U Ш9%e<Еу@sTdZddlmZmZmZmZmZmZmZm Z m Z mZmZm Z mZddlmZmZddДZddДZdd ДZd dДZdd ДZddДZddДZddДZddДZddДZddДZddДZddДZddДZd d!ДZ d"d#ДZ!d$d%ДZ"d&d'ДZ#d(d)ДZ$d*d+ДZ%d,d-ДZ&d.d/ДZ'd0d1ДZ(d2d3ДZ)d4d5ДZ*d6d7ДZ+d8d9ДZ,d:d;ДZ-dd?ДZ/d@dAДZ0dBdCДZ1dDdEДZ2dFdGДZ3dHdIДZ4dJdKДZ5dLdMДZ6dNdOДZ7dPdQДZ8dRdSДZ9dTdUДZ:dVdWДZ;dXdYДZd^d_ДZ?d`daДZ@dbdcДZAdddeДZBdfdgДZCdhdiДZDdjdkДZEdldmДZFdndoДZGdpdqДZHdrdsДZIdtduДZJdvdwДZKdxdyДZLdzd{ДZMd|d}ДZNd~dДZOdАdБДZPdВdГДZQdДS)ЕzEArithmetics for dense recursive polynomials in ``K[x]`` or ``K[X]``. щ) ┌ dup_slice┌dup_LC┌dmp_LC┌ dup_degree┌ dmp_degree┌ dup_strip┌ dmp_strip┌ dmp_zero_p┌dmp_zero┌ dmp_one_p┌dmp_one┌ dmp_ground┌ dmp_zeros)┌ExactQuotientFailed┌PolynomialDivisionFailedcCsЦ|s|St|Г}||d}||dkrFt|d|g|ddЕГS||krh|g|jg|||S|d|Е|||g||ddЕSdS)z╙ Add ``c*x**i`` to ``f`` in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_add_term(x**2 - 1, ZZ(2), 4) 2*x**4 + x**2 - 1 щrNй┌lenr┌zeroй┌f┌c┌i┌K┌n┌mйr·U/var/www/html/Darija-Ai-API/env/lib/python3.8/site-packages/sympy/polys/densearith.py┌dup_add_termsrcCs─|st||||ГS|d}t||Гr(|St|Г}||d}||dkrntt|d|||Гg|ddЕ|ГS||krР|gt||||Г|S|d|Еt|||||Гg||ddЕSdS)z▌ Add ``c(x_2..x_u)*x_0**i`` to ``f`` in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_add_term(x*y + 1, 2, 2) 2*x**2 + x*y + 1 rrN)rr rr┌dmp_addrйrrr┌ur┌vrrrrr┌dmp_add_term+s &r#cCsШ|s|St|Г}||d}||dkrFt|d|g|ddЕГS||krj|g|jg|||S|d|Е|||g||ddЕSdS)z┌ Subtract ``c*x**i`` from ``f`` in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_sub_term(2*x**4 + x**2 - 1, ZZ(2), 4) x**2 - 1 rrNrrrrr┌dup_sub_termMsr$cCs╬|st||||ГS|d}t||Гr*|St|Г}||d}||dkrptt|d|||Гg|ddЕ|ГS||krЪt|||Гgt||||Г|S|d|Еt|||||Гg||ddЕSdS)zф Subtract ``c(x_2..x_u)*x_0**i`` from ``f`` in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_sub_term(2*x**2 + x*y + 1, 2, 2) x*y + 1 rrN)rr rr┌dmp_sub┌dmp_negrr rrr┌dmp_sub_termjs &"r'cs.Иr|sgSЗfddД|DГ|jg|SdS)z╓ Multiply ``f`` by ``c*x**i`` in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_mul_term(x**2 - 1, ZZ(3), 2) 3*x**4 - 3*x**2 csg|]}|ИСqSrrй┌.0┌cfйrrr┌ Эsz dup_mul_term..Nйr)rrrrrr+r┌dup_mul_termМsr.cs`|st|И|ИГS|dЙt||Гr(|StИИГr:t|ГSЗЗЗfddД|DГt|ИИГSdS)zэ Multiply ``f`` by ``c(x_2..x_u)*x_0**i`` in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_mul_term(x**2*y + x, 3*y, 2) 3*x**4*y**2 + 3*x**3*y rcsg|]}t|ИИИГСqSr)┌dmp_mulr(йrrr"rrr,╕sz dmp_mul_term..N)r.r r r)rrrr!rrr0r┌dmp_mul_termаs r1cCst||d|ГS)zЁ Add an element of the ground domain to ``f``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_add_ground(x**3 + 2*x**2 + 3*x + 4, ZZ(4)) x**3 + 2*x**2 + 3*x + 8 r)rйrrrrrr┌dup_add_ground╗sr3cCst|t||dГd||ГS)zЇ Add an element of the ground domain