B ['@sdZddlmZmZddlmZmZmZddlm Z ddl m Z m Z ddl mZmZmZmZddlmZmZmZddlmZmZmZmZmZdd lmZdd lmZed d Z ed dZ!eddZ"eedfddZ#edddZ$dS)z/High-level polynomials manipulation functions. )print_functiondivision)poly_from_exprparallel_poly_from_exprPoly) allowed_flags)symmetric_polyinterpolating_poly)PolificationFailedComputationFailedMultivariatePolynomialError OptionError)numbered_symbolstakepublic)SBasicAddMulsymbols)range) factorialc s`t|ddgd}t|ds&d}|g}yt|f||\}}Wntk r}zlg}xb|jD],}|jrx||tjfq\t dt ||q\W|s|\}|j j s|S|r|gfS|gfSWdd}~XYnXg|j }} |j|j}} x>tt |D].} t| d|dd } |t| | | fqWttt |d} ttt |d d }g}x|D]}g}|js||||8}xD|rd \}}}xht|D]X\} \}tfd d| Drtddt|D}||kr||}}}qW|d kr||}nPg}x2tdddD]\}}|||q6Wddt||D}ddt||D} |t|f|| d |}x | ddD]}||}qW||8}qW|t||fqXWdd|D}|j s,x,t|D] \} \}}| ||f|| <qW|s8|\}|j sD|S|rR||fS||fSdS)a Rewrite a polynomial in terms of elementary symmetric polynomials. A symmetric polynomial is a multivariate polynomial that remains invariant under any variable permutation, i.e., if ``f = f(x_1, x_2, ..., x_n)``, then ``f = f(x_{i_1}, x_{i_2}, ..., x_{i_n})``, where ``(i_1, i_2, ..., i_n)`` is a permutation of ``(1, 2, ..., n)`` (an element of the group ``S_n``). Returns a tuple of symmetric polynomials ``(f1, f2, ..., fn)`` such that ``f = f1 + f2 + ... + fn``. Examples ======== >>> from sympy.polys.polyfuncs import symmetrize >>> from sympy.abc import x, y >>> symmetrize(x**2 + y**2) (-2*x*y + (x + y)**2, 0) >>> symmetrize(x**2 + y**2, formal=True) (s1**2 - 2*s2, 0, [(s1, x + y), (s2, x*y)]) >>> symmetrize(x**2 - y**2) (-2*x*y + (x + y)**2, -2*y**2) >>> symmetrize(x**2 - y**2, formal=True) (s1**2 - 2*s2, -2*y**2, [(s1, x + y), (s2, x*y)]) formalrT__iter__F symmetrizeN)polysr)rNNc3s"|]}||dkVqdS)rN).0i)monomr4lib/python3.7/site-packages/sympy/polys/polyfuncs.py nszsymmetrize..cSsg|]\}}||qSrr)rnmrrr" oszsymmetrize..)rcSsg|]\\}}}||qSrr)rs_r$rrr"r&~scSsg|]\\}}}||qSrr)rr(pr$rrr"r&scSsg|]\}}||fqSr)as_expr)rr'r)rrr"r&s)!rhasattrrr ZexprsZ is_NumberappendrZeror lenoptrrgensZdomainrrnextZ set_domainlistZis_homogeneousZTC enumerateZtermsallmaxziprmulrr*Zsubs)Fr0argsiterabler/excresultexprrrZdomr polyindicesZweightsfZ symmetricZ_heightZ_monomZ_coeffcoeffZheightZ exponentsZm1Zm2Ztermproductr)ZsymZnon_symr)r!r"rs!          rc Ost|gyt|f||\}}Wn$tk rF}z|jSd}~XYnXtj|j}}|jr|x`|D]}|||}qfWnBt |||dd}}x(|D]}||t |f||}qW|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] - http://en.wikipedia.org/wiki/Horner_scheme Nr) rrr r=rr-genZ is_univariate all_coeffsrhorner) r@r0r9r8r/r;ZformrCrArrr"rEs! rEc st|}d}t|tr:tt|\}}t|||}nt|dtrhtt|\}}t|||}nt|}tfddt d|dD}|ddkrt |d n t |d}g}x>> from sympy.polys.polyfuncs import interpolate >>> from sympy.abc import 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 Nrcsg|] }|qSrr)rr )xrr"r&szinterpolate..rcSsg|]\}}||qSrr)rrAyrrr"r&s)r. isinstancedictr2r6itemsr tuplerrrr,rexpand) datarFr$r>XYZnumertZdenomcoeffsr r)rFr" interpolates"  &rRrFc sTddlm}tt|\}}t|d}|dkr}x8t|dD]$} || |f|| || |df<q|WqfWx\t|dD]L}xFt|dD]2} || ||f || || |d|f<qWqW|dt fddtdDt fddt|dDS) 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 ========== Algorithm is adapted from: http://axiom-wiki.newsynthesis.org/RationalInterpolation r)onesrz'Too few values for the required degree.rGc3s|]}||VqdS)Nr)rr )rOrrr"r#Asz'rational_interpolate..c3s&|]}|d|VqdS)rNr)rr )rOdegnumrTrr"r#Bs) Zsympy.matrices.denserSr2r6r.r rr5Z nullspacesum) rNrUrOrSZxdataZydatakcjr r)rOrUrTr"rational_interpolate s' (6 rZNc Os>t|gt|tr$|f|d}}yt|f||\}}Wn.tk rj}ztdd|Wdd}~XYnX|jrztd|}|dkrt d|dkrt ddd}t ||}|t |krt d|t |f| |}}gd } } xLt|ddD]8\} } t| d|} | | |} | | | f| } qW| S) a# Generate Viete's formulas for ``f``. Examples ======== >>> from sympy.polys.polyfuncs import viete >>> from sympy import symbols >>> x, a, b, c, r1, r2 = symbols('x,a:c,r1:3') >>> viete(a*x**2 + b*x + c, [r1, r2], x) [(r1 + r2, -b/a), (r1*r2, c/a)] Nvieterz(multivariate polynomials are not allowedz7can't derive Viete's formulas for a constant polynomialrT)startzrequired %s roots, got %sr)rrIrrr r Zis_multivariater Zdegree ValueErrorrrr.ZLCrDr3rr,)r@rootsr0r9r/r;r$ZlcrQr<Zsignr rAr>rrr"r[Es6        r[)N)%__doc__Z __future__rrZsympy.polys.polytoolsrrrZsympy.polys.polyoptionsrZsympy.polys.specialpolysrr Zsympy.polys.polyerrorsr r r r Zsympy.utilitiesrrrZ sympy.corerrrrrZsympy.core.compatibilityrZ(sympy.functions.combinatorial.factorialsrrrErRrZr[rrrr"s"     6 89