632 lines
18 KiB
Python
632 lines
18 KiB
Python
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"""Tools and arithmetics for monomials of distributed polynomials. """
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from itertools import combinations_with_replacement, product
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from textwrap import dedent
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from sympy.core import Mul, S, Tuple, sympify
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from sympy.polys.polyerrors import ExactQuotientFailed
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from sympy.polys.polyutils import PicklableWithSlots, dict_from_expr
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from sympy.utilities import public
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from sympy.utilities.iterables import is_sequence, iterable
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@public
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def itermonomials(variables, max_degrees, min_degrees=None):
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r"""
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``max_degrees`` and ``min_degrees`` are either both integers or both lists.
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Unless otherwise specified, ``min_degrees`` is either ``0`` or
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``[0, ..., 0]``.
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A generator of all monomials ``monom`` is returned, such that
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either
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``min_degree <= total_degree(monom) <= max_degree``,
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or
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``min_degrees[i] <= degree_list(monom)[i] <= max_degrees[i]``,
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for all ``i``.
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Case I. ``max_degrees`` and ``min_degrees`` are both integers
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=============================================================
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Given a set of variables $V$ and a min_degree $N$ and a max_degree $M$
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generate a set of monomials of degree less than or equal to $N$ and greater
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than or equal to $M$. The total number of monomials in commutative
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variables is huge and is given by the following formula if $M = 0$:
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.. math::
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\frac{(\#V + N)!}{\#V! N!}
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For example if we would like to generate a dense polynomial of
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a total degree $N = 50$ and $M = 0$, which is the worst case, in 5
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variables, assuming that exponents and all of coefficients are 32-bit long
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and stored in an array we would need almost 80 GiB of memory! Fortunately
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most polynomials, that we will encounter, are sparse.
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Consider monomials in commutative variables $x$ and $y$
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and non-commutative variables $a$ and $b$::
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>>> from sympy import symbols
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>>> from sympy.polys.monomials import itermonomials
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>>> from sympy.polys.orderings import monomial_key
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>>> from sympy.abc import x, y
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>>> sorted(itermonomials([x, y], 2), key=monomial_key('grlex', [y, x]))
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[1, x, y, x**2, x*y, y**2]
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>>> sorted(itermonomials([x, y], 3), key=monomial_key('grlex', [y, x]))
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[1, x, y, x**2, x*y, y**2, x**3, x**2*y, x*y**2, y**3]
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>>> a, b = symbols('a, b', commutative=False)
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>>> set(itermonomials([a, b, x], 2))
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{1, a, a**2, b, b**2, x, x**2, a*b, b*a, x*a, x*b}
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>>> sorted(itermonomials([x, y], 2, 1), key=monomial_key('grlex', [y, x]))
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[x, y, x**2, x*y, y**2]
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Case II. ``max_degrees`` and ``min_degrees`` are both lists
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===========================================================
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If ``max_degrees = [d_1, ..., d_n]`` and
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``min_degrees = [e_1, ..., e_n]``, the number of monomials generated
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is:
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.. math::
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(d_1 - e_1 + 1) (d_2 - e_2 + 1) \cdots (d_n - e_n + 1)
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Let us generate all monomials ``monom`` in variables $x$ and $y$
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such that ``[1, 2][i] <= degree_list(monom)[i] <= [2, 4][i]``,
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``i = 0, 1`` ::
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>>> from sympy import symbols
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>>> from sympy.polys.monomials import itermonomials
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>>> from sympy.polys.orderings import monomial_key
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>>> from sympy.abc import x, y
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>>> sorted(itermonomials([x, y], [2, 4], [1, 2]), reverse=True, key=monomial_key('lex', [x, y]))
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[x**2*y**4, x**2*y**3, x**2*y**2, x*y**4, x*y**3, x*y**2]
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"""
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n = len(variables)
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if is_sequence(max_degrees):
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if len(max_degrees) != n:
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raise ValueError('Argument sizes do not match')
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if min_degrees is None:
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min_degrees = [0]*n
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elif not is_sequence(min_degrees):
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raise ValueError('min_degrees is not a list')
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else:
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if len(min_degrees) != n:
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raise ValueError('Argument sizes do not match')
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if any(i < 0 for i in min_degrees):
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raise ValueError("min_degrees cannot contain negative numbers")
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total_degree = False
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else:
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max_degree = max_degrees
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if max_degree < 0:
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raise ValueError("max_degrees cannot be negative")
