o 8VaJ@s dZddlmZmZddlmZddlmZmZm Z m Z ddl m Z ddl mZddlmZmZddlmZdd l mZed)d d Zd dZddZddZddZddZddZddZddZddZdd Zd!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)iterable)ExactQuotientFailed)PicklableWithSlotsdict_from_expr)public) is_sequenceNc#st|}tr?t|krtddurdg|nts%tdt|kr/tdtddDr>> 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.0irr7/usr/lib/python3/dist-packages/sympy/polys/monomials.py cz itermonomials..z*min_degrees can't contain negative numbersFzmax_degrees can't be negativezmin_degrees can't be negativeTcss|]}|jVqdSN)Zis_commutative)rvariablerrrrys)repeatc3s |] }||kVqdSrrr) max_degrees min_degreesrrrsz2min_degrees[i] must be <= max_degrees[i] for all icsg|]}|qSrrr)varrr z!itermonomials..)lenr ValueErroranyrZOnelistallrdictsumvaluesappendrsetrrangezip)Z variablesrrnZ total_degreeZ max_degreeZ min_degreeZmonomials_list_commitemZpowersrZmonomials_list_non_commZ power_listsZmin_dZmax_dr)rrrr itermonomialssJ      & r.cCs(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)Zsympyr/)VNr/rrrmonomial_counts r2cCtddt||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|]\}}||qSrrrabrrrrz monomial_mul..tupler+ABrrr monomial_mulr=cCs(t||}tdd|Drt|SdS)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$r9)r;r<Crrr monomial_divs rBcCr3)a Division of tuples representing monomials. Examples ======== Lets divide `x**3*y**4*z` by `x*y**2`:: >>> from sympy.polys.monomials import monomial_ldiv >>> monomial_ldiv((3, 4, 1), (1, 2, 0)) (2, 2, 1) which gives `x**2*y**2*z`. >>> monomial_ldiv((3, 4, 1), (1, 2, 2)) (2, 2, -1) which gives `x**2*y**2*z**-1`. cSsg|]\}}||qSrrr4rrrrr7z!monomial_ldiv..r8r:rrrr@sr@cstfdd|DS)z%Return the n-th pow of the monomial. csg|]}|qSrrrr5r,rrrrz monomial_pow..)r9)r;r,rrDr monomial_powsrEcCr3)a. Greatest common divisor of tuples representing monomials. Examples ======== Lets compute GCD of `x*y**4*z` and `x**3*y**2`:: >>> from sympy.polys.monomials import monomial_gcd >>> monomial_gcd((1, 4, 1), (3, 2, 0)) (1, 2, 0) which gives `x*y**2`. cSg|] \}}t||qSr)minr4rrrrz monomial_gcd..r8r:rrr monomial_gcdr>rIcCr3)a1 Least common multiple of tuples representing monomials. Examples ======== Lets compute LCM of `x*y**4*z` and `x**3*y**2`:: >>> from sympy.polys.monomials import monomial_lcm >>> monomial_lcm((1, 4, 1), (3, 2, 0)) (3, 4, 1) which gives `x**3*y**4*z`. cSrFr)maxr4rrrr+rHz monomial_lcm..r8r:rrr monomial_lcmr>rKcCr3)z Does there exist a monomial X such that XA == B? Examples ======== >>> from sympy.polys.monomials import monomial_divides >>> monomial_divides((1, 2), (3, 4)) True >>> monomial_divides((1, 2), (0, 2)) False css|] \}}||kVqdSrrr4rrrr:sz#monomial_divides..)r$r+r:rrrmonomial_divides-s rLcGJt|d}|ddD]}t|D] \}}t|||||<qq 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# enumeraterJr9ZmonomsMr1rr,rrr monomial_max< rQcGrM)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#rNrGr9rOrrr monomial_minUrRrScCst|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&)rPrrr monomial_degns rTcCs`|\}}|\}}t||}|jr|dur||||fSdS|dus.||s.||||fSdS)z,Division of two terms in over a ring/field. 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