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798 lines
24 KiB
798 lines
24 KiB
r"""Module that defines indexed objects.
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The classes ``IndexedBase``, ``Indexed``, and ``Idx`` represent a
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matrix element ``M[i, j]`` as in the following diagram::
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1) The Indexed class represents the entire indexed object.
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___|___
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' '
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M[i, j]
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/ \__\______
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| |
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| |
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| 2) The Idx class represents indices; each Idx can
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| optionally contain information about its range.
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3) IndexedBase represents the 'stem' of an indexed object, here `M`.
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The stem used by itself is usually taken to represent the entire
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array.
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There can be any number of indices on an Indexed object. No
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transformation properties are implemented in these Base objects, but
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implicit contraction of repeated indices is supported.
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Note that the support for complicated (i.e. non-atomic) integer
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expressions as indices is limited. (This should be improved in
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future releases.)
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Examples
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========
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To express the above matrix element example you would write:
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>>> from sympy import symbols, IndexedBase, Idx
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>>> M = IndexedBase('M')
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>>> i, j = symbols('i j', cls=Idx)
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>>> M[i, j]
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M[i, j]
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Repeated indices in a product implies a summation, so to express a
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matrix-vector product in terms of Indexed objects:
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>>> x = IndexedBase('x')
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>>> M[i, j]*x[j]
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M[i, j]*x[j]
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If the indexed objects will be converted to component based arrays, e.g.
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with the code printers or the autowrap framework, you also need to provide
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(symbolic or numerical) dimensions. This can be done by passing an
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optional shape parameter to IndexedBase upon construction:
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>>> dim1, dim2 = symbols('dim1 dim2', integer=True)
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>>> A = IndexedBase('A', shape=(dim1, 2*dim1, dim2))
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>>> A.shape
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(dim1, 2*dim1, dim2)
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>>> A[i, j, 3].shape
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(dim1, 2*dim1, dim2)
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If an IndexedBase object has no shape information, it is assumed that the
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array is as large as the ranges of its indices:
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>>> n, m = symbols('n m', integer=True)
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>>> i = Idx('i', m)
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>>> j = Idx('j', n)
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>>> M[i, j].shape
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(m, n)
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>>> M[i, j].ranges
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[(0, m - 1), (0, n - 1)]
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The above can be compared with the following:
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>>> A[i, 2, j].shape
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(dim1, 2*dim1, dim2)
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>>> A[i, 2, j].ranges
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[(0, m - 1), None, (0, n - 1)]
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To analyze the structure of indexed expressions, you can use the methods
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get_indices() and get_contraction_structure():
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>>> from sympy.tensor import get_indices, get_contraction_structure
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>>> get_indices(A[i, j, j])
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({i}, {})
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>>> get_contraction_structure(A[i, j, j])
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{(j,): {A[i, j, j]}}
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See the appropriate docstrings for a detailed explanation of the output.
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"""
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# TODO: (some ideas for improvement)
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#
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# o test and guarantee numpy compatibility
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# - implement full support for broadcasting
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# - strided arrays
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#
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# o more functions to analyze indexed expressions
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# - identify standard constructs, e.g matrix-vector product in a subexpression
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#
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# o functions to generate component based arrays (numpy and sympy.Matrix)
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# - generate a single array directly from Indexed
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# - convert simple sub-expressions
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#
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# o sophisticated indexing (possibly in subclasses to preserve simplicity)
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# - Idx with range smaller than dimension of Indexed
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# - Idx with stepsize != 1
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# - Idx with step determined by function call
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from collections.abc import Iterable
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from sympy.core.numbers import Number
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from sympy.core.assumptions import StdFactKB
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from sympy.core import Expr, Tuple, sympify, S
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from sympy.core.symbol import _filter_assumptions, Symbol
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from sympy.core.logic import fuzzy_bool, fuzzy_not
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from sympy.core.sympify import _sympify
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from sympy.functions.special.tensor_functions import KroneckerDelta
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from sympy.multipledispatch import dispatch
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from sympy.utilities.iterables import is_sequence, NotIterable
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from sympy.utilities.misc import filldedent
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class IndexException(Exception):
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pass
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class Indexed(Expr):
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"""Represents a mathematical object with indices.
