+ iHRt^RIHtHt^RIHt^RIHt^RIH t H t H t H t ^RI Ht^RIHtHt^RIHt^RIHtHt]RR l4tR tR tR tRtRtRtRtRtRt Rt!Rt"Rt#!RR4t$]!RR]44t%R #)z@Tools and arithmetics for monomials of distributed polynomials. )combinations_with_replacementproduct)dedent)cacheit)MulSTuplesympify)ExactQuotientFailed)PicklableWithSlotsdict_from_expr)public) is_sequenceiterableNc#aa"\S4'Ed~\V4p\S4V8wd \R4hSf ^.V,oM|\S4'g \R4h\S4V8wd \R4h\;QJdRS4F 'gK RM RM !RS44'd \R4h\;QJd)VV3Rl\ V44F 'gK RM RM!VV3Rl\ V444'd \R 4h.p\ VSS4F=wrVpTP \ Wg^,4Uu.uF qV,NK up4K? \V!Fp \V !xK R#Sp V ^8d \R 4hSf^p MS^8d \R 4hSp W8dR#V'dV ^8Xd\PxR#\V4\P.,p\;QJdR V4F 'dK RM RM !R V44'd \W 4p M \W R 7p \4p W, pV FBp^pVFpV^8XgK V^, pVV8gKK( V P\V!4KD T RjxL R#uupiL 5i)a ``max_degrees`` and ``min_degrees`` are either both integers or both lists. Unless otherwise specified, ``min_degrees`` is either ``0`` or ``[0, ..., 0]``. A generator of all monomials ``monom`` is returned, such that either ``min_degree <= total_degree(monom) <= max_degree``, or ``min_degrees[i] <= degree_list(monom)[i] <= max_degrees[i]``, for all ``i``. Case I. ``max_degrees`` and ``min_degrees`` are both integers ============================================================= Given a set of variables $V$ and a min_degree $N$ and a max_degree $M$ generate a set of monomials of degree less than or equal to $N$ and greater than or equal to $M$. The total number of monomials in commutative variables is huge and is given by the following formula if $M = 0$: .. math:: \frac{(\#V + N)!}{\#V! N!} For example if we would like to generate a dense polynomial of a total degree $N = 50$ and $M = 0$, which is the worst case, in 5 variables, assuming that exponents and all of coefficients are 32-bit long and stored in an array we would need almost 80 GiB of memory! Fortunately most polynomials, that we will encounter, are sparse. Consider monomials in commutative variables $x$ and $y$ and non-commutative variables $a$ and $b$:: >>> from sympy import symbols >>> from sympy.polys.monomials import itermonomials >>> from sympy.polys.orderings import monomial_key >>> from sympy.abc import x, y >>> sorted(itermonomials([x, y], 2), key=monomial_key('grlex', [y, x])) [1, x, y, x**2, x*y, y**2] >>> sorted(itermonomials([x, y], 3), key=monomial_key('grlex', [y, x])) [1, x, y, x**2, x*y, y**2, x**3, x**2*y, x*y**2, y**3] >>> a, b = symbols('a, b', commutative=False) >>> set(itermonomials([a, b, x], 2)) {1, a, a**2, b, b**2, x, x**2, a*b, b*a, x*a, x*b} >>> sorted(itermonomials([x, y], 2, 1), key=monomial_key('grlex', [y, x])) [x, y, x**2, x*y, y**2] Case II. ``max_degrees`` and ``min_degrees`` are both lists =========================================================== If ``max_degrees = [d_1, ..., d_n]`` and ``min_degrees = [e_1, ..., e_n]``, the number of monomials generated is: .. math:: (d_1 - e_1 + 1) (d_2 - e_2 + 1) \cdots (d_n - e_n + 1) Let us generate all monomials ``monom`` in variables $x$ and $y$ such that ``[1, 2][i] <= degree_list(monom)[i] <= [2, 4][i]``, ``i = 0, 1`` :: >>> from sympy import symbols >>> from sympy.polys.monomials import itermonomials >>> from sympy.polys.orderings import monomial_key >>> from sympy.abc import x, y >>> sorted(itermonomials([x, y], [2, 4], [1, 2]), reverse=True, key=monomial_key('lex', [x, y])) [x**2*y**4, x**2*y**3, x**2*y**2, x*y**4, x*y**3, x*y**2] zArgument sizes do not matchNzmin_degrees is not a listc3*"TF q^8xK R#5iN).0is& u/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/polys/monomials.py itermonomials..cs.+Qq5+TFz+min_degrees cannot contain negative numbersc3J<"TFpSV,SV,8xK R#5iNr)rr max_degrees min_degreess& rrresA1{1~ A.s #z2min_degrees[i] must be <= max_degrees[i] for all izmax_degrees cannot be negativezmin_degrees cannot be negativec38"TFqPxK R#5ir)is_commutative)rvariables& rrr}sAy8&&ys)repeat)rlen ValueErroranyrangezipappendrrrOnelistallrsetadd) variablesrrn power_listsvarmin_dmax_drpowers max_degree min_degreeit monomials_setditemcountr!s&ff r itermonomialsr<sNT;  N { q :; ;  #a%K[))89 9;1$ !>??s.+.sss.+... !NOO 3AaA333AaA A AQR R !$Y [!I C   eQY0GH0G1Q0GH I"J{+Fv, ,! >=> >  JQ !ABB$J  " J!O%%K Oquug- 3AyA333AyA A A.yEB6B  #DE q=QJE5y ! !!#t*-!  G IF !sbB KK"K=KK4'K?KK ,A&KAK.K :K K&KK  Kc^^RIHpV!W,4V!V4, V!V4, #)a Computes the number of monomials. The number of monomials is given by the following formula: .. math:: \frac{(\#V + N)!}{\#V! N!} where `N` is a total degree and `V` is a set of variables. Examples ======== >>> from sympy.polys.monomials import itermonomials, monomial_count >>> from sympy.polys.orderings import monomial_key >>> from sympy.abc import x, y >>> monomial_count(2, 2) 6 >>> M = list(itermonomials([x, y], 2)) >>> sorted(M, key=monomial_key('grlex', [y, x])) [1, x, y, x**2, x*y, y**2] >>> len(M) 6 ) factorial)(sympy.functions.combinatorial.factorialsr>)VNr>s&& rmonomial_countrBs'<C QU il *Yq\ 99cj\\W4UUu.uF wr#W#,NK upp4#uuppi)a Multiplication of tuples representing monomials. Examples ======== Lets multiply `x**3*y**4*z` with `x*y**2`:: >>> from sympy.polys.monomials import monomial_mul >>> monomial_mul((3, 4, 1), (1, 2, 0)) (4, 6, 1) which gives `x**4*y**5*z`. tupler'ABabs&& r monomial_mulrLs+" SY0YTQ155Y0 110/ c\W4p\;QJdRV4F 'dK RM RM !RV44'd \V4#R#)al Division of tuples representing monomials. Examples ======== Lets divide `x**3*y**4*z` by `x*y**2`:: >>> from sympy.polys.monomials import monomial_div >>> monomial_div((3, 4, 1), (1, 2, 0)) (2, 2, 1) which gives `x**2*y**2*z`. However:: >>> monomial_div((3, 4, 1), (1, 2, 2)) is None True `x*y**2*z**2` does not divide `x**3*y**4*z`. c3*"TF q^8xK R#5irr)rcs& rrmonomial_div..s 1a61rFTN) monomial_ldivr+rF)rHrICs&& r monomial_divrTs;, aA s 1 sss 1 QxrCcj\\W4UUu.uF wr#W#, NK upp4#uuppi)aU Division of tuples representing monomials. Examples ======== Lets divide `x**3*y**4*z` by `x*y**2`:: >>> from sympy.polys.monomials import monomial_ldiv >>> monomial_ldiv((3, 4, 1), (1, 2, 0)) (2, 2, 1) which gives `x**2*y**2*z`. >>> monomial_ldiv((3, 4, 1), (1, 2, 2)) (2, 2, -1) which