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if min_degrees is None:
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min_degree = 0
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else:
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if min_degrees < 0:
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raise ValueError("min_degrees cannot be negative")
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min_degree = min_degrees
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total_degree = True
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if total_degree:
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if min_degree > max_degree:
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return
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if not variables or max_degree == 0:
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yield S.One
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return
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# Force to list in case of passed tuple or other incompatible collection
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variables = list(variables) + [S.One]
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if all(variable.is_commutative for variable in variables):
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monomials_list_comm = []
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for item in combinations_with_replacement(variables, max_degree):
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powers = {variable: 0 for variable in variables}
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for variable in item:
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if variable != 1:
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powers[variable] += 1
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if sum(powers.values()) >= min_degree:
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monomials_list_comm.append(Mul(*item))
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yield from set(monomials_list_comm)
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else:
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monomials_list_non_comm = []
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for item in product(variables, repeat=max_degree):
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powers = {variable: 0 for variable in variables}
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for variable in item:
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if variable != 1:
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powers[variable] += 1
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if sum(powers.values()) >= min_degree:
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monomials_list_non_comm.append(Mul(*item))
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yield from set(monomials_list_non_comm)
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else:
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if any(min_degrees[i] > max_degrees[i] for i in range(n)):
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raise ValueError('min_degrees[i] must be <= max_degrees[i] for all i')
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power_lists = []
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for var, min_d, max_d in zip(variables, min_degrees, max_degrees):
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power_lists.append([var**i for i in range(min_d, max_d + 1)])
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for powers in product(*power_lists):
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yield Mul(*powers)
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def monomial_count(V, N):
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r"""
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Computes the number of monomials.
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The number of monomials is given by the following formula:
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.. math::
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\frac{(\#V + N)!}{\#V! N!}
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where `N` is a total degree and `V` is a set of variables.
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Examples
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========
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>>> from sympy.polys.monomials import itermonomials, monomial_count
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>>> from sympy.polys.orderings import monomial_key
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>>> from sympy.abc import x, y
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>>> monomial_count(2, 2)
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6
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>>> M = list(itermonomials([x, y], 2))
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>>> sorted(M, key=monomial_key('grlex', [y, x]))
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[1, x, y, x**2, x*y, y**2]
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>>> len(M)
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6
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"""
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from sympy.functions.combinatorial.factorials import factorial
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return factorial(V + N) / factorial(V) / factorial(N)
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def monomial_mul(A, B):
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"""
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Multiplication of tuples representing monomials.
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Examples
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========
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Lets multiply `x**3*y**4*z` with `x*y**2`::
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>>> from sympy.polys.monomials import monomial_mul
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>>> monomial_mul((3, 4, 1), (1, 2, 0))
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(4, 6, 1)
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which gives `x**4*y**5*z`.
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"""
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return tuple([ a + b for a, b in zip(A, B) ])
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def monomial_div(A, B):
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"""
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Division of tuples representing monomials.
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Examples
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========
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Lets divide `x**3*y**4*z` by `x*y**2`::
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>>> from sympy.polys.monomials import monomial_div
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>>> monomial_div((3, 4, 1), (1, 2, 0))
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(2, 2, 1)
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which gives `x**2*y**2*z`. However::
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>>> monomial_div((3, 4, 1), (1, 2, 2)) is None
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True
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`x*y**2*z**2` does not divide `x**3*y**4*z`.
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"""
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C = monomial_ldiv(A, B)
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if all(c >= 0 for c in C):
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return tuple(C)
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else:
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return None
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def monomial_ldiv(A, B):
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"""
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Division of tuples representing monomials.