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>>> from sympy import Indexed, IndexedBase, Idx, symbols
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>>> i, j = symbols('i j', cls=Idx)
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>>> Indexed('A', i, j)
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A[i, j]
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It is recommended that ``Indexed`` objects be created by indexing ``IndexedBase``:
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``IndexedBase('A')[i, j]`` instead of ``Indexed(IndexedBase('A'), i, j)``.
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>>> A = IndexedBase('A')
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>>> a_ij = A[i, j] # Prefer this,
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>>> b_ij = Indexed(A, i, j) # over this.
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>>> a_ij == b_ij
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True
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"""
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is_commutative = True
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is_Indexed = True
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is_symbol = True
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is_Atom = True
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def __new__(cls, base, *args, **kw_args):
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from sympy.tensor.array.ndim_array import NDimArray
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from sympy.matrices.matrices import MatrixBase
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if not args:
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raise IndexException("Indexed needs at least one index.")
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if isinstance(base, (str, Symbol)):
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base = IndexedBase(base)
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elif not hasattr(base, '__getitem__') and not isinstance(base, IndexedBase):
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raise TypeError(filldedent("""
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The base can only be replaced with a string, Symbol,
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IndexedBase or an object with a method for getting
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items (i.e. an object with a `__getitem__` method).
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"""))
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args = list(map(sympify, args))
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if isinstance(base, (NDimArray, Iterable, Tuple, MatrixBase)) and all(i.is_number for i in args):
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if len(args) == 1:
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return base[args[0]]
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else:
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return base[args]
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base = _sympify(base)
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obj = Expr.__new__(cls, base, *args, **kw_args)
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try:
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IndexedBase._set_assumptions(obj, base.assumptions0)
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except AttributeError:
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IndexedBase._set_assumptions(obj, {})
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return obj
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def _hashable_content(self):
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return super()._hashable_content() + tuple(sorted(self.assumptions0.items()))
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@property
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def name(self):
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return str(self)
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@property
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def _diff_wrt(self):
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"""Allow derivatives with respect to an ``Indexed`` object."""
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return True
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def _eval_derivative(self, wrt):
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from sympy.tensor.array.ndim_array import NDimArray
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if isinstance(wrt, Indexed) and wrt.base == self.base:
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if len(self.indices) != len(wrt.indices):
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msg = "Different # of indices: d({!s})/d({!s})".format(self,
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wrt)
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raise IndexException(msg)
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result = S.One
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for index1, index2 in zip(self.indices, wrt.indices):
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result *= KroneckerDelta(index1, index2)
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return result
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elif isinstance(self.base, NDimArray):
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from sympy.tensor.array import derive_by_array
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return Indexed(derive_by_array(self.base, wrt), *self.args[1:])
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else:
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if Tuple(self.indices).has(wrt):
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return S.NaN
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return S.Zero
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@property
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def assumptions0(self):
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return {k: v for k, v in self._assumptions.items() if v is not None}
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@property
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def base(self):
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"""Returns the ``IndexedBase`` of the ``Indexed`` object.
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Examples
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========
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>>> from sympy import Indexed, IndexedBase, Idx, symbols
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>>> i, j = symbols('i j', cls=Idx)
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>>> Indexed('A', i, j).base
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A
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>>> B = IndexedBase('B')
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>>> B == B[i, j].base
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True
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"""
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return self.args[0]
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@property
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def indices(self):
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"""
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Returns the indices of the ``Indexed`` object.
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Examples
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========
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>>> from sympy import Indexed, Idx, symbols
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>>> i, j = symbols('i j', cls=Idx)
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>>> Indexed('A', i, j).indices
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(i, j)
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"""
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return self.args[1:]
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@property
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def rank(self):
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"""
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Returns the rank of the ``Indexed`` object.
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Examples
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========
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>>> from sympy import Indexed, Idx, symbols
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>>> i, j, k, l, m = symbols('i:m', cls=Idx)
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>>> Indexed('A', i, j).rank
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2
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>>> q = Indexed('A', i, j, k, l, m)
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>>> q.rank
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5
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>>> q.rank == len(q.indices)
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True
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"""
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return len(self.args) - 1
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@property
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def shape(self):
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"""Returns a list with dimensions of each index.
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Dimensions is a property of the array, not of the indices. Still, if
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the ``IndexedBase`` does not define a shape attribute, it is assumed
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that the ranges of the indices correspond to the shape of the array.