gives `x**2*y**2*z**-1`. rErGs&& rrRrRs+, SY0YTQ155Y0 110rMcN\VUu.uF q"V,NK up4#uupi)z%Return the n-th pow of the monomial. )rF)rHr/rJs&& r monomial_powrWs# #1Q33# $$#s"c p\\W4UUu.uFwr#\W#4NK upp4#uuppi)a  Greatest common divisor of tuples representing monomials. Examples ======== Lets compute GCD of `x*y**4*z` and `x**3*y**2`:: >>> from sympy.polys.monomials import monomial_gcd >>> monomial_gcd((1, 4, 1), (3, 2, 0)) (1, 2, 0) which gives `x*y**2`. )rFr'minrGs&& r monomial_gcdrZ+" Q43q94 5542 c p\\W4UUu.uFwr#\W#4NK upp4#uuppi)a  Least common multiple of tuples representing monomials. Examples ======== Lets compute LCM of `x*y**4*z` and `x**3*y**2`:: >>> from sympy.polys.monomials import monomial_lcm >>> monomial_lcm((1, 4, 1), (3, 2, 0)) (3, 4, 1) which gives `x**3*y**4*z`. )rFr'maxrGs&& r monomial_lcmr_r[r\c\;QJd%R\W44F 'dK R# R#!R\W444#)z Does there exist a monomial X such that XA == B? Examples ======== >>> from sympy.polys.monomials import monomial_divides >>> monomial_divides((1, 2), (3, 4)) True >>> monomial_divides((1, 2), (0, 2)) False c3."TF wrW8*xK R#5irr)rrJrKs& rr#monomial_divides...s,)$!qv)sFT)r+r')rHrIs&&rmonomial_dividesrc!s5 3,#a),33,3,3,#a), ,,rCc\V^,4pVR,F+p\V4Fwr4\W,V4W&K K- \V4#)ai Returns maximal degree for each variable in a set of monomials. Examples ======== Consider monomials `x**3*y**4*z**5`, `y**5*z` and `x**6*y**3*z**9`. We wish to find out what is the maximal degree for each of `x`, `y` and `z` variables:: >>> from sympy.polys.monomials import monomial_max >>> monomial_max((3,4,5), (0,5,1), (6,3,9)) (6, 5, 9) NN)r* enumerater^rFmonomsMrArr/s* r monomial_maxrk0K" VAYA BZZaLDAqtQ>> from sympy.polys.monomials import monomial_min >>> monomial_min((3,4,5), (0,5,1), (6,3,9)) (0, 3, 1) re)r*rgrYrFrhs* r monomial_minrnIrlrCc\V4#)z Returns the total degree of a monomial. Examples ======== The total degree of `xy^2` is 3: >>> from sympy.polys.monomials import monomial_deg >>> monomial_deg((1, 2)) 3 )sum)rjs&r monomial_degrqbs  q6MrCcVwr4VwrV\W54pVP'dVeWrPWF43#R#Ve!WF,'gWrPWF43#R#)z,Division of two terms in over a ring/field. N)rTis_Fieldquo)rJrKdomaina_lma_lcb_lmb_lcmonoms&&& rterm_divr{qs^JDJD  $E   **T00 0 **T00 0rCcaa]tRtRtoRt]V3Rl4tRtRtRt ]R4t ]R4t ]R 4t ]R 4t ]R 4t]R 4t]R 4tRtVtV;t#) MonomialOpsiz6Code generator of fast monomial arithmetic functions. c2<\SV`V4pWnV#r)super__new__ngens)clsrobj __class__s&& rrMonomialOps.__new__sgoc"  rCcVP3#r)rselfs&r__getnewargs__MonomialOps.__getnewargs__s }rCc,/p\W4W2,#r)exec)rcodenamenss&&& r_buildMonomialOps._builds  TxrCc`\VP4Uu.uF q!: V: 2NK up#uupir)r&r)rrrs&& r_varsMonomialOps._varss*-24::->@->4#->@@@s+c TRp\R4pVPR4pVPR4p\W44UUu.uFwrVV: RV: 2NK pppVRVRRPV4RRPV4R RPV4/,pVP W4#uuppi) rLs def %(name)s(A, B): (%(A)s,) = A (%(B)s,) = B return (%(AB)s,) rJrK + rrH, rIABrrr'joinr rrtemplaterHrIrJrKrrs & rmulMonomialOps.muls   JJsO JJsO.1!i 9idaAq!i 964diilC1tUYU^U^_aUbcc{{4&&:B$c Rp\R4pVPR4pVUu.uF pRV,NK ppVRVRRPV4RRPV4/,pVPWa4#uupi)rWzZ def %(name)s(A, k): (%(A)s,) = A return (%(Ak)s,) rJz%s*krrHrAk)rrrr)rrrrHrJrrs& rpowMonomialOps.powsz   JJsO#$ &1avzz1 &64diilD$))B-PP{{4&&'sA5c VRp\R4pVPR4pVPR4p\W44UUu.uFwrVV: RV: R2NK pppVRVRRPV4R RPV4R RPV4/,pVP W4#uuppi) monomial_mulpowzw def %(name)s(A, B, k): (%(A)s,) = A (%(B)s,) = B return (%(ABk)s,) rJrKrz*krrHrrIABkr) rrrrHrIrJrKrrs & rmulpowMonomialOps.mulpows    JJsO JJsO14Q<q!