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Examples
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========
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Lets divide `x**3*y**4*z` by `x*y**2`::
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>>> from sympy.polys.monomials import monomial_ldiv
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>>> monomial_ldiv((3, 4, 1), (1, 2, 0))
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(2, 2, 1)
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which gives `x**2*y**2*z`.
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>>> monomial_ldiv((3, 4, 1), (1, 2, 2))
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(2, 2, -1)
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which gives `x**2*y**2*z**-1`.
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"""
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return tuple([ a - b for a, b in zip(A, B) ])
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def monomial_pow(A, n):
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"""Return the n-th pow of the monomial. """
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return tuple([ a*n for a in A ])
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def monomial_gcd(A, B):
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"""
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Greatest common divisor of tuples representing monomials.
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Examples
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========
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Lets compute GCD of `x*y**4*z` and `x**3*y**2`::
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>>> from sympy.polys.monomials import monomial_gcd
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>>> monomial_gcd((1, 4, 1), (3, 2, 0))
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(1, 2, 0)
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which gives `x*y**2`.
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"""
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return tuple([ min(a, b) for a, b in zip(A, B) ])
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def monomial_lcm(A, B):
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"""
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Least common multiple of tuples representing monomials.
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Examples
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========
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Lets compute LCM of `x*y**4*z` and `x**3*y**2`::
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>>> from sympy.polys.monomials import monomial_lcm
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>>> monomial_lcm((1, 4, 1), (3, 2, 0))
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(3, 4, 1)
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which gives `x**3*y**4*z`.
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"""
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return tuple([ max(a, b) for a, b in zip(A, B) ])
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def monomial_divides(A, B):
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"""
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Does there exist a monomial X such that XA == B?
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Examples
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========
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>>> from sympy.polys.monomials import monomial_divides
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>>> monomial_divides((1, 2), (3, 4))
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True
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>>> monomial_divides((1, 2), (0, 2))
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False
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"""
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return all(a <= b for a, b in zip(A, B))
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def monomial_max(*monoms):
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"""
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Returns maximal degree for each variable in a set of monomials.
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Examples
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========
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Consider monomials `x**3*y**4*z**5`, `y**5*z` and `x**6*y**3*z**9`.
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We wish to find out what is the maximal degree for each of `x`, `y`
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and `z` variables::
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>>> from sympy.polys.monomials import monomial_max
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>>> monomial_max((3,4,5), (0,5,1), (6,3,9))
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(6, 5, 9)
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"""
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M = list(monoms[0])
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for N in monoms[1:]:
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for i, n in enumerate(N):
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M[i] = max(M[i], n)
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return tuple(M)
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def monomial_min(*monoms):
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"""
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Returns minimal degree for each variable in a set of monomials.
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Examples
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========
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Consider monomials `x**3*y**4*z**5`, `y**5*z` and `x**6*y**3*z**9`.
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We wish to find out what is the minimal degree for each of `x`, `y`
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and `z` variables::
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>>> from sympy.polys.monomials import monomial_min
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>>> monomial_min((3,4,5), (0,5,1), (6,3,9))
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(0, 3, 1)
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"""
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M = list(monoms[0])
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for N in monoms[1:]:
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for i, n in enumerate(N):
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M[i] = min(M[i], n)
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return tuple(M)
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def monomial_deg(M):
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"""
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Returns the total degree of a monomial.
