Ed( dZddlmZmZmZmZmZmZddlm Z m Z m Z m Z ddl mZddlmZmZmZddlmZmZddlmZmZmZedZed Zed Zeed fd ZeddZd S)z/High-level polynomials manipulation functions. )SBasicAddMulsymbolsDummy)PolificationFailedComputationFailedMultivariatePolynomialError OptionError) allowed_flags)poly_from_exprparallel_poly_from_exprPoly)symmetric_polyinterpolating_poly)numbered_symbolstakepublicc  t|ddgd}t|dsd}|g} t|g|Ri|\}}n#t$r}g}|jD]H}|jr"||tjf+tdt|||s|\}|j j s|cYd}~S|r |gfcYd}~S|gfzcYd}~Sd}~wwxYwg|j } }|j|j} }t!t|D]N} t#| dz|d } |t%| | | fOt)t!t|dz } t)t!t|d d }g}|D]}g}|jsV|||||jz}|r_d \}}}t1|D]V\} \}t5fd | Dr3t7dt9|D}||kr||}}}W|d kr||c}nng}t9dddzD]\}}|||z dt9||D}dt9||D} |t;|g|R| d |}| ddD]}||}||z}|_|t?|| fd|D}|j s2t1|D]"\} \}}|!||f|| <#|s|\}|j s|S|r||fS||fzS)a Rewrite a polynomial in terms of elementary symmetric polynomials. A symmetric polynomial is a multivariate polynomial that remains invariant under any variable permutation, i.e., if `f = f(x_1, x_2, \dots, x_n)`, then `f = f(x_{i_1}, x_{i_2}, \dots, x_{i_n})`, where `(i_1, i_2, \dots, i_n)` is a permutation of `(1, 2, \dots, n)` (an element of the group `S_n`). Returns a tuple of symmetric polynomials ``(f1, f2, ..., fn)`` such that ``f = f1 + f2 + ... + fn``. Examples ======== >>> from sympy.polys.polyfuncs import symmetrize >>> from sympy.abc import x, y >>> symmetrize(x**2 + y**2) (-2*x*y + (x + y)**2, 0) >>> symmetrize(x**2 + y**2, formal=True) (s1**2 - 2*s2, 0, [(s1, x + y), (s2, x*y)]) >>> symmetrize(x**2 - y**2) (-2*x*y + (x + y)**2, -2*y**2) >>> symmetrize(x**2 - y**2, formal=True) (s1**2 - 2*s2, -2*y**2, [(s1, x + y), (s2, x*y)]) formalrT__iter__F symmetrizeN)polysr)rNNc3BK|]}||dzkVdSrN).0imonoms 5lib/python3.11/site-packages/sympy/polys/polyfuncs.py zsymmetrize..es4AAAuQx5Q</AAAAAAcg|] \}}||z Srr)r nms r# zsymmetrize..fs !F!F!F$!Q!A#!F!F!Fr%)rc$g|] \\}}}||zSrr)r s_r's r#r)zsymmetrize..u$@@@YVaQAqD@@@r%c$g|] \\}}}||zSrr)r r,pr's r#r)zsymmetrize..vr-r%c@g|]\}}||fSr)as_expr)r r+r/s r#r)zsymmetrize..s) 0 0 0$!Qa  0 0 0r%)"r hasattrrr exprs is_NumberappendrZeror lenoptrrgensdomainrangernext set_domainlistis_homogeneousTCas_poly enumeratetermsallmaxziprmulrr1subs)Fr9argsiterabler8excresultexprrrdomr!polyindicesweightsf symmetric_height_monom_coeffcoeffheight exponentsm1m2termproductr/symnon_symr"s @r#rrsB$9-...H 1j ! ! C&(:T:::T::33 &&&I C CD~ C tQVn----' c!ffcBBB GFw~ &MMMMMM &rz!!!!!!~%%%%%%%&(7E#*#D 3t99  <<a!eT666 d7mmT__S%9%9:;;;;5TQ''((G5TAr**++G F &6&6  (   QTTVV $ $ $ '' 'A &4 #GVV%.qwwyy%9%9 G G!>E5AAAAAAAAAG !F!F#gu2E2E!F!F!FGGF'G28%"} %v uuIeU122Y%566 * *B  b))))@@#eY*?*?@@@D@@#eY*?*?@@@D   S.... / / /1gkk%((G!""X ) )!++a.. LA; >  sI 45555 0 0% 0 0 0E :3!*6!2!2 3 3 A~W%'2F1II  :%  %5= UH$ $s6A C A%C /C5C :CC C Cct|g t|g|Ri|\}}n#t$r}|jcYd}~Sd}~wwxYwtj|j}}|jr |D] }||z|z} nGt|||dd}}|D]}||zt|g|Ri|z}|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 rr rNrr6gen is_univariate all_coeffsrhorner) rSr9rJrIr8rLformrbrXs r#reresB$1D111D1133 x#D;\\^^ $ $E8e#DD $q#,,QRR4\\^^ ; ;E8fU:T:::T:::DD Ks& A<AAcDt|}t|trE||vrt||St t |\}}nt|dtrFt t |\}}||vr(t|||Sn\|td|dzvrt||dz St |}t td|dz} t|||| S#t$rIt}t|||| ||cYSwxYw)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)r7 isinstancedictrr>rFitemstupleindexr;rexpand ValueErrorrrH)dataxr'XYds r# interpolatertsT D A$ & 9 T!W:: C&''11 d1gu % % &T ##DAqAv (1771::''' (E!QUOO# &a!e~~%T AU1a!e__%%AB!!Q1--44666 BBB GG!!Q1--4466;;AqAAAAABs(#E AFFrpc ddlm}tt|\}}t |z dz }|dkrt d||zdz|zdz}t t|D]5}t |zdzD]} || |f|| z|| |dzf<6t |dzD]?}t |zdzD]'} || ||z f || z|| |zdz|z f<(@|d t fdt dzDt fdt |dzDz S)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)onesrz'Too few values for the required degree.c34K|]}||zzVdSNr)r r!rqrs r#r$z'rational_interpolate..Ds/77!q!t 777777r%c3@K|]}|zdz|zzVdSrr)r r!rqdegnumrzs r#r$z'rational_interpolate..Es9AAq!AJN#ad*AAAAAAr%) sympy.matrices.denservr>rFr7r r;rE nullspacesum) ror|rqrvxdataydatakcjr!rzs `` @r#rational_interpolater sR*)))))T ##LE5 E VaA1uECDDD VaZ!^VaZ!^,,A 3vq>> " "++vzA~&& + +AAqD'%(*AaQhKK + 1q5\\==vzA~&& = =A()!QU( |E!H'>> 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 allowedz8Cannot derive Viete's formulas for a constant polynomialrz)startz required z roots, got r)r rhrrr r is_multivariater degreernrrr7LCrdrBrr5)rSrootsr9rJr8rLr'lccoeffsrMsignr!rXrPs r#rrHs"$%,hote11D111D1133 111C0001 8) 688 8  A1uH FHH H / A... 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