a ЯмddДZ?dd ДZ@d!d"ДZAd#d$ДZBd%d&ДZCd'd(ДZDd)d*ДZEd+d,ДZFd-d.ДZGd/d0ДZHd1d2ДZId3d4ДZJd5d6ДZKd7d8ДZLd9d:ДZMd;d<ДZNd=d>ДZOd?d@ДZPdAdBДZQdCdDДZRdEdFДZSdGdHДZTdIdJДZUdKdLДZVdMdNДZWdOdPДZXdQdRДZYdSdTДZZdUdVДZ[dcdYdZДZ\d[d\ДZ]ddd]d^ДZ^d_d`ДZ_dadbДZ`dWS)ezHAdvanced tools for dense recursive polynomials in ``K[x]`` or ``K[X]``. щ)┌dup_add_term┌dmp_add_term┌ dup_lshift┌dup_add┌dmp_add┌dup_sub┌dmp_sub┌dup_mul┌dmp_mul┌dup_sqr┌dup_div┌dup_rem┌dmp_rem┌ dmp_expand┌dup_mul_ground┌dmp_mul_ground┌dup_quo_ground┌dmp_quo_ground┌dup_exquo_ground┌dmp_exquo_ground)┌ dup_strip┌ dmp_strip┌dup_convert┌dmp_convert┌ dup_degree┌ dmp_degree┌dmp_to_dict┌ dmp_from_dict┌dup_LC┌dmp_LC┌ dmp_ground_LC┌dup_TC┌dmp_TC┌dmp_zero┌ dmp_ground┌ dmp_zero_p┌dup_to_raw_dict┌dup_from_raw_dict┌ dmp_zeros)┌MultivariatePolynomialError┌DomainError)┌ variations)┌ceil┌logc Csv|dks|s|S|jg|}tt|ГГD]H\}}|d}td|ГD]}|||d9}qB|аd|а|||Гббq(|S)a Computes the indefinite integral of ``f`` in ``K[x]``. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x = ring("x", QQ) >>> R.dup_integrate(x**2 + 2*x, 1) 1/3*x**3 + x**2 >>> R.dup_integrate(x**2 + 2*x, 2) 1/12*x**4 + 1/3*x**3 rщ)┌zero┌ enumerate┌reversed┌range┌insertZexquo)┌f┌m┌K┌g┌i┌c┌n┌jйr<·f/Users/vegardjervell/Documents/master/model/venv/lib/python3.9/site-packages/sympy/polys/densetools.py┌ dup_integrate'sr>c CsЬ|st|||ГS|dks"t||Гr&|St||d|Г|d}}tt|ГГD]J\}}|d}td|ГD]} ||| d9}qf|аdt|||Г||ГбqL|S)a& Computes the indefinite integral of ``f`` in ``x_0`` in ``K[X]``. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x,y = ring("x,y", QQ) >>> R.dmp_integrate(x + 2*y, 1) 1/2*x**2 + 2*x*y >>> R.dmp_integrate(x + 2*y, 2) 1/6*x**3 + x**2*y rr.)r>r%r(r0r1r2r3r) r4r5┌ur6r7┌vr8r9r:r;r<r<r=┌ dmp_integrateGsrAcsHИИkrt|И|ИГS|dИdЙЙtЗЗЗЗЗfddД|DГ|ГS)z.Recursive helper for :func:`dmp_integrate_in`.r.c sg|]}t|ИИИИИГСqSr<)┌_rec_integrate_inй┌.0r9йr6r8r;r5┌wr<r=┌ qєz%_rec_integrate_in..)rArйr7r5r@r8r;r6r<rEr=rBjsrBcCs2|dks||kr td||fГВt|||d||ГS)a+ Computes the indefinite integral of ``f`` in ``x_j`` in ``K[X]``. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x,y = ring("x,y", QQ) >>> R.dmp_integrate_in(x + 2*y, 1, 0) 1/2*x**2 + 2*x*y >>> R.dmp_integrate_in(x + 2*y, 1, 1) x*y + y**2 rz(0 <= j <= u expected, got u = %d, j = %d)┌ IndexErrorrBйr4r5r;r?r6r<r<r=┌dmp_integrate_intsrLcCs╕|dkr|St|Г}||kr gSg}|dkr\|d|ЕD]}|а||Г|б|d8}q:nT|d|ЕD]D}|}t|d||dГD]}||9}qЖ|а||Г|б|d8}qjt|ГS)a# ``m``-th order derivative of a polynomial in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_diff(x**3 + 2*x**2 + 3*x + 4, 1) 3*x**2 + 4*x + 3 >>> R.dup_diff(x**3 + 2*x**2 + 3*x + 4, 2) 6*x + 4 rr.Nщ )r┌appendr2r)r4r5r6r:┌deriv┌coeff┌kr8r<r<r=┌dup_diffКs" rRc