to ``f``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_add_ground(x**3 + 2*x**2 + 3*x + 4, ZZ(4)) x**3 + 2*x**2 + 3*x + 8 rr)r#r йrrr!rrrr┌dmp_add_ground╠sr5cCst||d|ГS)zє Subtract an element of the ground domain from ``f``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_sub_ground(x**3 + 2*x**2 + 3*x + 4, ZZ(4)) x**3 + 2*x**2 + 3*x r)r$r2rrr┌dup_sub_ground▌sr6cCst|t||dГd||ГS)zў Subtract an element of the ground domain from ``f``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_sub_ground(x**3 + 2*x**2 + 3*x + 4, ZZ(4)) x**3 + 2*x**2 + 3*x rr)r'r r4rrr┌dmp_sub_groundюsr7cs"Иr|sgSЗfddД|DГSdS)zт Multiply ``f`` by a constant value in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_mul_ground(x**2 + 2*x - 1, ZZ(3)) 3*x**2 + 6*x - 3 csg|]}|ИСqSrrr(r+rrr,sz"dup_mul_ground..Nrr2rr+r┌dup_mul_ground sr8cs.|st|ИИГS|dЙЗЗЗfddД|DГS)z┌ Multiply ``f`` by a constant value in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_mul_ground(2*x + 2*y, ZZ(3)) 6*x + 6*y rcsg|]}t|ИИИГСqSr)┌dmp_mul_groundr(r0rrr,&sz"dmp_mul_ground..)r8r4rr0rr9sr9csDИstdГВ|s|SИjr.ЗЗfddД|DГSЗfddД|DГSdS)a) Quotient by a constant in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ, QQ >>> R, x = ring("x", ZZ) >>> R.dup_quo_ground(3*x**2 + 2, ZZ(2)) x**2 + 1 >>> R, x = ring("x", QQ) >>> R.dup_quo_ground(3*x**2 + 2, QQ(2)) 3/2*x**2 + 1 ·polynomial divisioncsg|]}Иа|ИбСqSr)Zquor(йrrrrr,Asz"dup_quo_ground..csg|]}|ИСqSrrr(r+rrr,CsN)┌ZeroDivisionError┌is_Fieldr2rr;r┌dup_quo_ground)sr>cs.|st|ИИГS|dЙЗЗЗfddД|DГS)a= Quotient by a constant in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ, QQ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_quo_ground(2*x**2*y + 3*x, ZZ(2)) x**2*y + x >>> R, x,y = ring("x,y", QQ) >>> R.dmp_quo_ground(2*x**2*y + 3*x, QQ(2)) x**2*y + 3/2*x rcsg|]}t|ИИИГСqSr)┌dmp_quo_groundr(r0rrr,]sz"dmp_quo_ground..)r>r4rr0rr?Fsr?cs(ИstdГВ|s|SЗЗfddД|DГS)z╘ Exact quotient by a constant in ``K[x]``. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x = ring("x", QQ) >>> R.dup_exquo_ground(x**2 + 2, QQ(2)) 1/2*x**2 + 1 r:csg|]}Иа|ИбСqSr)┌exquor(r;rrr,ssz$dup_exquo_ground..)r<r2rr;r┌dup_exquo_ground`s rAcs.|st|ИИГS|dЙЗЗЗfddД|DГS)z▐ Exact quotient by a constant in ``K[X]``. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x,y = ring("x,y", QQ) >>> R.dmp_exquo_ground(x**2*y + 2*x, QQ(2)) 1/2*x**2*y + x rcsg|]}t|ИИИГСqSr)┌dmp_exquo_groundr(r0rrr,Йsz$dmp_exquo_ground..)rAr4rr0rrBvsrBcCs|s|S||jg|SdS)z╙ Efficiently multiply ``f`` by ``x**n`` in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_lshift(x**2 + 1, 2) x**4 + x**2 Nr-йrrrrrr┌ dup_lshiftМsrDcCs|d|ЕS)a