355 lines
11 KiB
Python
355 lines
11 KiB
Python
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from sympy.concrete.expr_with_limits import ExprWithLimits
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from sympy.core.singleton import S
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from sympy.core.relational import Eq
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class ReorderError(NotImplementedError):
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"""
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Exception raised when trying to reorder dependent limits.
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"""
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def __init__(self, expr, msg):
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super().__init__(
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"%s could not be reordered: %s." % (expr, msg))
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class ExprWithIntLimits(ExprWithLimits):
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"""
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Superclass for Product and Sum.
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See Also
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========
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sympy.concrete.expr_with_limits.ExprWithLimits
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sympy.concrete.products.Product
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sympy.concrete.summations.Sum
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"""
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__slots__ = ()
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def change_index(self, var, trafo, newvar=None):
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r"""
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Change index of a Sum or Product.
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Perform a linear transformation `x \mapsto a x + b` on the index variable
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`x`. For `a` the only values allowed are `\pm 1`. A new variable to be used
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after the change of index can also be specified.
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Explanation
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===========
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``change_index(expr, var, trafo, newvar=None)`` where ``var`` specifies the
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index variable `x` to transform. The transformation ``trafo`` must be linear
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and given in terms of ``var``. If the optional argument ``newvar`` is
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provided then ``var`` gets replaced by ``newvar`` in the final expression.
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Examples
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========
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>>> from sympy import Sum, Product, simplify
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>>> from sympy.abc import x, y, a, b, c, d, u, v, i, j, k, l
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>>> S = Sum(x, (x, a, b))
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>>> S.doit()
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-a**2/2 + a/2 + b**2/2 + b/2
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>>> Sn = S.change_index(x, x + 1, y)
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>>> Sn
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Sum(y - 1, (y, a + 1, b + 1))
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>>> Sn.doit()
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-a**2/2 + a/2 + b**2/2 + b/2
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>>> Sn = S.change_index(x, -x, y)
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>>> Sn
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Sum(-y, (y, -b, -a))
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>>> Sn.doit()
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-a**2/2 + a/2 + b**2/2 + b/2
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>>> Sn = S.change_index(x, x+u)
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>>> Sn
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Sum(-u + x, (x, a + u, b + u))
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>>> Sn.doit()
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-a**2/2 - a*u + a/2 + b**2/2 + b*u + b/2 - u*(-a + b + 1) + u
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>>> simplify(Sn.doit())
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-a**2/2 + a/2 + b**2/2 + b/2
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>>> Sn = S.change_index(x, -x - u, y)
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>>> Sn
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Sum(-u - y, (y, -b - u, -a - u))
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>>> Sn.doit()
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-a**2/2 - a*u + a/2 + b**2/2 + b*u + b/2 - u*(-a + b + 1) + u
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>>> simplify(Sn.doit())
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-a**2/2 + a/2 + b**2/2 + b/2
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>>> P = Product(i*j**2, (i, a, b), (j, c, d))
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>>> P
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Product(i*j**2, (i, a, b), (j, c, d))
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>>> P2 = P.change_index(i, i+3, k)
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>>> P2
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Product(j**2*(k - 3), (k, a + 3, b + 3), (j, c, d))
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>>> P3 = P2.change_index(j, -j, l)
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>>> P3
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Product(l**2*(k - 3), (k, a + 3, b + 3), (l, -d, -c))
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When dealing with symbols only, we can make a
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general linear transformation:
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>>> Sn = S.change_index(x, u*x+v, y)
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>>> Sn
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Sum((-v + y)/u, (y, b*u + v, a*u + v))
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>>> Sn.doit()
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-v*(a*u - b*u + 1)/u + (a**2*u**2/2 + a*u*v + a*u/2 - b**2*u**2/2 - b*u*v + b*u/2 + v)/u
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>>> simplify(Sn.doit())
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a**2*u/2 + a/2 - b**2*u/2 + b/2
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However, the last result can be inconsistent with usual
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summation where the index increment is always 1. This is
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obvious as we get back the original value only for ``u``
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equal +1 or -1.