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Examples
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========
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The total degree of `xy^2` is 3:
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>>> from sympy.polys.monomials import monomial_deg
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>>> monomial_deg((1, 2))
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3
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"""
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||
|
|
return sum(M)
|
||
|
|
|
||
|
|
def term_div(a, b, domain):
|
||
|
|
"""Division of two terms in over a ring/field. """
|
||
|
|
a_lm, a_lc = a
|
||
|
|
b_lm, b_lc = b
|
||
|
|
|
||
|
|
monom = monomial_div(a_lm, b_lm)
|
||
|
|
|
||
|
|
if domain.is_Field:
|
||
|
|
if monom is not None:
|
||
|
|
return monom, domain.quo(a_lc, b_lc)
|
||
|
|
else:
|
||
|
|
return None
|
||
|
|
else:
|
||
|
|
if not (monom is None or a_lc % b_lc):
|
||
|
|
return monom, domain.quo(a_lc, b_lc)
|
||
|
|
else:
|
||
|
|
return None
|
||
|
|
|
||
|
|
class MonomialOps:
|
||
|
|
"""Code generator of fast monomial arithmetic functions. """
|
||
|
|
|
||
|
|
def __init__(self, ngens):
|
||
|
|
self.ngens = ngens
|
||
|
|
|
||
|
|
def _build(self, code, name):
|
||
|
|
ns = {}
|
||
|
|
exec(code, ns)
|
||
|
|
return ns[name]
|
||
|
|
|
||
|
|
def _vars(self, name):
|
||
|
|
return [ "%s%s" % (name, i) for i in range(self.ngens) ]
|
||
|
|
|
||
|
|
def mul(self):
|
||
|
|
name = "monomial_mul"
|
||
|
|
template = dedent("""\
|
||
|
|
def %(name)s(A, B):
|
||
|
|
(%(A)s,) = A
|
||
|
|
(%(B)s,) = B
|
||
|
|
return (%(AB)s,)
|
||
|
|
""")
|
||
|
|
A = self._vars("a")
|
||
|
|
B = self._vars("b")
|
||
|
|
AB = [ "%s + %s" % (a, b) for a, b in zip(A, B) ]
|
||
|
|
code = template % {"name": name, "A": ", ".join(A), "B": ", ".join(B), "AB": ", ".join(AB)}
|
||
|
|
return self._build(code, name)
|
||
|
|
|
||
|
|
def pow(self):
|
||
|
|
name = "monomial_pow"
|
||
|
|
template = dedent("""\
|
||
|
|
def %(name)s(A, k):
|
||
|
|
(%(A)s,) = A
|
||
|
|
return (%(Ak)s,)
|
||
|
|
""")
|
||
|
|
A = self._vars("a")
|
||
|
|
Ak = [ "%s*k" % a for a in A ]
|
||
|
|
code = template % {"name": name, "A": ", ".join(A), "Ak": ", ".join(Ak)}
|
||
|
|
return self._build(code, name)
|
||
|
|
|
||
|
|
def mulpow(self):
|
||
|
|
name = "monomial_mulpow"
|
||
|
|
template = dedent("""\
|
||
|
|
def %(name)s(A, B, k):
|
||
|
|
(%(A)s,) = A
|
||
|
|
(%(B)s,) = B
|
||
|
|
return (%(ABk)s,)
|
||
|
|
""")
|
||
|
|
A = self._vars("a")
|
||
|
|
B = self._vars("b")
|
||
|
|
ABk = [ "%s + %s*k" % (a, b) for a, b in zip(A, B) ]
|
||
|
|
code = template % {"name": name, "A": ", ".join(A), "B": ", ".join(B), "ABk": ", ".join(ABk)}
|
||
|
|
return self._build(code, name)
|
||
|
|
|
||
|
|
def ldiv(self):
|
||
|
|
name = "monomial_ldiv"
|
||
|
|
template = dedent("""\
|
||
|
|
def %(name)s(A, B):
|
||
|
|
(%(A)s,) = A
|
||
|
|
(%(B)s,) = B
|
||
|
|
return (%(AB)s,)
|
||
|