Efficiently divide ``f`` by ``x**n`` in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_rshift(x**4 + x**2, 2) x**2 + 1 >>> R.dup_rshift(x**4 + x**2 + 2, 2) x**2 + 1 NrrCrrr┌ dup_rshiftаsrEcsЗfddД|DГS)z┬ Make all coefficients positive in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_abs(x**2 - 1) x**2 + 1 csg|]}Иа|бСqSr)┌absйr)┌coeffйrrrr,┴szdup_abs..rйrrrrIr┌dup_abs│srKcs*|st|ИГS|dЙЗЗfddД|DГS)z╩ Make all coefficients positive in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_abs(x**2*y - x) x**2*y + x rcsg|]}t|ИИГСqSr)┌dmp_absr(йrr"rrr,╫szdmp_abs..)rKйrr!rrrMrrL─s rLcCsddД|DГS)z╕ Negate a polynomial in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_neg(x**2 - 1) -x**2 + 1 cSsg|] }|СqSrrrGrrrr,шszdup_neg..rrJrrr┌dup_neg┌srOcs*|st|ИГS|dЙЗЗfddД|DГS)z└ Negate a polynomial in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_neg(x**2*y - x) -x**2*y + x rcsg|]}t|ИИГСqSr)r&r(rMrrr,■szdmp_neg..)rOrNrrMrr&ыs r&cCsж|s|S|s|St|Г}t|Г}||kr@tddДt||ГDГГSt||Г}||krp|d|Е||dЕ}}n|d|Е||dЕ}}|ddДt||ГDГSdS)z─ Add dense polynomials in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_add(x**2 - 1, x - 2) x**2 + x - 3 cSsg|]\}}||СqSrrйr)┌a┌brrrr,szdup_add..NcSsg|]\}}||СqSrrrPrrrr,!s)rr┌ziprFйr┌gr┌df┌dg┌k┌hrrr┌dup_addsrZcs╪|st||ИГSt||Г}|dkr&|St||Г}|dkr<|S|dЙ||krltЗЗfddДt||ГDГ|ГSt||Г}||krЬ|d|Е||dЕ}}n|d|Е||dЕ}}|ЗЗfddДt||ГDГSdS)z╓ Add dense polynomials in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_add(x**2 + y, x**2*y + x) x**2*y + x**2 + x + y rrcsg|]\}}t||ИИГСqSrйrrPrMrrr,Bszdmp_add..Ncsg|]\}}t||ИИГСqSrr[rPrMrrr,Ks)rZrrrSrFйrrUr!rrVrWrXrYrrMrr$s rcCs▓|st||ГS|s|St|Г}t|Г}||krFtddДt||ГDГГSt||Г}||krv|d|Е||dЕ}}n t|d|Е|Г||dЕ}}|ddДt||ГDГSdS)z╔ Subtract dense polynomials in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_sub(x**2 - 1, x - 2) x**2 - x + 1 cSsg|]\}}||СqSrrrPrrrr,eszdup_sub..NcSsg|]\}}||СqSrrrPrrrr,ns)rOrrrSrFrTrrr┌dup_subNs r]csш|st||ИГSt||Г}|dkr.t||ИГSt||Г}|dkrD|S|dЙ||krttЗЗfddДt||ГDГ|ГSt||Г}||krд|d|Е||dЕ}}n"t|d|Е|ИГ||dЕ}}|ЗЗfddДt||ГDГSdS)z▄ Subtract dense polynomials in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_sub(x**2 + y, x**2*y + x) -x**2*y + x**2 - x + y rrcsg|]\}}t||ИИГСqSrйr%rPrMrrr,Пszdmp_sub..Ncsg|]\}}t||ИИГСqSrr^rPrMrrr,Шs)r]rr&rrSrFr\rrMrr%qs "r%cCst|t|||Г|ГS)zс Returns ``f + g*h`` where ``f, g, h`` are in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_add_mul(x**2 - 1, x - 2, x + 2) 2*x**2 - 5 )rZ┌dup_mulйrrUrYrrrr┌dup_add_mulЫsracCst|t||||Г||ГS)zч Returns ``f + g*h`` where ``f, g, h`` are in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_add_mul(x**2 + y, x, x + 2) 2*x**2 + 2*x + y )rr/йrrUrYr!rrrr┌dmp_add_mulмsrccCst|t|||Г|ГS)z╪ Returns ``f - g*h`` where ``f, g, h`` are in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_sub_mul(x**2 - 1, x - 2, x + 2) 3 )r]r_r`rrr┌dup_sub_mul╜srdcCst|t||||Г||ГS)z▀ Returns ``f - g*h`` where ``f, g, h`` are in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_sub_mul(x**2 + y, x, x + 2) -2*x + y )r%r/rbrrr┌dmp_sub_mul╬srecCs|||krt||ГS|r|sgSt|Г}t|Г}t||Гd}|dkr┤g}td||dГD]P}|j}ttd||Гt||ГdГD]} ||| ||| 7}qВ|а|бqZt|ГS|d} t|d| |Гt|d| |Г}}t t|| ||Г| |Г} t t|| ||Г| |Г}t |||Гt | ||Г}}t t|| |Гt|||Г|Г}t|t|||Г|Г}tt|t || |Г|Гt |d| |Г|ГSdS)z┬ Multiply dense polynomials in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_mul(x - 2, x + 2) x**2 - 4 rщdrщN)┌dup_sqrr┌max┌ranger┌min┌appendrrrEr_rZr]rD)rrUrrVrWrrYrrH┌jZn2┌fl┌gl┌fh┌gh┌lo┌hi┌midrrrr_▀s4 " r_c Cs┌|st|||ГS||kr$t|||ГSt||Г}|dkr:|St||Г}|dkrP|Sg|d}}td||dГD]^}t|Г} ttd||Гt||ГdГD](} t| t|| ||| ||Г||Г} qЪ|а | бqpt ||ГS)z╞ Multiply dense polynomials in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_mul(x*y + 1, x) x**2*y + x rr)r_┌dmp_sqrrrjr rirkrr/rlr)rrUr!rrVrWrYr"rrHrmrrrr/s" "&r/cCs╩t|Гdg}}tdd|dГD]Ь}|j}td||Г}t||Г}||d}||dd}t||dГD]} ||| ||| 7}qp||7}|d@r╢||d} || d7}|а|бq$t|ГS)z┼ Square dense polynomials in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_sqr(x**2 + 1) x**4 + 2*x**2 + 1 rrrg)rrjrrirkrlr)rrrVrYrr┌jmin┌jmaxrrm┌elemrrrrhCs rhc Cs|st||ГSt||Г}|dkr$|Sg|d}}tdd|dГD]╛}t|Г}td||Г}t||Г} | |d} || dd} t|| dГD](}t|t|||||||Г||Г}qТt||dГ||Г}| d@r°t || d||Г}t||||Г}|а |бqDt||ГS)zЁ Square dense polynomials in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_sqr(x**2 + x*y + y**2) x**4 + 2*x**3*y + 3*x**2*y**2 + 2*x*y**3 + y**4 rrrg)rhrrjr rirkrr/r9rurlr) rr!rrVrYr"rrrvrwrrmrxrrrruks( &rucCsx|s|jgS|dkrtdГВ|dks4|r4||jgkr8|S|jg}|d|}}|drht|||Г}|shqtt||Г}q@|S)z╒ Raise ``f`` to the ``n``-th power in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_pow(x - 2, 3) x**3 - 6*x**2 + 12*x - 8 r·+Cannot raise polynomial to a negative powerrrg)┌one┌ ValueErrorr_rh)rrrrUrrrr┌dup_powЫsr|cCsЦ|st|||ГS|st||ГS|dkr.tdГВ|dksLt||ГsLt|||ГrP|St||Г}|d|}}|d@rДt||||Г}|sДqТt|||Г}qZ|S)zц Raise ``f`` to the ``n``-th power in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_pow(x*y + 1, 3) x**3*y**3 + 3*x**2*y**2 + 3*x*y + 1 rryrrg)r|rr{r rr/ru)rrr!rrUrrrr┌dmp_pow└s r}cCst|Г}t|Г}g||}}}|s.tdГВn||kr>||fS||d}t||Г} t||Г} |||d}}t|| |Г}t|| ||Г}t|| |Г} t|| ||Г}t| ||Г}|t|Г}}||kr╩qрqT||ksTt|||ГВqT| |}t|||Г}t|||Г}||fS)z═ Polynomial pseudo-division in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_pdiv(x**2 + 1, 2*x - 4) (2*x + 4, 20) r:r)rr<rr8rr.r]r)rrUrrVrW┌q┌r┌dr┌N┌lc_g┌lc_rrm┌Q┌R┌G┌_drrrrr┌dup_pdivшs2 