a RG5dI@sdZddlmZmZddlmZddlmZmZm Z m Z ddl m Z ddl mZmZddlmZddlmZmZed(d d Zd d ZddZddZddZddZddZddZddZddZddZd d!Z d"d#Z!Gd$d%d%Z"eGd&d'd'eZ#d S))z@Tools and arithmetics for monomials of distributed polynomials. )combinations_with_replacementproduct)dedent)MulSTuplesympify)ExactQuotientFailed)PicklableWithSlotsdict_from_expr)public) is_sequenceiterableNc#st|}tr~t|kr$tddur8dg|n@tsJtdn.t|kr^tdtddDrxtdd}n:}|dkrtd durd}ndkrtd }d }|r||krdS|r|dkrtjVdSt|tjg}td d|Drg}t||D]Z}d d|D} |D] } | dkr*| | d7<q*t | |kr| t |qt |EdHnzg} t||dD]Z}dd|D} |D] } | dkr| | d7<qt | |kr| t |qt | EdHntfddt|Dr"tdg} t|D].\} }| fddt| |dDq2t| D]} t | VqjdS)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 listcss|]}|dkVqdSrN.0irrQ/var/www/html/django/DPS/env/lib/python3.9/site-packages/sympy/polys/monomials.py bz itermonomials..z+min_degrees cannot contain negative numbersFzmax_degrees cannot be negativezmin_degrees cannot be negativeTcss|] }|jVqdSN)is_commutativervariablerrrrxrcSsi|] }|dqSrrrrrr {rz!itermonomials..)repeatcSsi|] }|dqSrrrrrrrrc3s|]}||kVqdSrrr) max_degrees min_degreesrrrrz2min_degrees[i] must be <= max_degrees[i] for all icsg|] }|qSrrr)varrr rz!itermonomials..)lenr ValueErroranyrOnelistallrsumvaluesappendrsetrrangezip) variablesrr n total_degree max_degreeZ min_degreeZmonomials_list_commitempowersrZmonomials_list_non_commZ power_listsZmin_dZmax_dr)rr r!r itermonomials snJ       & r5cCs(ddlm}|||||||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.factorialsr6)VNr6rrrmonomial_counts r:cCstddt||DS)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`. cSsg|]\}}||qSrrrabrrrr"rz monomial_mul..tupler.ABrrr monomial_mulsrCcCs,t||}tdd|Dr$t|SdSdS)a Division of tuples representing monomials. Examples ======== Lets divide `x**3*y**4*z` by `x*y**2`:: >>> from sympy.polys.monomials import monomial_div >>> monomial_div((3, 4, 1), (1, 2, 0)) (2, 2, 1) which gives `x**2*y**2*z`. However:: >>> monomial_div((3, 4, 1), (1, 2, 2)) is None True `x*y**2*z**2` does not divide `x**3*y**4*z`. css|]}|dkVqdSrr)rcrrrrrzmonomial_div..N) monomial_ldivr(r?)rArBCrrr monomial_divs rGcCstddt||DS)a Division of tuples representing monomials. Examples ======== Lets divide `x**3*y**4*z` by `x*y**2`:: >>> from sympy.polys.monomials import monomial_ldiv >>> monomial_ldiv((3, 4, 1), (1, 2, 0)) (2, 2, 1) which gives `x**2*y**2*z`. >>> monomial_ldiv((3, 4, 1), (1, 2, 2)) (2, 2, -1) which gives `x**2*y**2*z**-1`. cSsg|]\}}||qSrrr;rrrr"rz!monomial_ldiv..r>r@rrrrEsrEcstfdd|DS)z%Return the n-th pow of the monomial. csg|] }|qSrrrr<r0rrr"rz monomial_pow..)r?)rAr0rrIr monomial_powsrJcCstddt||DS)a. Greatest common divisor of tuples representing monomials. Examples ======== Lets compute GCD of `x*y**4*z` and `x**3*y**2`:: >>> from sympy.polys.monomials import monomial_gcd >>> monomial_gcd((1, 4, 1), (3, 2, 0)) (1, 2, 0) which gives `x*y**2`. cSsg|]\}}t||qSr)minr;rrrr"rz monomial_gcd..r>r@rrr monomial_gcdsrLcCstddt||DS)a1 Least common multiple of tuples representing monomials. Examples ======== Lets compute LCM of `x*y**4*z` and `x**3*y**2`:: >>> from sympy.polys.monomials import monomial_lcm >>> monomial_lcm((1, 4, 1), (3, 2, 0)) (3, 4, 1) which gives `x**3*y**4*z`. cSsg|]\}}t||qSr)maxr;rrrr"&rz monomial_lcm..r>r@rrr monomial_lcmsrNcCstddt||DS)z Does there exist a monomial X such that XA == B? Examples ======== >>> from sympy.polys.monomials import monomial_divides >>> monomial_divides((1, 2), (3, 4)) True >>> monomial_divides((1, 2), (0, 2)) False css|]\}}||kVqdSrrr;rrrr5rz#monomial_divides..)r(r.r@rrrmonomial_divides(s rOcGsJt|d}|ddD](}t|D]\}}t|||||<q$qt|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' enumeraterMr?monomsMr9rr0rrr monomial_max7s  rTcGsJt|d}|ddD](}t|D]\}}t|||||<q$qt|S)a Returns minimal 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 minimal degree for each of `x`, `y` and `z` variables:: >>> from sympy.polys.monomials import monomial_min >>> monomial_min((3,4,5), (0,5,1), (6,3,9)) (0, 3, 1) rrN)r'rPrKr?rQrrr monomial_minPs  rUcCst|S)z Returns the total degree of a monomial. 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