o 3IhH@sdZddlmZmZddlmZddlmZddlm Z m Z m Z m Z ddl mZddlmZmZddlmZdd lmZmZed)d d Zd dZddZddZddZddZddZddZddZddZ dd Z!d!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)cacheit)MulSTuplesympify)ExactQuotientFailed)PicklableWithSlotsdict_from_expr)public) is_sequenceiterableNc#str{t|}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.0irrU/home/air/sanwanet/gpt-api/venv/lib/python3.10/site-packages/sympy/polys/monomials.py cz itermonomials..z+min_degrees cannot contain negative numbersc3s |] }||kVqdSNrr) max_degrees min_degreesrrresz2min_degrees[i] must be <= max_degrees[i] for all icsg|]}|qSrrr)varrr iz!itermonomials..zmax_degrees cannot be negativezmin_degrees cannot be negativecss|]}|jVqdSr)is_commutative)rvariablerrrr}s)repeat)rlen ValueErroranyrangezipappendrrrOnelistallrsetadd) variablesrrn power_listsmin_dmax_dpowers max_degree min_degreeit monomials_setditemcountr r)rrrr itermonomialssfJ   $     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)(sympy.functions.combinatorial.factorialsr;)VNr;rrrmonomial_counts r?cCtddt||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_mulrJcCs(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*rF)rHrICrrr monomial_divs rOcCr@)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|]\}}||qSrrrArrrrrDz!monomial_ldiv..rErGrrrrMsrMcstfdd|DS)z%Return the n-th pow of the monomial. csg|]}|qSrrrrBr.rrrrz monomial_pow..)rF)rHr.rrQr monomial_powsrRcCr@)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)minrArrrr z monomial_gcd..rErGrrr monomial_gcdrKrVcCr@)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`. cSrSr)maxrArrrrrUz monomial_lcm..rErGrrr monomial_lcmrKrXcCr@)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|] \}}||kVqdSrrrArrrr.sz#monomial_divides..)r*r&rGrrrmonomial_divides!s rYcGJt|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) enumeraterWrFmonomsMr>rr.rrr monomial_max0 r_cGrZ)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)r[rTrFr\rrr monomial_minIr`racCst|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 )sum)r^rrr monomial_degbs rccCs`|\}}|\}}t||}|jr|dur||||fSdS|dus.||s.||||fSdS)z,Division of two terms in over a ring/field. N)rOis_Fieldquo)rBrCdomaina_lma_lcb_lmb_lcmonomrrrterm_divqs rlcseZdZdZefddZddZddZdd Zed d Z ed d Z eddZ eddZ eddZ eddZeddZZS) MonomialOpsz6Code generator of fast monomial arithmetic functions. cst|}||_|Sr)super__new__ngens)clsrpobj __class__rrros zMonomialOps.__new__cCs|jfSr)rpselfrrr__getnewargs__szMonomialOps.