rИcCs╠t|Г}t|Г}||}}|s(tdГВn||kr4|S||d}t||Г}t||Г} |||d} }t|||Г}t|| | |Г}t|||Г}|t|Г} }||krжq╝qJ|| ksJt|||ГВqJt||||ГS)z├ Polynomial pseudo-remainder in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_prem(x**2 + 1, 2*x - 4) 20 r:r)rr<rr8r.r]r)rrUrrVrWrrАrБrВrГrmrЕrЖrЗrrr┌dup_prems( rЙcCst|||ГdS)a Polynomial exact pseudo-quotient in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_pquo(x**2 - 1, 2*x - 2) 2*x + 2 >>> R.dup_pquo(x**2 + 1, 2*x - 4) 2*x + 4 r)rИйrrUrrrr┌dup_pquoJsrЛcCs&t|||Г\}}|s|St||ГВdS)a\ Polynomial pseudo-quotient in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_pexquo(x**2 - 1, 2*x - 2) 2*x + 2 >>> R.dup_pexquo(x**2 + 1, 2*x - 4) Traceback (most recent call last): ... ExactQuotientFailed: [2, -4] does not divide [1, 0, 1] N)rИrйrrUrr~rrrr┌ dup_pexquo^srНcCsF|st|||ГSt||Г}t||Г}|dkr4tdГВt|Г||}}}||krX||fS||d} t||Г} t||Г}||| d}} t|| d||Г} t| ||||Г}t|| d||Г}t|||||Г}t||||Г}|t||Г}}||krЎРqqn||ksnt|||ГВqnt | | |d|Г}t||d||Г}t||d||Г}||fS)z▀ Polynomial pseudo-division in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_pdiv(x**2 + x*y, 2*x + 2) (2*x + 2*y - 2, -4*y + 4) rr:r) rИrr<r rr1r#r%rr})rrUr!rrVrWr~rrАrБrВrГrmrДrЕrЖrЗrrrr┌dmp_pdivys6 rОcCs■|st|||ГSt||Г}t||Г}|dkr4tdГВ||}}||krJ|S||d}t||Г} t||Г} |||d}}t|| d||Г}t|| |||Г} t|| ||Г}|t||Г}}||kr╞q▄q`||ks`t|||ГВq`t| ||d|Г}t||d||ГS)z╧ Polynomial pseudo-remainder in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_prem(x**2 + x*y, 2*x + 2) -4*y + 4 rr:r)rЙrr<rr1r%rr})rrUr!rrVrWrrАrБrВrГrmrЕrЖrЗrrrr┌dmp_prem▓s. rПcCst||||ГdS)a. Polynomial exact pseudo-quotient in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = x**2 + x*y >>> g = 2*x + 2*y >>> h = 2*x + 2 >>> R.dmp_pquo(f, g) 2*x >>> R.dmp_pquo(f, h) 2*x + 2*y - 2 r)rОйrrUr!rrrr┌dmp_pquoхsrСcCs.t||||Г\}}t||Гr |St||ГВdS)aП Polynomial pseudo-quotient in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = x**2 + x*y >>> g = 2*x + 2*y >>> h = 2*x + 2 >>> R.dmp_pexquo(f, g) 2*x >>> R.dmp_pexquo(f, h) Traceback (most recent call last): ... ExactQuotientFailed: [[2], [2]] does not divide [[1], [1, 0], []] N)rОr rйrrUr!rr~rrrr┌ dmp_pexquo¤s rУcCs╨t|Г}t|Г}g||}}}|s.tdГВn||kr>||fSt||Г}t||Г} | |r\q╚|а| |б} ||}t|| ||Г}t|| ||Г}t|||Г}|t|Г} }||kr▓q╚qH|| ksHt|||ГВqH||fS)z╫ Univariate division with remainder over a ring. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_rr_div(x**2 + 1, 2*x - 4) (0, x**2 + 1) r:)rr<rr@rr.r]rйrrUrrVrWr~rrАrВrГrrmrYrЗrrr┌ dup_rr_divs, rХcCs|st|||ГSt||Г}t||Г}|dkr4tdГВt|Г||}}}||krX||fSt||Г|d} } t||Г}t|| | |Г\}} t| | ГsЦРq||}t|||||Г}t|||||Г}t ||||Г}|t||Г}}||krъРqql||kslt |||ГВql||fS)zс Multivariate division with remainder over a ring. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_rr_div(x**2 + x*y, 2*x + 2) (0, x**2 + x*y) rr:r)rХrr<r r┌ dmp_rr_divr r#r1r%rйrrUr!rrVrWr~rrАrВr"rГrrЕrmrYrЗrrrrЦMs0 rЦcCs°t|Г}t|Г}g||}}}|s.tdГВn||kr>||fSt||Г}t||Г} |а| |б} ||}t|| ||Г}t|| ||Г}t|||Г}|t|Г} }||krиqЁqH|| kr┌|js┌t|ddЕГ}t|Г}||krюqЁqH|| ksHt |||ГВqH||fS)z┘ Polynomial division with remainder over a field. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x = ring("x", QQ) >>> R.dup_ff_div(x**2 + 1, 2*x - 4) (1/2*x + 1, 5) r:rN) rr<rr@rr.r]Zis_ExactrrrФrrr┌ dup_ff_divВs2 rШcCs|st|||ГSt||Г}t||Г}|dkr4tdГВt|Г||}}}||krX||fSt||Г|d} } t||Г}t|| | |Г\}} t| | ГsЦРq||}t|||||Г}t|||||Г}t ||||Г}|t||Г}}||krъРqql||kslt |||ГВql||fS)zю Polynomial division with remainder over a field. Examples ======== >>> from sympy.polys import ring, QQ >>> R, x,y = ring("x,y", QQ) >>> R.dmp_ff_div(x**2 + x*y, 2*x + 2) (1/2*x + 1/2*y - 1/2, -y + 1) rr:r)rШrr<r r┌ dmp_ff_divr r#r1r%rrЧrrrrЩ╢s0 rЩcCs"|jrt|||ГSt|||ГSdS)a. Polynomial division with remainder in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ, QQ >>> R, x = ring("x", ZZ) >>> R.dup_div(x**2 + 1, 2*x - 4) (0, x**2 + 1) >>> R, x = ring("x", QQ) >>> R.dup_div(x**2 + 1, 2*x - 4) (1/2*x + 1, 5) N)r=rШrХrКrrr┌dup_divыsrЪcCst|||ГdS)a Returns polynomial remainder in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ, QQ >>> R, x = ring("x", ZZ) >>> R.dup_rem(x**2 + 1, 2*x - 4) x**2 + 1 >>> R, x = ring("x", QQ) >>> R.dup_rem(x**2 + 1, 2*x - 4) 5 rйrЪrКrrr┌dup_remsrЬcCst|||ГdS)a Returns exact polynomial quotient in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ, QQ >>> R, x = ring("x", ZZ) >>> R.dup_quo(x**2 + 1, 2*x - 4) 0 >>> R, x = ring("x", QQ) >>> R.dup_quo(x**2 + 1, 2*x - 4) 1/2*x + 1 rrЫrКrrr┌dup_quosrЭcCs&t|||Г\}}|s|St||ГВdS)aW Returns polynomial quotient in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_exquo(x**2 - 1, x - 1) x + 1 >>> R.dup_exquo(x**2 + 1, 2*x - 4) Traceback (most recent call last): ... ExactQuotientFailed: [2, -4] does not divide [1, 0, 1] N)rЪrrМrrr┌ dup_exquo-srЮcCs&|jrt||||ГSt||||ГSdS)aK Polynomial division with remainder in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ, QQ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_div(x**2 + x*y, 2*x + 2) (0, x**2 + x*y) >>> R, x,y = ring("x,y", QQ) >>> R.dmp_div(x**2 + x*y, 2*x + 2) (1/2*x + 1/2*y - 1/2, -y + 1) N)r=rЩrЦrРrrr┌dmp_divHsrЯcCst||||ГdS)a) Returns polynomial remainder in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ, QQ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_rem(x**2 + x*y, 2*x + 2) x**2 + x*y >>> R, x,y = ring("x,y", QQ) >>> R.dmp_rem(x**2 + x*y, 2*x + 2) -y + 1 rйrЯrРrrr┌dmp_rem`srбcCst||||ГdS)a2 Returns exact polynomial quotient in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ, QQ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_quo(x**2 + x*y, 2*x + 2) 0 >>> R, x,y = ring("x,y", QQ) >>> R.dmp_quo(x**2 + x*y, 2*x + 2) 1/2*x + 1/2*y - 1/2 rrаrРrrr┌dmp_quousrвcCs.t||||Г\}}t||Гr |St||ГВdS)aИ Returns polynomial quotient in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = x**2 + x*y >>> g = x + y >>> h = 2*x + 2 >>> R.dmp_exquo(f, g) x >>> R.dmp_exquo(f, h) Traceback (most recent call last): ... ExactQuotientFailed: [[2], [2]] does not divide [[1], [1, 0], []] N)rЯr rrТrrr┌ dmp_exquoКs rгcCs|s |jStt||ГГSdS)z═ Returns maximum norm of a polynomial in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_max_norm(-x**2 + 2*x - 3) 3 N)rrirKrJrrr┌dup_max_normйsrдcs.|st|ИГS|dЙtЗЗfddД|DГГS)z╧ Returns maximum norm of a polynomial in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_max_norm(2*x*y - x - 3) 3 rcsg|]}t|ИИГСqSr)┌dmp_max_normйr)rrMrrr,╨sz dmp_max_norm..)rдrirNrrMrrе╜s rеcCs|s |jStt||ГГSdS)z╦ Returns l1 norm of a polynomial in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_l1_norm(2*x**3 - 3*x**2 + 1) 6 N)r┌sumrKrJrrr┌dup_l1_norm╙srиcs.|st|ИГS|dЙtЗЗfddД|DГГS)z╔ Returns l1 norm of a polynomial in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_l1_norm(2*x*y - x - 3) 6 rcsg|]}t|ИИГСqSr)┌dmp_l1_normrжrMrrr,·szdmp_l1_norm..)rиrзrNrrMrrйчs rйcCstddД|DГ|jГS)z▄ Returns squared l2 norm of a polynomial in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_l2_norm_squared(2*x**3 - 3*x**2 + 1) 14 cSsg|]}|dСqS)rgrrGrrrr,sz'dup_l2_norm_squared..)rзrrJrrr┌dup_l2_norm_squared¤srкcs.|st|ИГS|dЙtЗЗfddД|DГГS)z┌ Returns squared l2 norm of a polynomial in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_l2_norm_squared(2*x*y - x - 3) 14 rcsg|]}t|ИИГСqSr)┌dmp_l2_norm_squaredrжrMrrr,!sz'dmp_l2_norm_squared..)rкrзrNrrMrrлs rлcCs6|s|jgS|d}|ddЕD]}t|||Г}q |S)z╪ Multiply together several polynomials in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_expand([x**2 - 1, x, 2]) 2*x**3 - 2*x rrN)rzr_)┌polysrrrUrrr┌ dup_expand$srнcCs:|st||ГS|d}|ddЕD]}t||||Г}q"|S)zя Multiply together several polynomials in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_expand([x**2 + y**2, x + 1]) x**3 + x**2 + x*y**2 + y**2 rrN)rr/)rмr!rrrUrrr┌ dmp_expand=s rоN)R┌__doc__Zsympy.polys.densebasicrrrrrrrr r rrr rZsympy.polys.polyerrorsrrrr#r$r'r.r1r3r5r6r7r8r9r>r?rArBrDrErKrLrOr&rZrr]r%rarcrdrer_r/rhrur|r}rИrЙrЛrНrОrПrСrУrХrЦrШrЩrЪrЬrЭrЮrЯrбrвrгrдrеrиrйrкrлrнrоrrrr┌sД<""#*#*9+(0%(5-931545