Csц|st|||ГS|dkr|St||Г}||kr6t|ГSg|d}}|dkrВ|d|ЕD]$}|аt|||Г||Гб|d8}qZnZ|d|ЕD]J}|}t|d||dГD]} || 9}qм|аt|||Г||Гб|d8}qРt||ГS)a3 ``m``-th order derivative in ``x_0`` of a polynomial in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = x*y**2 + 2*x*y + 3*x + 2*y**2 + 3*y + 1 >>> R.dmp_diff(f, 1) y**2 + 2*y + 3 >>> R.dmp_diff(f, 2) 0 rr.NrM)rRrr#rNrr2r) r4r5r?r6r:rOr@rPrQr8r<r<r=┌dmp_diff╡s& rScsHИИkrt|И|ИГS|dИdЙЙtЗЗЗЗЗfddД|DГ|ГS)z)Recursive helper for :func:`dmp_diff_in`.r.c sg|]}t|ИИИИИГСqSr<)┌_rec_diff_inrCrEr<r=rGыrHz _rec_diff_in..)rSrrIr<rEr=rTфsrTcCs2|dks||kr td||fГВt|||d||ГS)aS ``m``-th order derivative in ``x_j`` of a polynomial in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = x*y**2 + 2*x*y + 3*x + 2*y**2 + 3*y + 1 >>> R.dmp_diff_in(f, 1, 0) y**2 + 2*y + 3 >>> R.dmp_diff_in(f, 1, 1) 2*x*y + 2*x + 4*y + 3 r·0 <= j <= %s expected, got %s)rJrTrKr<r<r=┌dmp_diff_inюsrVcCs2|st||ГS|j}|D]}||9}||7}q|S)z▐ Evaluate a polynomial at ``x = a`` in ``K[x]`` using Horner scheme. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_eval(x**2 + 2*x + 3, 2) 11 )r!r/)r4┌ar6┌resultr9r<r<r=┌dup_evals rYcCsd|st|||ГS|st||ГSt||Г|d}}|ddЕD] }t||||Г}t||||Г}q>|S)zЄ Evaluate a polynomial at ``x_0 = a`` in ``K[X]`` using the Horner scheme. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_eval(2*x*y + 3*x + y + 2, 2) 5*y + 8 r.N)rYr"rrr)r4rWr?r6rXr@rPr<r<r=┌dmp_eval s rZcsHИИkrt|ИИИГSИdИdЙЙtЗЗЗЗЗfddД|DГИГS)z)Recursive helper for :func:`dmp_eval_in`.r.c sg|]}t|ИИИИИГСqSr<)┌_rec_eval_inrCйr6rWr8r;r@r<r=rGDrHz _rec_eval_in..)rZr)r7rWr@r8r;r6r<r\r=r[=sr[cCs2|dks||kr td||fГВt|||d||ГS)a2 Evaluate a polynomial at ``x_j = a`` in ``K[X]`` using the Horner scheme. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = 2*x*y + 3*x + y + 2 >>> R.dmp_eval_in(f, 2, 0) 5*y + 8 >>> R.dmp_eval_in(f, 2, 1) 7*x + 4 rrU)rJr[)r4rWr;r?r6r<r<r=┌dmp_eval_inGsr]csfИИkrt|ИdИГSЗЗЗЗfddД|DГ}ИИtИГdkrH|St|ИИИdИГSdS)z+Recursive helper for :func:`dmp_eval_tail`.rMcs g|]}t|ИdИИИГСqSйr.)┌_rec_eval_tailrCй┌Ar6r8r?r<r=rGdrHz"_rec_eval_tail..r.N)rY┌len)r7r8rar?r6┌hr<r`r=r__sr_cCs\|s|St||Гr"t|t|ГГSt|d|||Г}|t|ГdkrF|St||t|ГГSdS)a! Evaluate a polynomial at ``x_j = a_j, ...`` in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = 2*x*y + 3*x + y + 2 >>> R.dmp_eval_tail(f, [2]) 7*x + 4 >>> R.dmp_eval_tail(f, [2, 2]) 18 rr.N)r%r#rbr_r)r4rar?r6┌er<r<r=┌ dmp_eval_taills recsTИИkr tt|ИИИГИИИГSИdИdЙЙtЗЗЗЗЗЗfddД|DГИГS)z+Recursive helper for :func:`dmp_diff_eval`.r.c s g|]}t|ИИИИИИГСqSr<)┌_rec_diff_evalrCйr6rWr8r;r5r@r<r=rGУrHz"_rec_diff_eval..)rZrSr)r7r5rWr@r8r;r6r<rgr=rfМsrfcCsJ||krtd|||fГВ|s6tt||||Г|||ГSt||||d||ГS)a] Differentiate and evaluate a polynomial in ``x_j`` at ``a`` in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = x*y**2 + 2*x*y + 3*x + 2*y**2 + 3*y + 1 >>> R.dmp_diff_eval_in(f, 1, 2, 0) y**2 + 2*y + 3 >>> R.dmp_diff_eval_in(f, 1, 2, 1) 6*x + 11 z-%s <= j < %s expected, got %sr)rJrZrSrf)r4r5rWr;r?r6r<r<r=┌dmp_diff_eval_inЦs rhcs^|jrDg}|D]2}|И}|Иdkr6|а|Ибq|а|бqnЗfddД|DГ}t|ГS)zЇ Reduce a ``K[x]`` polynomial modulo a constant ``p`` in ``K``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_trunc(2*x**3 + 3*x**2 + 5*x + 7, ZZ(3)) -x**3 - x + 1 щcsg|]}|ИСqSr<r<rCй┌pr<r=rG╔rHzdup_trunc..)┌is_ZZrNr)r4rkr6r7r9r<rjr=┌ dup_trunc░srmcstЗЗЗfddД|DГИГS)a9 Reduce a ``K[X]`` polynomial modulo a polynomial ``p`` in ``K[Y]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = 3*x**2*y + 8*x**2 + 5*x*y + 6*x + 2*y + 3 >>> g = (y - 1).drop(x) >>> R.dmp_trunc(f, g) 11*x**2 + 11*x + 5 csg|]}t|ИИdИГСqSr^)rrCйr6rkr?r<r=rG▀rHzdmp_trunc..)rйr4rkr?r6r<rnr=┌ dmp_trunc╬srpcs4|st|ИИГS|dЙtЗЗЗfddД|DГ|ГS)a Reduce a ``K[X]`` polynomial modulo a constant ``p`` in ``K``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = 3*x**2*y + 8*x**2 + 5*x*y + 6*x + 2*y + 3 >>> R.dmp_ground_trunc(f, ZZ(3)) -x**2 - x*y - y r.csg|]}t|ИИИГСqSr<)┌dmp_ground_truncrCйr6rkr@r<r=rGўrHz$dmp_ground_trunc..)rmrror<rrr=rqтsrqcCs0|s|St||Г}|а|бr |St|||ГSdS)a7 Divide all coefficients by ``LC(f)`` in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ, QQ >>> R, x = ring("x", ZZ) >>> R.dup_monic(3*x**2 + 6*x + 9) x**2 + 2*x + 3 >>> R, x = ring("x", QQ) >>> R.dup_monic(3*x**2 + 4*x + 2) x**2 + 4/3*x + 2/3 N)r┌is_oner)r4r6┌lcr<r<r=┌ dup_monic·s rucCsH|st||ГSt||Гr|St|||Г}|а|бr6|St||||ГSdS)a├ Divide all coefficients by ``LC(f)`` in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ, QQ >>> R, x,y = ring("x,y", ZZ) >>> f = 3*x**2*y + 6*x**2 + 3*x*y + 9*y + 3 >>> R.dmp_ground_monic(f) x**2*y + 2*x**2 + x*y + 3*y + 1 >>> R, x,y = ring("x,y", QQ) >>> f = 3*x**2*y + 8*x**2 + 5*x*y + 6*x + 2*y + 3 >>> R.dmp_ground_monic(f) x**2*y + 8/3*x**2 + 5/3*x*y + 2*x + 2/3*y + 1 N)rur%r rsr)r4r?r6rtr<r<r=┌dmp_ground_monics rvcCsdddlm}|s|jS|j}||kr<|D]}|а||б}q(n$|D]}|а||б}|а|бr@q`q@|S)aA Compute the GCD of coefficients of ``f`` in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ, QQ >>> R, x = ring("x", ZZ) >>> f = 6*x**2 + 8*x + 12 >>> R.dup_content(f) 2 >>> R, x = ring("x", QQ) >>> f = 6*x**2 + 8*x + 12 >>> R.dup_content(f) 2 rй┌QQ)┌sympy.polys.domainsrxr/┌gcdrs)r4r6rx┌contr9r<r<r=┌dup_content;s r|cCsТddlm}|st||ГSt||Гr*|jS|j|d}}||krb|D]}|а|t|||Гб}qFn,|D]&}|а|t|||Гб}|а|бrfqОqf|S)aa Compute the GCD of coefficients of ``f`` in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ, QQ >>> R, x,y = ring("x,y", ZZ) >>> f = 2*x*y + 6*x + 4*y + 12 >>> R.dmp_ground_content(f) 2 >>> R, x,y = ring("x,y", QQ) >>> f = 2*x*y + 6*x + 4*y + 12 >>> R.dmp_ground_content(f) 2 rrwr.)ryrxr|r%r/rz┌dmp_ground_contentrs)r4r?r6rxr{r@r9r<r<r=r}es r}cCs>|s|j|fSt||Г}|а|бr*||fS|t|||ГfSdS)at Compute content and the primitive form of ``f`` in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ, QQ >>> R, x = ring("x", ZZ) >>> f = 6*x**2 + 8*x + 12 >>> R.dup_primitive(f) (2, 3*x**2 + 4*x + 6) >>> R, x = ring("x", QQ) >>> f = 6*x**2 + 8*x + 12 >>> R.dup_primitive(f) (2, 3*x**2 + 4*x + 6) N)r/r|rsr)r4r6r{r<r<r=┌ dup_primitiveТs r~cCsV|st||ГSt||Гr"|j|fSt|||Г}|а|бr@||fS|t||||ГfSdS)aЪ Compute content and the primitive form of ``f`` in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ, QQ >>> R, x,y = ring("x,y", ZZ) >>> f = 2*x*y + 6*x + 4*y + 12 >>> R.dmp_ground_primitive(f) (2, x*y + 3*x + 2*y + 6) >>> R, x,y = ring("x,y", QQ) >>> f = 2*x*y + 6*x + 4*y + 12 >>> R.dmp_ground_primitive(f) (2, x*y + 3*x + 2*y + 6) N)r~r%r/r}rsr)r4r?r6r{r<r<r=┌dmp_ground_primitive│s rcCsLt||Г}t||Г}|а||б}|а|бsBt|||Г}t|||Г}|||fS)a Extract common content from a pair of polynomials in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_extract(6*x**2 + 12*x + 18, 4*x**2 + 8*x + 12) (2, 3*x**2 + 6*x + 9, 2*x**2 + 4*x + 6) )r|rzrsr)r4r7r6┌fc┌gcrzr<r<r=┌dup_extract╫s rВcCsTt|||Г}t|||Г}|а||б}|а|бsJt||||Г}t||||Г}|||fS)a Extract common content from a pair of polynomials in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_ground_extract(6*x*y + 12*x + 18, 4*x*y + 8*x + 12) (2, 3*x*y + 6*x + 9, 2*x*y + 4*x + 6) )r}rzrsr)r4r7r?r6rАrБrzr<r<r=┌dmp_ground_extractёs rГc Cs|js|jstd|ГВtdГ}tdГ}|s4||fS|j|jgg|jgggg}t|ddГ}|ddЕD](}t||d|Г}t|t|dГdd|Г}qht |Г}|а бD]b\}}|d} | s╞t||d|Г}qв| dkr▐t||d|Г}qв| dkrЎt||d|Г}qвt||d|Г}qв||fS)a4 Return bivariate polynomials ``f1`` and ``f2``, such that ``f = f1 + f2*I``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dup_real_imag(x**3 + x**2 + x + 1) (x**3 + x**2 - 3*x*y**2 + x - y**2 + 1, 3*x**2*y + 2*x*y - y**3 + y) z;computing real and imaginary parts is not supported over %sr.rriNщ) rlZis_QQr*r#┌oner/r$r rr&┌itemsrr) r4r6┌f1┌f2r7rcr9┌HrQr5r<r<r=┌ dup_real_imags,rКcCs4t|Г}tt|ГdddГD]}||||<q|S)zЇ Evaluate efficiently the composition ``f(-x)`` in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_mirror(x**3 + 2*x**2 - 4*x + 2) -x**3 + 2*x**2 + 4*x + 2 rirMщ■ )┌listr2rb)r4r6r8r<r<r=┌ dup_mirror:srНcCsPt|Гt|Гd|}}}t|dddГD]}|||||||<}q,|S)zц Evaluate efficiently composition ``f(a*x)`` in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_scale(x**2 - 2*x + 1, ZZ(2)) 4*x**2 - 4*x + 1 r.rMйrМrbr2)r4rWr6r:┌br8r<r<r=┌ dup_scalePsrРcCsXt|Гt|Гd}}t|ddГD]0}td|ГD] }||d|||7<q0q"|S)zч Evaluate efficiently Taylor shift ``f(x + a)`` in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_shift(x**2 - 2*x + 1, ZZ(2)) x**2 + 2*x + 1 r.rrMrО)r4rWr6r:r8r;r<r<r=┌ dup_shiftfs rСc CsЬ|sgSt|Гd}|dg|jgg}}td|ГD]}|аt|d||Гбq4t|ddЕ|ddЕГD],\}}t|||Г}t|||Г}t|||Г}qj|S)a Evaluate functional transformation ``q**n * f(p/q)`` in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_transform(x**2 - 2*x + 1, x**2 + 1, x - 1) x**4 - 2*x**3 + 5*x**2 - 4*x + 4 r.rrMN)rbrЕr2rNr ┌ziprr) r4rk┌qr6r:rc┌Qr8r9r<r<r=┌ dup_transform}s"rХcCsft|Гdkr$tt|t||Г|ГgГS|s,gS|dg}|ddЕD]}t|||Г}t||d|Г}qB|S)z╫ Evaluate functional composition ``f(g)`` in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_compose(x**2 + x, x - 1) x**2 - x r.rN)rbrrYrr r)r4r7r6rcr9r<r<r=┌dup_composeЬs rЦcCs\|st|||ГSt||Гr|S|dg}|ddЕD]"}t||||Г}t||d||Г}q4|S)z▐ Evaluate functional composition ``f(g)`` in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_compose(x*y + 2*x + y, y) y**2 + 3*y rr.N)rЦr%r r)r4r7r?r6rcr9r<r<r=┌dmp_compose╣s rЧcCs╥t|Гd}t||Г}t|Г}||ji}||}td|ГD]М}|j}td|ГD]Z} || ||vrdqN|| |vrrqN||| |||| } }|||| | |7}qN|а||||б|||<q:t||ГS)·+Helper function for :func:`_dup_decompose`.r.r)rbrr&rЕr2r/Zquor')r4┌sr6r:rtr7┌rr8rPr;rАrБr<r<r=┌_dup_right_decompose╓s rЫcCsVid}}|rLt|||Г\}}t|Гdkr.dSt||Г||<||d}}q t||ГS)rШrNr.)rrrr')r4rcr6r7r8rУrЪr<r<r=┌_dup_left_decomposeЄs rЬcCsbt|Гd}td|ГD]F}||dkr(qt|||Г}|durt|||Г}|dur||fSqdS)z*Helper function for :func:`dup_decompose`.r.rirN)rbr2rЫrЬ)r4r6ZdfrЩrcr7r<r<r=┌_dup_decomposesrЭcCs8g}t||Г}|dur.|\}}|g|}qq.q|g|S)ae Computes functional decomposition of ``f`` in ``K[x]``. Given a univariate polynomial ``f`` with coefficients in a field of characteristic zero, returns list ``[f_1, f_2, ..., f_n]``, where:: f = f_1 o f_2 o ... f_n = f_1(f_2(... f_n)) and ``f_2, ..., f_n`` are monic and homogeneous polynomials of at least second degree. Unlike factorization, complete functional decompositions of polynomials are not unique, consider examples: 1. ``f o g = f(x + b) o (g - b)`` 2. ``x**n o x**m = x**m o x**n`` 3. ``T_n o T_m = T_m o T_n`` where ``T_n`` and ``T_m`` are Chebyshev polynomials. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_decompose(x**4 - 2*x**3 + x**2) [x**2, x**2 - x] References ========== .. [1] [Kozen89]_ N)rЭ)r4r6┌FrXrcr<r<r=┌ dup_decomposes$ rЯc Csт|jr |аб}t||||Г}|}|js.tdГВt||Гgg}}}|абD]\}}|jsL|а|бqLt ddgt |ГddН} | D]H} t|Г}t| |ГD]\}}|dkrЦ||||<qЦ|аt |||ГбqАtt|||Г|||jГS)a^ Convert algebraic coefficients to integers in ``K[X]``. Examples ======== >>> from sympy.polys import ring, QQ >>> from sympy import I >>> K = QQ.algebraic_field(I) >>> R, x = ring("x", K) >>> f = x**2 + K([QQ(1), QQ(0)])*x + K([QQ(2), QQ(0)]) >>> R.dmp_lift(f) x**8 + 2*x**6 + 9*x**4 - 8*x**2 + 16 z3computation can be done only in an algebraic domainrMr.T)Z repetition)Zis_GaussianFieldZas_AlgebraicFieldrZis_Algebraicr*rrЖZ is_groundrNr+rb┌dictrТrr┌dom) r4r?r6┌K1rЮZmonomsZpolysZmonomrPZperms┌perm┌G┌signr<r<r=┌dmp_liftGs( rжcCs8|jd}}|D]"}|а||бr*|d7}|r|}q|S)zт Compute the number of sign variations of ``f`` in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_sign_variations(x**4 - x**2 - x + 1) 2 rr.)r/Zis_negative)r4r6┌prevrQrPr<r<r=┌dup_sign_variationswsrиNFcCst|dur|jr|аб}n|}|j}|D]}|а||а|бб}q&|а|бsTt|||Г}|s`||fS|t|||ГfSdS)a@ Clear denominators, i.e. transform ``K_0`` to ``K_1``. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x = ring("x", QQ) >>> f = QQ(1,2)*x + QQ(1,3) >>> R.dup_clear_denoms(f, convert=False) (6, 3*x + 2) >>> R.dup_clear_denoms(f, convert=True) (6, 3*x + 2) N)┌has_assoc_Ring┌get_ringrЕ┌lcm┌denomrsrr)r4┌K0rв┌convert┌commonr9r<r<r=┌dup_clear_denomsСs r░c CsT|j}|s(|D]}|а||а|бб}qn(|d}|D]}|а|t||||Гб}q4|S)z.Recursive helper for :func:`dmp_clear_denoms`.r.)rЕrлrм┌_rec_clear_denoms)r7r@rнrвrпr9rFr<r<r=r▒╖sr▒cCsx|st||||dНS|dur0|jr,|аб}n|}t||||Г}|а|бsVt||||Г}|sb||fS|t||||ГfSdS)aV Clear denominators, i.e. transform ``K_0`` to ``K_1``. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x,y = ring("x,y", QQ) >>> f = QQ(1,2)*x + QQ(1,3)*y + 1 >>> R.dmp_clear_denoms(f, convert=False) (6, 3*x + 2*y + 6) >>> R.dmp_clear_denoms(f, convert=True) (6, 3*x + 2*y + 6) )rоN)r░rйrкr▒rsrr)r4r?rнrвrоrпr<r<r=┌dmp_clear_denoms╟s r▓c CsТ|аt||Гбg}|j|j|jg}ttt|dГГГ}td|dГD]J}t||dГ|Г}t |t ||Г|Г}tt|||Г||Г}t |t|Г|Г}qB|S)aў Compute ``f**(-1)`` mod ``x**n`` using Newton iteration. This function computes first ``2**n`` terms of a polynomial that is a result of inversion of a polynomial modulo ``x**n``. This is useful to efficiently compute series expansion of ``1/f``. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x = ring("x", QQ) >>> f = -QQ(1,720)*x**6 + QQ(1,24)*x**4 - QQ(1,2)*x**2 + 1 >>> R.dup_revert(f, 8) 61/720*x**6 + 5/24*x**4 + 1/2*x**2 + 1 rir.)┌revertr!rЕr/┌int┌_ceil┌_logr2rr rr rrr) r4r:r6r7rc┌Nr8rWrПr<r<r=┌ dup_revertэsr╕cCs|st|||ГSt||ГВdS)zй Compute ``f**(-1)`` mod ``x**n`` using Newton iteration. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x,y = ring("x,y", QQ) N)r╕r))r4r7r?r6r<r<r=┌ dmp_revertsr╣)NF)NF)a┌__doc__Zsympy.polys.densearithrrrrrrrr r rrr rrrrrrrrZsympy.polys.densebasicrrrrrrrrrrr r!r"r#r$r%r&r'r(Zsympy.polys.polyerrorsr)r*Zsympy.utilitiesr+┌mathr,r╡r-r╢r>rArBrLrRrSrTrVrYrZr[r]r_rerfrhrmrprqrurvr|r}r~rrВrГrКrНrРrСrХrЦrЧrЫrЬrЭrЯrжrиr░r▒r▓r╕r╣r<r<r<r=┌sdXT # +/ $*-!$/20 & &"