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>>> from sympy import IndexedBase, Idx, symbols
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>>> n, m = symbols('n m', integer=True)
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>>> i = Idx('i', m)
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>>> j = Idx('j', m)
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>>> A = IndexedBase('A', shape=(n, n))
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>>> B = IndexedBase('B')
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>>> A[i, j].shape
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(n, n)
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>>> B[i, j].shape
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(m, m)
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"""
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if self.base.shape:
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return self.base.shape
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sizes = []
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for i in self.indices:
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upper = getattr(i, 'upper', None)
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lower = getattr(i, 'lower', None)
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if None in (upper, lower):
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raise IndexException(filldedent("""
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Range is not defined for all indices in: %s""" % self))
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try:
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size = upper - lower + 1
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except TypeError:
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raise IndexException(filldedent("""
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Shape cannot be inferred from Idx with
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undefined range: %s""" % self))
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sizes.append(size)
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return Tuple(*sizes)
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@property
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def ranges(self):
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"""Returns a list of tuples with lower and upper range of each index.
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If an index does not define the data members upper and lower, the
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corresponding slot in the list contains ``None`` instead of a tuple.
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Examples
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========
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>>> from sympy import Indexed,Idx, symbols
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>>> Indexed('A', Idx('i', 2), Idx('j', 4), Idx('k', 8)).ranges
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[(0, 1), (0, 3), (0, 7)]
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>>> Indexed('A', Idx('i', 3), Idx('j', 3), Idx('k', 3)).ranges
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[(0, 2), (0, 2), (0, 2)]
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>>> x, y, z = symbols('x y z', integer=True)
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>>> Indexed('A', x, y, z).ranges
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[None, None, None]
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"""
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ranges = []
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sentinel = object()
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for i in self.indices:
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upper = getattr(i, 'upper', sentinel)
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lower = getattr(i, 'lower', sentinel)
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if sentinel not in (upper, lower):
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ranges.append((lower, upper))
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else:
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ranges.append(None)
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return ranges
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def _sympystr(self, p):
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indices = list(map(p.doprint, self.indices))
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return "%s[%s]" % (p.doprint(self.base), ", ".join(indices))
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@property
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def free_symbols(self):
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base_free_symbols = self.base.free_symbols
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indices_free_symbols = {
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fs for i in self.indices for fs in i.free_symbols}
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if base_free_symbols:
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return {self} | base_free_symbols | indices_free_symbols
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else:
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return indices_free_symbols
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@property
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def expr_free_symbols(self):
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from sympy.utilities.exceptions import sympy_deprecation_warning
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sympy_deprecation_warning("""
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The expr_free_symbols property is deprecated. Use free_symbols to get
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the free symbols of an expression.
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""",
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deprecated_since_version="1.9",
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active_deprecations_target="deprecated-expr-free-symbols")
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return {self}
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class IndexedBase(Expr, NotIterable):
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"""Represent the base or stem of an indexed object
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The IndexedBase class represent an array that contains elements. The main purpose
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of this class is to allow the convenient creation of objects of the Indexed
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class. The __getitem__ method of IndexedBase returns an instance of
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Indexed. Alone, without indices, the IndexedBase class can be used as a
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notation for e.g. matrix equations, resembling what you could do with the
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Symbol class. But, the IndexedBase class adds functionality that is not
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available for Symbol instances:
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- An IndexedBase object can optionally store shape information. This can
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be used in to check array conformance and conditions for numpy
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broadcasting. (TODO)
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- An IndexedBase object implements syntactic sugar that allows easy symbolic
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representation of array operations, using implicit summation of
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repeated indices.
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- The IndexedBase object symbolizes a mathematical structure equivalent
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to arrays, and is recognized as such for code generation and automatic
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compilation and wrapping.
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>>> from sympy.tensor import IndexedBase, Idx
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>>> from sympy import symbols
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>>> A = IndexedBase('A'); A
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A
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>>> type(A)
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<class 'sympy.tensor.indexed.IndexedBase'>
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When an IndexedBase object receives indices, it returns an array with named
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axes, represented by an Indexed object:
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>>> i, j = symbols('i j', integer=True)
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>>> A[i, j, 2]
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A[i, j, 2]
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>>> type(A[i, j, 2])
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<class 'sympy.tensor.indexed.Indexed'>
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The IndexedBase constructor takes an optional shape argument. If given,
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it overrides any shape information in the indices. (But not the index
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ranges!)
