wgIdZddlmZmZddlmZddlmZmZm Z m Z ddl m Z ddl mZmZddlmZddlmZmZedd Zd Zd Zd ZdZdZdZdZdZdZdZdZ dZ!GddZ"eGddeZ#y )z@Tools and arithmetics for monomials of distributed polynomials. )combinations_with_replacementproduct)dedent)MulSTuplesympify)ExactQuotientFailed)PicklableWithSlotsdict_from_expr)public) is_sequenceiterableNc#hKt|}trqt|k7r tddg|znLts tdt|k7r tdtdDr tdd}n+}|dkr tdd}ndkr td }d }|rfkDry|r|dk(rtj yt |tj gz}td |Drg}t||D]f}tj|d} |D]} | d k7s | | xxd z cc<t| j|k\sP|jt|ht|Ed{yg} t!|| D]f}tj|d} |D]} | d k7s | | xxd z cc<t| j|k\sP| jt|ht| Ed{ytfdt#|Dr tdg} t%|D]5\} }}| jt#||d zDcgc]}| |z c}7t!| D] } t| y77cc}ww)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 matchNrzmin_degrees is not a listc3&K|] }|dk ywrN).0is Z/home/mcse/projects/flask/flask-venv/lib/python3.12/site-packages/sympy/polys/monomials.py z itermonomials..bs.Q1q5.z+min_degrees cannot contain negative numbersFzmax_degrees cannot be negativezmin_degrees cannot be negativeTc34K|]}|jywN)is_commutative)rvariables rrz itermonomials..xsA8x&&As)repeatc34K|]}||kDywrr)rr max_degrees min_degreess rrz itermonomials..sA1{1~ A.Asz2min_degrees[i] must be <= max_degrees[i] for all i)lenr ValueErroranyrOnelistallrdictfromkeyssumvaluesappendrsetrrangezip) variablesr r!n total_degree max_degree min_degreemonomials_list_commitempowersrmonomials_list_non_comm power_listsvarmin_dmax_drs `` r itermonomialsr= sT IA; { q :; ;  #a%K[)89 9;1$ !>??.+.. !NOO  >=> >  JQ !ABB$J   " J!O%%K Oquug- AyA A"$ 5iL ;y!4 $.H1}x(A-(.v}}':5'..sDz:  ;./ / /&( # *= ?y!4 $.H1}x(A-(.v}}':5+223:>  ?23 3 3 AaA AQR R !$Y [!I J C   eUQY0GH1QH I J{+ Fv,  # 0 4 IsOD6J2;+J2''J2J(7J2+J23'J2J+A J2; J- "J2+J2-J2cHddlm}|||z||z ||z S)aW 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 r) factorial)(sympy.functions.combinatorial.factorialsr?)VNr?s rmonomial_countrCs)<C QU il *Yq\ 99cdtt||Dcgc] \}}||z c}}Scc}}w)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_mulrMs+" SAY0TQ1q50 110, cVt||}td|Dr t|Sy)a 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&K|] }|dk\ ywrr)rcs rrzmonomial_div..s a16 rN) monomial_ldivr'rG)rIrJCs r monomial_divrTs*, aA 1 QxrDcdtt||Dcgc] \}}||z  c}}Scc}}w)a 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`. rFrHs rrRrRs+, SAY0TQ1q50 110rNcDt|Dcgc]}||z c}Scc}w)z%Return the n-th pow of the monomial. )rG)rIr1rKs r monomial_powrWs #11Q3# $$#s c rtt||Dcgc]\}}t||c}}Scc}}w)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`. )rGr/minrHs r monomial_gcdrZ-" Q4A3q!94 5543 c rtt||Dcgc]\}}t||c}}Scc}}w)a1 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`. )rGr/maxrHs r monomial_lcmr_r[r\c:tdt||DS)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,K|] \}}||kywrr)rrKrLs rrz#monomial_divides..5s,$!QqAv,s)r'r/)rIrJs rmonomial_dividesrb(s ,#a), ,,rDct|d}|ddD]'}t|D]\}}t|||||<)t|S)a Returns maximal degree for each variable in a set of monomials. Examples ======== Consider monomials `x**3*y**4*z**5`, `y**5*z` and `x**6*y**3*z**9`. We wish to find out what is the maximal degree for each of `x`, `y` and `z` variables:: >>> from sympy.polys.monomials import monomial_max >>> monomial_max((3,4,5), (0,5,1), (6,3,9)) (6, 5, 9) rrN)r& enumerater^rGmonomsMrBrr1s r monomial_maxrh7[" VAYA ABZ aL DAqqtQ>> from sympy.polys.monomials import monomial_min >>> monomial_min((3,4,5), (0,5,1), (6,3,9)) (0, 3, 1) rrN)r&rdrYrGres r monomial_minrkPrirDct|S)z Returns the total