MZdIdZddlmZmZddlmZddlmZmZm 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#dKt|}trqt|k7r tddg|znLts tdt|k7r tdtdDr tdd}n+}|dkr tdd}ndkr td }d }|rZkDry|r|dk(rtj yt |tj gz}td |Drg}t||D]`}|D cic]} | d} } |D]} | d k7s | | xxd z cc<t| j|k\sJ|jt|bt|Ed{yg} t|| D]`}|D cic]} | d} } |D]} | d k7s | | xxd z cc<t| j|k\sJ| jt|bt| Ed{ytfdt|Dr tdg} t!|D]5\} }}| jt||d zDcgc]}| |z c}7t| D] } t| ycc} w7cc} w7cc}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 7/usr/lib/python3/dist-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 ValueErroranyrOnelistallrsumvaluesappendrsetrrangezip) variablesr r!n total_degree max_degree min_degreemonomials_list_commitemrpowersmonomials_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 ;6?@((A+@@ $.H1}x(A-(.v}}':5'..sDz:  ;./ / /&( # *= ?6?@((A+@@ $.H1}x(A-(.v}}':5+223:>  ?23 3 3 AaA AQR R !$Y [!I J C   eUQY0GH1QH I J{+ Fv,  /A 0A 4 IsgDJ0 J( J05+J0!'J0J! J0$ J$. J0;+J0''J0J)A J0/ J+ ;'J0$J0+J0cHddlm}|||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_countrAs)<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_mulrKs+" 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'rE)rGrHCs r monomial_divrRs*, aA 1 QxrBcdtt||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`. rDrFs rrPrPs+, SAY0TQ1q50 110rLcDt|Dcgc]}||z c}Scc}w)z%Return the n-th pow of the monomial. )rE)rGr/rIs r monomial_powrUs #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`. )rEr-minrFs r monomial_gcdrX-" 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`. )rEr-maxrFs r monomial_lcmr]rYrZc: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)rrIrJs rrz#monomial_divides..5s,$!QqAv,s)r'r-)rGrHs rmonomial_dividesr`(s ,#a), ,,rBct|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\rEmonomsMr@rr/s r monomial_maxrf7[" 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&rbrWrErcs r monomial_minriPrgrBct|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()res r monomial_degrkis  q6MrBc|\}}|\}}t||}|jr|||j||fSy|||zs||j||fSy)z,Division of two terms in over a ring/field. 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