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See Also
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========
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sympy.concrete.expr_with_intlimits.ExprWithIntLimits.index,
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reorder_limit,
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sympy.concrete.expr_with_intlimits.ExprWithIntLimits.reorder,
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sympy.concrete.summations.Sum.reverse_order,
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sympy.concrete.products.Product.reverse_order
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"""
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if newvar is None:
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newvar = var
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limits = []
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for limit in self.limits:
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if limit[0] == var:
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p = trafo.as_poly(var)
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if p.degree() != 1:
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raise ValueError("Index transformation is not linear")
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alpha = p.coeff_monomial(var)
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beta = p.coeff_monomial(S.One)
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if alpha.is_number:
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if alpha == S.One:
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limits.append((newvar, alpha*limit[1] + beta, alpha*limit[2] + beta))
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elif alpha == S.NegativeOne:
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limits.append((newvar, alpha*limit[2] + beta, alpha*limit[1] + beta))
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else:
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raise ValueError("Linear transformation results in non-linear summation stepsize")
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else:
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# Note that the case of alpha being symbolic can give issues if alpha < 0.
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limits.append((newvar, alpha*limit[2] + beta, alpha*limit[1] + beta))
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else:
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limits.append(limit)
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function = self.function.subs(var, (var - beta)/alpha)
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function = function.subs(var, newvar)
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return self.func(function, *limits)
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def index(expr, x):
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"""
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Return the index of a dummy variable in the list of limits.
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Explanation
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===========
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``index(expr, x)`` returns the index of the dummy variable ``x`` in the
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limits of ``expr``. Note that we start counting with 0 at the inner-most
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limits tuple.
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Examples
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========
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>>> from sympy.abc import x, y, a, b, c, d
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>>> from sympy import Sum, Product
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>>> Sum(x*y, (x, a, b), (y, c, d)).index(x)
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0
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>>> Sum(x*y, (x, a, b), (y, c, d)).index(y)
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1
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>>> Product(x*y, (x, a, b), (y, c, d)).index(x)
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0
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>>> Product(x*y, (x, a, b), (y, c, d)).index(y)
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1
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See Also
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========
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reorder_limit, reorder, sympy.concrete.summations.Sum.reverse_order,
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sympy.concrete.products.Product.reverse_order
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"""
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variables = [limit[0] for limit in expr.limits]
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if variables.count(x) != 1:
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raise ValueError(expr, "Number of instances of variable not equal to one")
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else:
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return variables.index(x)
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def reorder(expr, *arg):
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"""
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Reorder limits in a expression containing a Sum or a Product.
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Explanation
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===========
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``expr.reorder(*arg)`` reorders the limits in the expression ``expr``
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according to the list of tuples given by ``arg``. These tuples can
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contain numerical indices or index variable names or involve both.
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Examples
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========
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>>> from sympy import Sum, Product
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>>> from sympy.abc import x, y, z, a, b, c, d, e, f
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>>> Sum(x*y, (x, a, b), (y, c, d)).reorder((x, y))
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Sum(x*y, (y, c, d), (x, a, b))
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>>> Sum(x*y*z, (x, a, b), (y, c, d), (z, e, f)).reorder((x, y), (x, z), (y, z))
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Sum(x*y*z, (z, e, f), (y, c, d), (x, a, b))
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>>> P = Product(x*y*z, (x, a, b), (y, c, d), (z, e, f))
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>>> P.reorder((x, y), (x, z), (y, z))
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Product(x*y*z, (z, e, f), (y, c, d), (x, a, b))
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We can also select the index variables by counting them, starting
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with the inner-most one:
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>>> Sum(x**2, (x, a, b), (x, c, d)).reorder((0, 1))
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Sum(x**2, (x, c, d), (x, a, b))
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And of course we can mix both schemes:
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>>> Sum(x*y, (x, a, b), (y, c, d)).reorder((y, x))
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Sum(x*y, (y, c, d), (x, a, b))
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>>> Sum(x*y, (x, a, b), (y, c, d)).reorder((y, 0))
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Sum(x*y, (y, c, d), (x, a, b))
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See Also
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========
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reorder_limit, index, sympy.concrete.summations.Sum.reverse_order,
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sympy.concrete.products.Product.reverse_order
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"""
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new_expr = expr
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for r in arg:
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if len(r) != 2:
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raise ValueError(r, "Invalid number of arguments")
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index1 = r[0]
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index2 = r[1]
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if not isinstance(r[0], int):
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index1 = expr.index(r[0])
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if not isinstance(r[1], int):
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index2 = expr.index(r[1])
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new_expr = new_expr.reorder_limit(index1, index2)
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return new_expr
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def reorder_limit(expr, x, y):
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"""
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Interchange two limit tuples of a Sum or Product expression.