__getnewargs__cCsi}t||||Sr)exec)rvcodenamensrrr_builds zMonomialOps._buildcsfddt|jDS)Ncsg|]}d|fqS)z%s%srrrzrrrrDz%MonomialOps._vars..)r%rp)rvrzrr}r_varsszMonomialOps._varscCdd}td}|d}|d}ddt||D}||d|d|d|d}|||S) NrJs def %(name)s(A, B): (%(A)s,) = A (%(B)s,) = B return (%(AB)s,) rBrCcSg|] \}}d||fqS)z%s + %srrArrrrz#MonomialOps.mul.., rzrHrIABrr~r&joinr|rvrztemplaterHrIrryrrrmul  $ zMonomialOps.mulcCsLd}td}|d}dd|D}||d|d|d}|||S)NrRzZ def %(name)s(A, k): (%(A)s,) = A return (%(Ak)s,) rBcSsg|]}d|qS)z%s*krrPrrrrrz#MonomialOps.pow..r)rzrHAk)rr~rr|)rvrzrrHrryrrrpows   zMonomialOps.powcCr) Nmonomial_mulpowzw def %(name)s(A, B, k): (%(A)s,) = A (%(B)s,) = B return (%(ABk)s,) rBrCcSr)z %s + %s*krrArrrrrz&MonomialOps.mulpow..r)rzrHrIABkr)rvrzrrHrIrryrrrmulpowrzMonomialOps.mulpowcCr) NrMrrBrCcSr)z%s - %srrArrrrrz$MonomialOps.ldiv..rrrrrrrldivrzMonomialOps.ldivcCsvd}td}|d}|d}ddt|jD}|d}||d|d|d |d|d }|||S) NrOz def %(name)s(A, B): (%(A)s,) = A (%(B)s,) = B %(RAB)s return (%(R)s,) rBrCcSsg|]}dd|iqS)z7r%(i)s = a%(i)s - b%(i)s if r%(i)s < 0: return NonerrrrrrrrDz#MonomialOps.div..rrz )rzrHrIRABR)rr~r%rprr|)rvrzrrHrIrrryrrrdivs   , zMonomialOps.divcCr) NrXrrBrCcS g|] \}}d||||fqS)z%s if %s >= %s else %srrArrrr z#MonomialOps.lcm..rrrrrrrlcmrzMonomialOps.lcmcCr) NrVrrBrCcSr)z%s if %s <= %s else %srrArrrrrz#MonomialOps.gcd..rrrrrrrgcdrzMonomialOps.gcd)__name__ __module__ __qualname____doc__rrorwr|r~rrrrrrr __classcell__rrrsrrms*      rmc@seZdZdZdZd"ddZd"ddZdd Zd d Zd d Z ddZ ddZ ddZ ddZ ddZddZddZeZddZddZd d!ZdS)#Monomialz9Class representing a monomial, i.e. a product of powers. ) exponentsgensNcCsvt|s.tt||d\}}t|dkr't|ddkr't|d}ntd|t t t ||_ ||_ dS)N)rrrzExpected a monomial got {})rr r r"r)valueskeysr#formatrFmapintrr)rvrkrreprrr__init__s  zMonomial.__init__cCs|||p|jSr)rtr)rvrrrrrrebuildszMonomial.rebuildcC t|jSr)r"rrurrr__len__ zMonomial.__len__cCrr)iterrrurrr__iter__rzMonomial.__iter__cCs |j|Sr)r)rvr8rrr __getitem__rzMonomial.__getitem__cCst|jj|j|jfSr)hashrtrrrrurrr__hash__szMonomial.__hash__cCs6|jrdddt|j|jDSd|jj|jfS)N*cSr)z%s**%srrgenexprrrr"rz$Monomial.__str__..z%s(%s))rrr&rrtrrurrr__str__ szMonomial.__str__cGs4|p|j}|s td|tddt||jDS)z3Convert a monomial instance to a SymPy expression. z5Cannot convert %s to an expression without generatorscSsg|]\}}||qSrrrrrrr.rDz$Monomial.as_expr..)rr#rr&r)rvrrrras_expr&s zMonomial.as_exprcCs4t|tr |j}n t|ttfr|}ndS|j|kS)NF) isinstancerrrFrrvotherrrrr__eq__0s  zMonomial.__eq__cCs ||k Srr)rvrrrr__ne__:rzMonomial.__ne__cCs<t|tr |j}n t|ttfr|}nt|t|j|Sr)rrrrFrNotImplementedErrorrrJrrrr__mul__=s zMonomial.__mul__cCsVt|tr |j}n t|ttfr|}ntt|j|}|dur$||St|t|r) rrrrFrrrOrr )rvrrresultrrr __truediv__Gs   zMonomial.__truediv__cCs.t|}|dkrtd||t|j|S)Nrz'a non-negative integer expected, got %s)rr#rrRr)rvrr.rrr__pow__Xs zMonomial.__pow__cCDt|tr |j}nt|ttfr|}ntd||t|j|S)z&Greatest common divisor of monomials. .an instance of Monomial class expected, got %s)rrrrFr TypeErrorrrVrrrrr^ z Monomial.gcdcCr)z$Least common multiple of monomials. r)rrrrFrrrrXrrrrrjrz Monomial.lcmr)rrrr __slots__rrrrrrrrrrrr __floordiv__rrrrrrrrs&      rr)&r itertoolsrrtextwraprsympy.core.cacher sympy.corerrrr sympy.polys.polyerrorsr sympy.polys.polyutilsr r sympy.utilitiesr sympy.utilities.iterablesrrr:r?rJrOrMrRrVrXrYr_rarcrlrmrrrrrs4     !}