U ZʹjI@sdZddlmZmZddlmZddlmZmZm Z m Z ddl m Z ddl mZmZddlmZddlmZmZed(d d Zd d ZddZddZddZddZddZddZddZddZddZd d!Z d"d#Z!Gd$d%d%Z"eGd&d'd'eZ#d S))z@Tools and arithmetics for monomials of distributed polynomials. )combinations_with_replacementproduct)dedent)MulSTuplesympify)ExactQuotientFailed)PicklableWithSlotsdict_from_expr)public) is_sequenceiterableNc#s|t|}tr~t|kr$tddkr8dg|n@tsJtdn.t|kr^tdtddDrxtdd}n:}|dkrtd dkrd}ndkrtd }d }|r||krdS|r|dkrtjVdSt|tjg}td d|Dr~g}t||D]X}t |d} |D] } | d kr(| | d 7<q(t | |kr| t|qt|EdHnxg} t||dD]X}t |d} |D] } | d kr| | d 7<qt | |kr| t|qt| EdHntfddt|Drtdg} t|D].\} }| fddt| |d Dq.t| D]} t| VqfdS)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.0irrT/var/www/html/bwave/python/venv/lib/python3.8/site-packages/sympy/polys/monomials.py bsz itermonomials..z+min_degrees cannot contain negative numbersFzmax_degrees cannot be negativezmin_degrees cannot be negativeTcss|] }|jVqdSN)Zis_commutative)rvariablerrrrxs)repeatc3s|]}||kVqdSrrr) max_degrees min_degreesrrrsz2min_degrees[i] must be <= max_degrees[i] for all icsg|] }|qSrrr)varrr sz!itermonomials..)lenr ValueErroranyrZOnelistallrdictfromkeyssumvaluesappendrsetrrangezip) variablesrrnZ total_degreeZ max_degreeZ min_degreeZmonomials_list_commitemZpowersrZmonomials_list_non_commZ power_listsZmin_dZmax_dr)rrrr itermonomials snJ         & 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)Z(sympy.functions.combinatorial.factorialsr/)VNr/rrrmonomial_counts r2cCstddt||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|]\}}||qSrrrabrrrrsz monomial_mul..tupler*ABrrr monomial_mulsr;cCs,t||}tdd|Dr$t|SdSdS)a Division of tuples representing monomials. Examples ======== Lets divide `x**3*y**4*z` by `x*y**2`:: >>> from sympy.polys.monomials import monomial_div >>> monomial_div((3, 4, 1), (1, 2, 0)) (2, 2, 1) which gives `x**2*y**2*z`. However:: >>> monomial_div((3, 4, 1), (1, 2, 2)) is None True `x*y**2*z**2` does not divide `x**3*y**4*z`. css|]}|dkVqdSrr)rcrrrrszmonomial_div..N) monomial_ldivr"r7)r9r:Crrr monomial_divs r?cCstddt||DS)a Division of tuples representing monomials. Examples ======== Lets divide `x**3*y**4*z` by `x*y**2`:: >>> from sympy.polys.monomials import monomial_ldiv >>> monomial_ldiv((3, 4, 1), (1, 2, 0)) (2, 2, 1) which gives `x**2*y**2*z`. >>> monomial_ldiv((3, 4, 1), (1, 2, 2)) (2, 2, -1) which gives `x**2*y**2*z**-1`. cSsg|]\}}||qSrrr3rrrrsz!monomial_ldiv..r6r8rrrr=sr=cstfdd|DS)z%Return the n-th pow of the monomial. csg|] }|qSrrrr4r,rrrsz monomial_pow..)r7)r9r,rrAr monomial_powsrBcCstddt||DS)a. Greatest common divisor of tuples representing monomials. Examples ======== Lets compute GCD of `x*y**4*z` and `x**3*y**2`:: >>> from sympy.polys.monomials import monomial_gcd >>> monomial_gcd((1, 4, 1), (3, 2, 0)) (1, 2, 0) which gives `x*y**2`. cSsg|]\}}t||qSr)minr3rrrrsz monomial_gcd..r6r8rrr monomial_gcdsrDcCstddt||DS)a1 Least common multiple of tuples representing monomials. Examples ======== Lets compute LCM of `x*y**4*z` and `x**3*y**2`:: >>> from sympy.polys.monomials import monomial_lcm >>> monomial_lcm((1, 4, 1), (3, 2, 0)) (3, 4, 1) which gives `x**3*y**4*z`. cSsg|]\}}t||qSr)maxr3rrrr&sz monomial_lcm..r6r8rrr monomial_lcmsrFcCstddt||DS)z Does there exist a monomial X such that XA == B? Examples ======== >>> from sympy.polys.monomials import monomial_divides >>> monomial_divides((1, 2), (3, 4)) True >>> monomial_divides((1, 2), (0, 2)) False css|]\}}||kVqdSrrr3rrrr5sz#monomial_divides..)r"r*r8rrrmonomial_divides(s rGcGsJt|d}|ddD](}t|D]\}}t|||||<q$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! enumeraterEr7ZmonomsMr1rr,rrr monomial_max7s  rKcGsJt|d}|ddD](}t|D]\}}t|||||<q$qt|S)a Returns minimal degree for each variable in a set of monomials. Examples ======== Consider monomials `x**3*y**4*z**5`, `y**5*z` and `x**6*y**3*z**9`. We wish to find out what is the minimal degree for each of `x`, `y` and `z` variables:: >>> from sympy.polys.monomials import monomial_min >>> monomial_min((3,4,5), (0,5,1), (6,3,9)) (0, 3, 1) rrN)r!rHrCr7rIrrr monomial_minPs  rLcCst|S)z Returns the total degree of a monomial. Examples ======== The total degree of `xy^2` is 3: >>> from sympy.polys.monomials import monomial_deg >>> monomial_deg((1, 2)) 3 )r%)rJrrr monomial_degis rMcCsf|\}}|\}}t||}|jr>|dk r8||||fSdSn$|dks^||s^||||fSdSdS)z,Division of two terms in over a ring/field. N)r?Zis_FieldZquo)r4r5domainZa_lmZa_lcZb_lmZb_lcmonomrrrterm_divxs rPc@s`eZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ ddZ ddZ dS) MonomialOpsz6Code generator of fast monomial arithmetic functions. cCs ||_dSr)ngens)selfrRrrr__init__szMonomialOps.__init__cCsi}t||||Sr)exec)rScodenamensrrr_builds zMonomialOps._buildcsfddt|jDS)Ncsg|]}d|fqS)z%s%srrrWrrrsz%MonomialOps._vars..)r)rR)rSrWrrZr_varsszMonomialOps._varscCsdd}td}|d}|d}ddt||D}||d|d|d|d}|||S) Nr;s def %(name)s(A, B): (%(A)s,) = A (%(B)s,) = B return (%(AB)s,) r4r5cSsg|]\}}d||fqS)z%s + %srr3rrrrsz#MonomialOps.mul.., rWr9r:ABrr[r*joinrYrSrWtemplater9r:r_rVrrrmuls  $zMonomialOps.mulcCsLd}td}|d}dd|D}||d|d|d}|||S)NrBzZ def %(name)s(A, k): (%(A)s,) = A return (%(Ak)s,) r4cSsg|] }d|qS)z%s*krr@rrrrsz#MonomialOps.pow..r])rWr9Ak)rr[rarY)rSrWrcr9rerVrrrpows  zMonomialOps.powcCsdd}td}|d}|d}ddt||D}||d|d|d|d}|||S) NZmonomial_mulpowzw def %(name)s(A, B, k): (%(A)s,) = A (%(B)s,) = B return (%(ABk)s,) r4r5cSsg|]\}}d||fqS)z %s + %s*krr3rrrrsz&MonomialOps.mulpow..r])rWr9r:ABkr`)rSrWrcr9r:rgrVrrrmulpows  $zMonomialOps.mulpowcCsdd}td}|d}|d}ddt||D}||d|d|d|d}|||S) Nr=r\r4r5cSsg|]\}}d||fqS)z%s - %srr3rrrrsz$MonomialOps.ldiv..r]r^r`rbrrrldivs  $zMonomialOps.ldivcCsvd}td}|d}|d}ddt|jD}|d}||d|d|d |d|d }|||S) Nr?z def %(name)s(A, B): (%(A)s,) = A (%(B)s,) = B %(RAB)s return (%(R)s,) r4r5cSsg|]}dd|iqS)z7r%(i)s = a%(i)s - b%(i)s if r%(i)s < 0: return Nonerrrrrrrsz#MonomialOps.div..rr]z )rWr9r:RABR)rr[r)rRrarY)rSrWrcr9r:rkrlrVrrrdivs   ,zMonomialOps.divcCsdd}td}|d}|d}ddt||D}||d|d|d|d}|||S) NrFr\r4r5cSs g|]\}}d||||fqS)z%s if %s >= %s else %srr3rrrrsz#MonomialOps.lcm..r]r^r`rbrrrlcms  $zMonomialOps.lcmcCsdd}td}|d}|d}ddt||D}||d|d|d|d}|||S) NrDr\r4r5cSs g|]\}}d||||fqS)z%s if %s <= %s else %srr3rrrrsz#MonomialOps.gcd..r]r^r`rbrrrgcds  $zMonomialOps.gcdN)__name__ __module__ __qualname____doc__rTrYr[rdrfrhrirmrnrorrrrrQs rQc@seZdZdZdZd"ddZd#ddZdd Zd d Zd d Z ddZ ddZ ddZ ddZ ddZddZddZeZddZddZd d!ZdS)$Monomialz9Class representing a monomial, i.e. a product of powers. ) exponentsgensNcCsvt|s\tt||d\}}t|dkrNt|ddkrNt|d}ntd|t t t ||_ ||_ dS)N)rvrrzExpected a monomial got {})rr rrr!r&keysrformatr7mapintrurv)rSrOrvreprrrrTs zMonomial.__init__cCs|||p|jSr) __class__rv)rSrurvrrrrebuild szMonomial.rebuildcCs t|jSr)rrurSrrr__len__szMonomial.__len__cCs t|jSr)iterrur~rrr__iter__szMonomial.__iter__cCs |j|Sr)ru)rSr-rrr __getitem__szMonomial.__getitem__cCst|jj|j|jfSr)hashr|rprurvr~rrr__hash__szMonomial.__hash__cCs:|jr$dddt|j|jDSd|jj|jfSdS)N*cSsg|]\}}d||fqS)z%s**%srrgenexprrrrsz$Monomial.__str__..z%s(%s))rvrar*rur|rpr~rrr__str__szMonomial.__str__cGs4|p|j}|std|tddt||jDS)z3Convert a monomial instance to a SymPy expression. z5Cannot convert %s to an expression without generatorscSsg|]\}}||qSrrrrrrr(sz$Monomial.as_expr..)rvrrr*ru)rSrvrrras_expr s  zMonomial.as_exprcCs4t|tr|j}nt|ttfr&|}ndS|j|kS)NF) isinstancertrur7rrSotherrurrr__eq__*s  zMonomial.__eq__cCs ||k Srr)rSrrrr__ne__4szMonomial.__ne__cCs<t|tr|j}nt|ttfr&|}nt|t|j|Sr)rrtrur7rNotImplementedErrorr}r;rrrr__mul__7s  zMonomial.__mul__cCsZt|tr|j}nt|ttfr&|}ntt|j|}|dk rH||St|t|dSr) rrtrur7rrr?r}r )rSrruresultrrr __truediv__As   zMonomial.__truediv__cCs.t|}|dkrtd||t|j|S)Nrz'a non-negative integer expected, got %s)rzrr}rBru)rSrr,rrr__pow__Rs zMonomial.__pow__cCsDt|tr|j}n t|ttfr&|}n td||t|j|S)z&Greatest common divisor of monomials. .an instance of Monomial class expected, got %s)rrtrur7r TypeErrorr}rDrrrrroXs z Monomial.gcdcCsDt|tr|j}n t|ttfr&|}n td||t|j|S)z$Least common multiple of monomials. r)rrtrur7rrr}rFrrrrrnds z Monomial.lcm)N)N)rprqrrrs __slots__rTr}rrrrrrrrrr __floordiv__rrornrrrrrts$     rt)N)$rs itertoolsrrtextwraprZ sympy.corerrrrZsympy.polys.polyerrorsr Zsympy.polys.polyutilsr r Zsympy.utilitiesr Zsympy.utilities.iterablesr rr.r2r;r?r=rBrDrFrGrKrLrMrPrQrtrrrrs2    !p