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Explanation
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===========
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``expr.reorder_limit(x, y)`` interchanges two limit tuples. The
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arguments ``x`` and ``y`` are integers corresponding to the index
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variables of the two limits which are to be interchanged. The
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expression ``expr`` has to be either a Sum or a Product.
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Examples
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========
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>>> from sympy.abc import x, y, z, a, b, c, d, e, f
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>>> from sympy import Sum, Product
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>>> Sum(x*y*z, (x, a, b), (y, c, d), (z, e, f)).reorder_limit(0, 2)
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Sum(x*y*z, (z, e, f), (y, c, d), (x, a, b))
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>>> Sum(x**2, (x, a, b), (x, c, d)).reorder_limit(1, 0)
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Sum(x**2, (x, c, d), (x, a, b))
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>>> Product(x*y*z, (x, a, b), (y, c, d), (z, e, f)).reorder_limit(0, 2)
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Product(x*y*z, (z, e, f), (y, c, d), (x, a, b))
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See Also
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========
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index, reorder, sympy.concrete.summations.Sum.reverse_order,
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sympy.concrete.products.Product.reverse_order
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"""
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var = {limit[0] for limit in expr.limits}
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limit_x = expr.limits[x]
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limit_y = expr.limits[y]
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if (len(set(limit_x[1].free_symbols).intersection(var)) == 0 and
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len(set(limit_x[2].free_symbols).intersection(var)) == 0 and
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len(set(limit_y[1].free_symbols).intersection(var)) == 0 and
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len(set(limit_y[2].free_symbols).intersection(var)) == 0):
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limits = []
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for i, limit in enumerate(expr.limits):
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if i == x:
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limits.append(limit_y)
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elif i == y:
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limits.append(limit_x)
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else:
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limits.append(limit)
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return type(expr)(expr.function, *limits)
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else:
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raise ReorderError(expr, "could not interchange the two limits specified")
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@property
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def has_empty_sequence(self):
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"""
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Returns True if the Sum or Product is computed for an empty sequence.
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Examples
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========
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>>> from sympy import Sum, Product, Symbol
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>>> m = Symbol('m')
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>>> Sum(m, (m, 1, 0)).has_empty_sequence
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True
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>>> Sum(m, (m, 1, 1)).has_empty_sequence
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False
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>>> M = Symbol('M', integer=True, positive=True)
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>>> Product(m, (m, 1, M)).has_empty_sequence
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False
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>>> Product(m, (m, 2, M)).has_empty_sequence
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>>> Product(m, (m, M + 1, M)).has_empty_sequence
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True
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>>> N = Symbol('N', integer=True, positive=True)
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>>> Sum(m, (m, N, M)).has_empty_sequence
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>>> N = Symbol('N', integer=True, negative=True)
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>>> Sum(m, (m, N, M)).has_empty_sequence
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False
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See Also
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========
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has_reversed_limits
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has_finite_limits
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"""
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ret_None = False
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for lim in self.limits:
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dif = lim[1] - lim[2]
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eq = Eq(dif, 1)
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if eq == True:
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return True
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elif eq == False:
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continue
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else:
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ret_None = True
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if ret_None:
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return None
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return False
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