$<64diilC1uVZV_V_`cVdee{{4&&=sB%c TRp\R4pVPR4pVPR4p\W44UUu.uFwrVV: RV: 2NK pppVRVRRPV4RRPV4R RPV4/,pVP W4#uuppi) rRrrJrKz - rrHrrIrrrs & rldivMonomialOps.ldivs   JJsO JJsO.1!i 9idaAq!i 964diilC1tUYU^U^_aUbcc{{4&&:rc Rp\R4pVPR4pVPR4p\VP4Uu.uFpRRV/,NK ppVPR4pVRVRR P V4R R P V4R R P V4R R P V4/,pVP W4#uupi)rTz def %(name)s(A, B): (%(A)s,) = A (%(B)s,) = B %(RAB)s return (%(R)s,) rJrKz7r%(i)s = a%(i)s - b%(i)s if r%(i)s < 0: return NonerrrrHrrIRABz R)rrr&rrr) rrrrHrIrrrrs & rdivMonomialOps.divs   JJsO JJsO_deieoeo_pr_pZ[JcSTXUU_pr JJsO64diilC1uV^VcVcdgVhjmosoxoxyzo{||{{4&&ssCc dRp\R4pVPR4pVPR4p\W44UUu.uFwrVV: RV: RV: RV: 2NK pppVRVRR PV4R R PV4R R PV4/,pVP W4#uuppi) r_rrJrK if z >=  else rrHrrIrrrs & rlcmMonomialOps.lcm   JJsO JJsOCFq9 N9411aA69 N64diilC1tUYU^U^_aUbcc{{4&&OB,c dRp\R4pVPR4pVPR4p\W44UUu.uFwrVV: RV: RV: RV: 2NK pppVRVRR PV4R R PV4R R PV4/,pVP W4#uuppi) rZrrJrKrz <= rrrHrrIrrrs & rgcdMonomialOps.gcdrrr)__name__ __module__ __qualname____firstlineno____doc__rrrrrrrrrrrr__static_attributes____classdictcell__ __classcell__)r __classdict__s@@rr}r}s@    A  '  '  '  '  '  '  '  ' ' '   '  '  '  ' 'rCr}ca]tRtRtoRtRtRRltRRltRtRt Rt R t R t R t R tR tRtRt]tRtRtRtRtVtR#)Monomializ9Class representing a monomial, i.e. a product of powers. Nc\V4'g\\V4VR7wr2\V4^8XdG\ VP 44^,^8Xd"\ VP 44^,pM\RPV44h\\\V44Vn W n R#))genszExpected a monomial got {}N)rr r r#r*valueskeysr$formatrFmapint exponentsr)rrzrreps&&& r__init__Monomial.__init__s&wu~DAIC3x1}cjjl!3A!6!!;SXXZ(+ !=!D!DU!KLLs3/ rCcLTPY;'g VP4#r)rr)rrrs&&&rrebuildMonomial.rebuilds~~i):):;;rCc,\VP4#r)r#rrs&r__len__Monomial.__len__s4>>""rCc,\VP4#r)iterrrs&r__iter__Monomial.__iter__sDNN##rCc(VPV,#r)r)rr:s&&r __getitem__Monomial.__getitem__s~~d##rCcn\VPPVPVP34#r)hashrrrrrs&r__hash__Monomial.__hash__s&T^^,,dnndiiHIIrCc VP'dKRP\VPVP4UUu.uFwrV: RV: 2NK upp4#VPP : RVP: R2#uuppi)*z**())rrr'rrr)rgenexps& r__str__Monomial.__str__ sc 99988C SWSaSaDbdDb#s3Dbde e#~~66G GesB c T;'g VPpV'g\RV,4h\\WP4UUu.uF wr#W#,NK upp!#uuppi)z3Convert a monomial instance to a SymPy expression. z5Cannot convert %s to an expression without generators)rr$rr'r)rrrrs&* ras_exprMonomial.as_expr&s]  tyyG$NP Ps4/HJ/H83chh/HJKKJsA' c\V\4'dVPpM!\V\\34'dTpMR#VPV8H#)F) isinstancerrrFrrotherrs&& r__eq__Monomial.__eq__0s@ eX & &I u~ . .I~~**rCc W8X*#rr)rrs&&r__ne__Monomial.__ne__:s   rCc\V\4'dVPpM%\V\\34'dTpM\ hVP \VPV44#r)rrrrFrNotImplementedErrorrrLrs&& r__mul__Monomial.__mul__=sN eX & &I u~ . .I% %||LCDDrCc\V\4'dVPpM%\V\\34'dTpM\ h\ VPV4pVeVPV4#\V\V44hr) rrrrFrrrTrr )rrrresults&& r __truediv__Monomial.__truediv__Gsj eX & &I u~ . .I% %dnni8  <<' '%dHUO< rsF=$--6D";}!}!~:B2&:20%6&6& -22 ${'{'zsE!sEsErC