degree of a monomial. Examples ======== The total degree of `xy^2` is 3: >>> from sympy.polys.monomials import monomial_deg >>> monomial_deg((1, 2)) 3 )r*)rgs r monomial_degrmis  q6MrDc|\}}|\}}t||}|jr|||j||fSy|||zs||j||fSy)z,Division of two terms in over a ring/field. N)rTis_Fieldquo)rKrLdomaina_lma_lcb_lmb_lcmonoms rterm_divrwxsjJD$JD$ t $E   &**T400 0 &**T400 0rDcLeZdZdZdZdZdZdZdZdZ dZ d Z d Z d Z y ) MonomialOpsz6Code generator of fast monomial arithmetic functions. c||_yr)ngens)selfr{s r__init__zMonomialOps.__init__s  rDc(i}t||||Sr)exec)r|codenamenss r_buildzMonomialOps._builds  T2$xrDcZt|jDcgc]}|| c}Scc}wr)r.r{)r|rrs r_varszMonomialOps._varss$-24::->@4#@@@s (cDd}td}|jd}|jd}t||Dcgc] \}}|d|}}}||dj|dj|dj|dz}|j ||Scc}}w)NrMs def %(name)s(A, B): (%(A)s,) = A (%(B)s,) = B return (%(AB)s,) rKrL + , rrIrJABrrr/joinr r|rtemplaterIrJrKrLrrs rmulzMonomialOps.muls   JJsO JJsO.1!Qi 9daAq! 9 94diil1UYU^U^_aUbcc{{4&&:Bcd}td}|jd}|Dcgc]}d|z }}||dj|dj|dz}|j||Scc}w)NrWzZ def %(name)s(A, k): (%(A)s,) = A return (%(Ak)s,) rKz%s*kr)rrIAk)rrrr)r|rrrIrKrrs rpowzMonomialOps.powss   JJsO#$ &avz & &4diil$))B-PP{{4&&'s A*cFd}td}|jd}|jd}t||Dcgc] \}}|d|d}}}||dj|dj|dj|dz}|j ||Scc}}w) Nmonomial_mulpowzw def %(name)s(A, B, k): (%(A)s,) = A (%(B)s,) = B return (%(ABk)s,) rKrLrz*kr)rrIrJABkr) r|rrrIrJrKrLrrs rmulpowzMonomialOps.mulpows    JJsO JJsO14Q<Aq!$<<4diil1VZV_V_`cVdee{{4&&=sBcDd}td}|jd}|jd}t||Dcgc] \}}|d|}}}||dj|dj|dj|dz}|j ||Scc}}w)NrRrrKrLz - rrrrs rldivzMonomialOps.ldivs   JJsO JJsO.1!Qi 9daAq! 9 94diil1UYU^U^_aUbcc{{4&&:rcd}td}|jd}|jd}t|jDcgc] }dd|iz }}|jd}||dj |dj |d j |dj |d z}|j ||Scc}w) NrTz def %(name)s(A, B): (%(A)s,) = A (%(B)s,) = B %(RAB)s return (%(R)s,) rKrLz7r%(i)s = a%(i)s - b%(i)s if r%(i)s < 0: return Nonerrrz )rrIrJRABR)rrr.r{rr) r|rrrIrJrrrrs rdivzMonomialOps.divs   JJsO JJsO_deieoeo_prZ[JcSTXUrr JJsO4diil1V^VcVcdgVhosoxoxyzo{||{{4&&ssCc Pd}td}|jd}|jd}t||Dcgc]\}}|d|d|d|}}}||dj|dj|dj|d z}|j ||Scc}}w) Nr_rrKrL if z >=  else rrrrs rlcmzMonomialOps.lcm   JJsO JJsOCFq!9 N41a1aA6 N N4diil1UYU^U^_aUbcc{{4&&OB"c Pd}td}|jd}|jd}t||Dcgc]\}}|d|d|d|}}}||dj|dj|dj|d z}|j ||Scc}}w) NrZrrKrLrz <= rrrrrs rgcdzMonomialOps.gcdrrN)__name__ __module__ __qualname____doc__r}rrrrrrrrrrrDrryrys8@ A ' ' ' '' ' 'rDrycveZdZdZdZddZddZdZdZdZ d Z d Z d Z d Z d ZdZdZeZdZdZdZy)Monomialz9Class representing a monomial, i.e. a product of powers. ) exponentsgensNc^t|s}tt||\}}t|dk(r>""rDc,t|jSr)iterrrs r__iter__zMonomial.__iter__sDNN##rDc |j|Sr)r)r|r6s r __getitem__zMonomial.__getitem__s~~d##rDcnt|jj|j|jfSr)hashrrrrrs r__hash__zMonomial.__hash__s&T^^,,dnndiiHIIrDc |jrGdjt|j|jDcgc] \}}|d|c}}S|jj d|jdScc}}w)N*z**())rrr/rrr)r|genexps r__str__zMonomial.__str__s] 9988C SWSaSaDbdS#s3de e#~~66G GesA: c|xs |j}|std|ztt||jDcgc] \}}||z c}}Scc}}w)z3Convert a monomial instance to a SymPy expression. z5Cannot convert %s to an expression without generators)rr#rr/r)r|rrrs ras_exprzMonomial.as_expr sX tyyG$NP Ps4/HJ83c3hJKKJsA ct|tr |j}nt|ttfr|}ny|j|k(S)NF) isinstancerrrGrr|otherrs r__eq__zMonomial.__eq__*s: eX &I u~ .I~~**rDc||k( Srr)r|rs r__ne__zMonomial.__ne__4s5=  rDct|tr |j}nt|ttfr|}nt |j t|j|Sr)rrrrGrNotImplementedErrorrrMrs r__mul__zMonomial.__mul__7sH eX &I u~ .I% %||LCDDrDct|tr |j}nt|ttfr|}nt t |j|}||j|St|t|r) rrrrGrrrTrr )r|rrresults r __truediv__zMonomial.__truediv__Asd eX &I u~ .I% %dnni8  <<' '%dHUO< rsF=--6D";EEN:B2&:20%6&6& -22 $n'n'`sE!sEsErD