  rLrGc sDddlm}tt|\}}t|d}|dkr>> from sympy.polys.polyfuncs import rational_interpolate >>> data = [(1, -210), (2, -35), (3, 105), (4, 231), (5, 350), (6, 465)] >>> rational_interpolate(data, 2) (105*x**2 - 525)/(x + 1) Values do not need to be integers: >>> from sympy import sympify >>> x = [1, 2, 3, 4, 5, 6] >>> y = sympify("[-1, 0, 2, 22/5, 7, 68/7]") >>> rational_interpolate(zip(x, y), 2) (3*x**2 - 7*x + 2)/(x + 1) The symbol for the variable can be changed if needed: >>> from sympy import symbols >>> z = symbols('z') >>> rational_interpolate(data, 2, X=z) (105*z**2 - 525)/(z + 1) References ========== .. [1] Algorithm is adapted from: http://axiom-wiki.newsynthesis.org/RationalInterpolation r)onesr6z'Too few values for the required degree.c3s|]}||VqdS)Nrr)rIr3rr sz'rational_interpolate..c3s&|]}|d|VqdS)r6NrrrIdegnumr3rrrOs) Zsympy.matrices.denserMr@r'r$r r#maxZ nullspacesum) rFrQrIrMZxdataZydatakcjrrrPrrational_interpolates ) $2  rWNc Os:t|gt|tr$|f|d}}zt|f||\}}Wn.tk rj}ztdd|W5d}~XYnX|jrztd|}|dkrt d|dkrt ddd}t ||}|t |krt d|t |f| |}}gd } } t|ddD]8\} } 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)] Nvieter6z(multivariate polynomials are not allowedz8Cannot derive Viete's formulas for a constant polynomialr3)startzrequired %s roots, got %s)r r>rr rrZis_multivariaterZdegreerErrr$ZLCr9r(rr&)r1rootsr+r,r/r;rHlcZcoeffsr0signrr=ZpolyrrrrX s:        rX)N)__doc__Z sympy.corerrrrZsympy.polys.polyerrorsrrrr Zsympy.polys.polyoptionsr r Zsympy.polys.polytoolsr r Zsympy.polys.specialpolysrrZsympy.polys.ringsrZsympy.utilitiesrrrr%r:rLrWrXrrrrs"  G 5 A;