|
""")
|
||
|
|
A = self._vars("a")
|
||
|
|
B = self._vars("b")
|
||
|
|
AB = [ "%s - %s" % (a, b) for a, b in zip(A, B) ]
|
||
|
|
code = template % {"name": name, "A": ", ".join(A), "B": ", ".join(B), "AB": ", ".join(AB)}
|
||
|
|
return self._build(code, name)
|
||
|
|
|
||
|
|
def div(self):
|
||
|
|
name = "monomial_div"
|
||
|
|
template = dedent("""\
|
||
|
|
def %(name)s(A, B):
|
||
|
|
(%(A)s,) = A
|
||
|
|
(%(B)s,) = B
|
||
|
|
%(RAB)s
|
||
|
|
return (%(R)s,)
|
||
|
|
""")
|
||
|
|
A = self._vars("a")
|
||
|
|
B = self._vars("b")
|
||
|
|
RAB = [ "r%(i)s = a%(i)s - b%(i)s\n if r%(i)s < 0: return None" % {"i": i} for i in range(self.ngens) ]
|
||
|
|
R = self._vars("r")
|
||
|
|
code = template % {"name": name, "A": ", ".join(A), "B": ", ".join(B), "RAB": "\n ".join(RAB), "R": ", ".join(R)}
|
||
|
|
return self._build(code, name)
|
||
|
|
|
||
|
|
def lcm(self):
|
||
|
|
name = "monomial_lcm"
|
||
|
|
template = dedent("""\
|
||
|
|
def %(name)s(A, B):
|
||
|
|
(%(A)s,) = A
|
||
|
|
(%(B)s,) = B
|
||
|
|
return (%(AB)s,)
|
||
|
|
""")
|
||
|
|
A = self._vars("a")
|
||
|
|
B = self._vars("b")
|
||
|
|
AB = [ "%s if %s >= %s else %s" % (a, a, b, b) for a, b in zip(A, B) ]
|
||
|
|
code = template % {"name": name, "A": ", ".join(A), "B": ", ".join(B), "AB": ", ".join(AB)}
|
||
|
|
return self._build(code, name)
|
||
|
|
|
||
|
|
def gcd(self):
|
||
|
|
name = "monomial_gcd"
|
||
|
|
template = dedent("""\
|
||
|
|
def %(name)s(A, B):
|
||
|
|
(%(A)s,) = A
|
||
|
|
(%(B)s,) = B
|
||
|
|
return (%(AB)s,)
|
||
|
|
""")
|
||
|
|
A = self._vars("a")
|
||
|
|
B = self._vars("b")
|
||
|
|
AB = [ "%s if %s <= %s else %s" % (a, a, b, b) for a, b in zip(A, B) ]
|
||
|
|
code = template % {"name": name, "A": ", ".join(A), "B": ", ".join(B), "AB": ", ".join(AB)}
|
||
|
|
return self._build(code, name)
|
||
|
|
|
||
|
|
@public
|
||
|
|
class Monomial(PicklableWithSlots):
|
||
|
|
"""Class representing a monomial, i.e. a product of powers. """
|
||
|
|
|
||
|
|
__slots__ = ('exponents', 'gens')
|
||
|
|
|
||
|
|
def __init__(self, monom, gens=None):
|
||
|
|
if not iterable(monom):
|
||
|
|
rep, gens = dict_from_expr(sympify(monom), gens=gens)
|
||
|
|
if len(rep) == 1 and list(rep.values())[0] == 1:
|
||
|
|
monom = list(rep.keys())[0]
|
||
|
|
else:
|
||
|
|
raise ValueError("Expected a monomial got {}".format(monom))
|
||
|
|
|
||
|
|
self.exponents = tuple(map(int, monom))
|
||
|
|
self.gens = gens
|
||
|
|
|
||
|
|
def rebuild(self, exponents, gens=None):
|
||
|
|
return self.__class__(exponents, gens or self.gens)
|
||
|
|
|
||
|
|
def __len__(self):
|
||
|
|
return len(self.exponents)
|
||
|
|
|
||
|
|
def __iter__(self):
|
||
|
|
return iter(self.exponents)
|
||
|
|
|
||
|
|
def __getitem__(self, item):
|
||
|
|
return self.exponents[item]
|
||
|
|
|
||
|
|
def __hash__(self):
|
||
|
|
return hash((self.__class__.__name__, self.exponents, self.gens))
|
||
|
|
|
||
|
|
def __str__(self):
|
||
|
|
if self.gens:
|
||
|
|
