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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.cache import cacheit |
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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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if is_sequence(max_degrees): |
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n = len(variables) |
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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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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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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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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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variables = list(variables) + [S.One] |
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if all(variable.is_commutative for variable in variables): |
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it = combinations_with_replacement(variables, max_degree) |
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else: |
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it = product(variables, repeat=max_degree) |
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monomials_set = set() |
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d = max_degree - min_degree |
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for item in it: |
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count = 0 |
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for variable in item: |
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if variable == 1: |
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count += 1 |
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if d < count: |
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break |
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else: |
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monomials_set.add(Mul(*item)) |
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yield from monomials_set |
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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) |
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def term_div(a, b, domain): |
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"""Division of two terms in over a ring/field. """ |
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a_lm, a_lc = a |
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b_lm, b_lc = b |
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monom = monomial_div(a_lm, b_lm) |
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if domain.is_Field: |
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if monom is not None: |
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return monom, domain.quo(a_lc, b_lc) |
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else: |
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return None |
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else: |
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if not (monom is None or a_lc % b_lc): |
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return monom, domain.quo(a_lc, b_lc) |
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else: |
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return None |
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class MonomialOps: |
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"""Code generator of fast monomial arithmetic functions. """ |
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@cacheit |
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def __new__(cls, ngens): |
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obj = super().__new__(cls) |
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obj.ngens = ngens |
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return obj |
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def __getnewargs__(self): |
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return (self.ngens,) |
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def _build(self, code, name): |
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ns = {} |
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exec(code, ns) |
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return ns[name] |
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def _vars(self, name): |
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return [ "%s%s" % (name, i) for i in range(self.ngens) ] |
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@cacheit |
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def mul(self): |
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name = "monomial_mul" |
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template = dedent("""\ |
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def %(name)s(A, B): |
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(%(A)s,) = A |
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(%(B)s,) = B |
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return (%(AB)s,) |
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""") |
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A = self._vars("a") |
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B = self._vars("b") |
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AB = [ "%s + %s" % (a, b) for a, b in zip(A, B) ] |
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code = template % {"name": name, "A": ", ".join(A), "B": ", ".join(B), "AB": ", ".join(AB)} |
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return self._build(code, name) |
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@cacheit |
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def pow(self): |
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name = "monomial_pow" |
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template = dedent("""\ |
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def %(name)s(A, k): |
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(%(A)s,) = A |
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return (%(Ak)s,) |
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""") |
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A = self._vars("a") |
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Ak = [ "%s*k" % a for a in A ] |
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code = template % {"name": name, "A": ", ".join(A), "Ak": ", ".join(Ak)} |
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return self._build(code, name) |
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@cacheit |
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def mulpow(self): |
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name = "monomial_mulpow" |
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template = dedent("""\ |
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def %(name)s(A, B, k): |
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(%(A)s,) = A |
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(%(B)s,) = B |
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return (%(ABk)s,) |
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""") |
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A = self._vars("a") |
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B = self._vars("b") |
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ABk = [ "%s + %s*k" % (a, b) for a, b in zip(A, B) ] |
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code = template % {"name": name, "A": ", ".join(A), "B": ", ".join(B), "ABk": ", ".join(ABk)} |
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return self._build(code, name) |
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@cacheit |
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def ldiv(self): |
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name = "monomial_ldiv" |
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template = dedent("""\ |
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def %(name)s(A, B): |
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(%(A)s,) = A |
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(%(B)s,) = B |
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return (%(AB)s,) |
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""") |
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A = self._vars("a") |
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B = self._vars("b") |
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AB = [ "%s - %s" % (a, b) for a, b in zip(A, B) ] |
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code = template % {"name": name, "A": ", ".join(A), "B": ", ".join(B), "AB": ", ".join(AB)} |
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return self._build(code, name) |
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@cacheit |
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def div(self): |
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name = "monomial_div" |
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template = dedent("""\ |
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def %(name)s(A, B): |
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(%(A)s,) = A |
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(%(B)s,) = B |
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%(RAB)s |
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return (%(R)s,) |
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""") |
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A = self._vars("a") |
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B = self._vars("b") |
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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) ] |
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R = self._vars("r") |
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code = template % {"name": name, "A": ", ".join(A), "B": ", ".join(B), "RAB": "\n ".join(RAB), "R": ", ".join(R)} |
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return self._build(code, name) |
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@cacheit |
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def lcm(self): |
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name = "monomial_lcm" |
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template = dedent("""\ |
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def %(name)s(A, B): |
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(%(A)s,) = A |
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(%(B)s,) = B |
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return (%(AB)s,) |
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""") |
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A = self._vars("a") |
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B = self._vars("b") |
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AB = [ "%s if %s >= %s else %s" % (a, a, b, b) for a, b in zip(A, B) ] |
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code = template % {"name": name, "A": ", ".join(A), "B": ", ".join(B), "AB": ", ".join(AB)} |
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return self._build(code, name) |
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@cacheit |
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def gcd(self): |
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name = "monomial_gcd" |
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template = dedent("""\ |
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def %(name)s(A, B): |
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(%(A)s,) = A |
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(%(B)s,) = B |
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return (%(AB)s,) |
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""") |
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A = self._vars("a") |
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B = self._vars("b") |
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AB = [ "%s if %s <= %s else %s" % (a, a, b, b) for a, b in zip(A, B) ] |
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code = template % {"name": name, "A": ", ".join(A), "B": ", ".join(B), "AB": ", ".join(AB)} |
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return self._build(code, name) |
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@public |
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class Monomial(PicklableWithSlots): |
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"""Class representing a monomial, i.e. a product of powers. """ |
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__slots__ = ('exponents', 'gens') |
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def __init__(self, monom, gens=None): |
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if not iterable(monom): |
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rep, gens = dict_from_expr(sympify(monom), gens=gens) |
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if len(rep) == 1 and list(rep.values())[0] == 1: |
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monom = list(rep.keys())[0] |
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else: |
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raise ValueError("Expected a monomial got {}".format(monom)) |
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self.exponents = tuple(map(int, monom)) |
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self.gens = gens |
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def rebuild(self, exponents, gens=None): |
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return self.__class__(exponents, gens or self.gens) |
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def __len__(self): |
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return len(self.exponents) |
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def __iter__(self): |
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return iter(self.exponents) |
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def __getitem__(self, item): |
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return self.exponents[item] |
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def __hash__(self): |
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return hash((self.__class__.__name__, self.exponents, self.gens)) |
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def __str__(self): |
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if self.gens: |
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return "*".join([ "%s**%s" % (gen, exp) for gen, exp in zip(self.gens, self.exponents) ]) |
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else: |
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return "%s(%s)" % (self.__class__.__name__, self.exponents) |
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def as_expr(self, *gens): |
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"""Convert a monomial instance to a SymPy expression. """ |
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gens = gens or self.gens |
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if not gens: |
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raise ValueError( |
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"Cannot convert %s to an expression without generators" % self) |
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return Mul(*[ gen**exp for gen, exp in zip(gens, self.exponents) ]) |
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def __eq__(self, other): |
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if isinstance(other, Monomial): |
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exponents = other.exponents |
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elif isinstance(other, (tuple, Tuple)): |
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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 n < 0: |
|
raise ValueError("a non-negative integer expected, got %s" % other) |
|
return self.rebuild(monomial_pow(self.exponents, n)) |
|
|
|
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)) |
|
|