return "*".join([ "%s**%s" % (gen, exp) for gen, exp in zip(self.gens, self.exponents) ])
|
||
|
|
else:
|
||
|
|
return "%s(%s)" % (self.__class__.__name__, self.exponents)
|
||
|
|
|
||
|
|
def as_expr(self, *gens):
|
||
|
|
"""Convert a monomial instance to a SymPy expression. """
|
||
|
|
gens = gens or self.gens
|
||
|
|
|
||
|
|
if not gens:
|
||
|
|
raise ValueError(
|
||
|
|
"Cannot convert %s to an expression without generators" % self)
|
||
|
|
|
||
|
|
return Mul(*[ gen**exp for gen, exp in zip(gens, self.exponents) ])
|
||
|
|
|
||
|
|
def __eq__(self, other):
|
||
|
|
if isinstance(other, Monomial):
|
||
|
|
exponents = other.exponents
|
||
|
|
elif isinstance(other, (tuple, Tuple)):
|
||
|
|
exponents = other
|
||
|
|
else:
|
||
|
|
return False
|
||
|
|
|
||
|
|
return self.exponents == exponents
|
||
|
|
|
||
|
|
def __ne__(self, other):
|
||
|
|
return not self == other
|
||
|
|
|
||
|
|
def __mul__(self, other):
|
||
|
|
if isinstance(other, Monomial):
|
||
|
|
exponents = other.exponents
|
||
|
|
elif isinstance(other, (tuple, Tuple)):
|
||
|
|
exponents = other
|
||
|
|
else:
|
||
|
|
raise NotImplementedError
|
||
|
|
|
||
|
|
return self.rebuild(monomial_mul(self.exponents, exponents))
|
||
|
|
|
||
|
|
def __truediv__(self, other):
|
||
|
|
if isinstance(other, Monomial):
|
||
|
|
exponents = other.exponents
|
||
|
|
elif isinstance(other, (tuple, Tuple)):
|
||
|
|
exponents = other
|
||
|
|
else:
|
||
|
|
raise NotImplementedError
|
||
|
|
|
||
|
|
result = monomial_div(self.exponents, exponents)
|
||
|
|
|
||
|
|
if result is not None:
|
||
|
|
return self.rebuild(result)
|
||
|
|
else:
|
||
|
|
raise ExactQuotientFailed(self, Monomial(other))
|
||
|
|
|
||
|
|
__floordiv__ = __truediv__
|
||
|
|
|
||
|
|
def __pow__(self, other):
|
||
|
|
n = int(other)
|
||
|
|
|
||
|
|
if not n:
|
||
|
|
return self.rebuild([0]*len(self))
|
||
|
|
elif n > 0:
|
||
|
|
exponents = self.exponents
|
||
|
|
|
||
|
|
for i in range(1, n):
|
||
|
|
exponents = monomial_mul(exponents, self.exponents)
|
||
|
|
|
||
|
|
return self.rebuild(exponents)
|
||
|
|
else:
|
||
|
|
raise ValueError("a non-negative integer expected, got %s" % other)
|
||
|
|
|
||
|
|
def gcd(self, other):
|
||
|
|
"""Greatest common divisor of monomials. """
|
||
|
|
if isinstance(other, Monomial):
|
||
|
|
exponents = other.exponents
|
||
|
|
elif isinstance(other, (tuple, Tuple)):
|
||
|
|
exponents = other
|
||
|
|
else:
|
||
|
|
raise TypeError(
|
||
|
|
"an instance of Monomial class expected, got %s" % other)
|
||
|
|
|
||
|
|
return self.rebuild(monomial_gcd(self.exponents, exponents))
|
||
|
|
|
||
|
|
def lcm(self, other):
|
||
|
|
"""Least common multiple of monomials. """
|
||
|
|
if isinstance(other, Monomial):
|
||
|
|
exponents = other.exponents
|
||
|
|
elif isinstance(other, (tuple, Tuple)):
|
||
|
|
exponents = other
|
||
|
|
else:
|
||
|
|
raise TypeError(
|
||
|
|
"an instance of Monomial class expected, got %s" % other)
|
||
|
|
|
||
|
|
return self.rebuild(monomial_lcm(self.exponents, exponents))
|