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>>> m, n, o, p = symbols('m n o p', integer=True)
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>>> i = Idx('i', m)
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>>> j = Idx('j', n)
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>>> A[i, j].shape
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(m, n)
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>>> B = IndexedBase('B', shape=(o, p))
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>>> B[i, j].shape
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(o, p)
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Assumptions can be specified with keyword arguments the same way as for Symbol:
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>>> A_real = IndexedBase('A', real=True)
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>>> A_real.is_real
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True
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>>> A != A_real
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True
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Assumptions can also be inherited if a Symbol is used to initialize the IndexedBase:
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>>> I = symbols('I', integer=True)
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>>> C_inherit = IndexedBase(I)
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>>> C_explicit = IndexedBase('I', integer=True)
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>>> C_inherit == C_explicit
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True
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"""
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is_commutative = True
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is_symbol = True
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is_Atom = True
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@staticmethod
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def _set_assumptions(obj, assumptions):
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"""Set assumptions on obj, making sure to apply consistent values."""
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tmp_asm_copy = assumptions.copy()
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is_commutative = fuzzy_bool(assumptions.get('commutative', True))
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assumptions['commutative'] = is_commutative
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obj._assumptions = StdFactKB(assumptions)
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obj._assumptions._generator = tmp_asm_copy # Issue #8873
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def __new__(cls, label, shape=None, *, offset=S.Zero, strides=None, **kw_args):
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from sympy.matrices.matrices import MatrixBase
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from sympy.tensor.array.ndim_array import NDimArray
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assumptions, kw_args = _filter_assumptions(kw_args)
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if isinstance(label, str):
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label = Symbol(label, **assumptions)
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elif isinstance(label, Symbol):
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assumptions = label._merge(assumptions)
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elif isinstance(label, (MatrixBase, NDimArray)):
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return label
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elif isinstance(label, Iterable):
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return _sympify(label)
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else:
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label = _sympify(label)
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if is_sequence(shape):
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shape = Tuple(*shape)
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elif shape is not None:
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shape = Tuple(shape)
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if shape is not None:
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obj = Expr.__new__(cls, label, shape)
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else:
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obj = Expr.__new__(cls, label)
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obj._shape = shape
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obj._offset = offset
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obj._strides = strides
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obj._name = str(label)
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IndexedBase._set_assumptions(obj, assumptions)
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return obj
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@property
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def name(self):
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return self._name
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def _hashable_content(self):
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return super()._hashable_content() + tuple(sorted(self.assumptions0.items()))
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@property
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def assumptions0(self):
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return {k: v for k, v in self._assumptions.items() if v is not None}
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def __getitem__(self, indices, **kw_args):
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if is_sequence(indices):
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# Special case needed because M[*my_tuple] is a syntax error.
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if self.shape and len(self.shape) != len(indices):
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raise IndexException("Rank mismatch.")
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return Indexed(self, *indices, **kw_args)
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else:
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if self.shape and len(self.shape) != 1:
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raise IndexException("Rank mismatch.")
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return Indexed(self, indices, **kw_args)
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@property
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def shape(self):
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"""Returns the shape of the ``IndexedBase`` object.
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Examples
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========
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>>> from sympy import IndexedBase, Idx
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>>> from sympy.abc import x, y
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>>> IndexedBase('A', shape=(x, y)).shape
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(x, y)
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Note: If the shape of the ``IndexedBase`` is specified, it will override
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any shape information given by the indices.
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>>> A = IndexedBase('A', shape=(x, y))
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>>> B = IndexedBase('B')
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>>> i = Idx('i', 2)
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>>> j = Idx('j', 1)
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>>> A[i, j].shape
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(x, y)
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>>> B[i, j].shape
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(2, 1)
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"""
|
|
return self._shape
|
|
|
|
@property
|
|
def strides(self):
|
|
"""Returns the strided scheme for the ``IndexedBase`` object.
|
|
|
|
Normally this is a tuple denoting the number of
|
|
steps to take in the respective dimension when traversing
|
|
an array. For code generation purposes strides='C' and
|
|
strides='F' can also be used.
|
|
|
|
strides='C' would mean that code printer would unroll
|
|
in row-major order and 'F' means unroll in column major
|
|
order.
|
|
|
|
"""
|
|
|
|
return self._strides
|
|
|
|
@property
|
|
def offset(self):
|
|
"""Returns the offset for the ``IndexedBase`` object.
|
|
|
|
This is the value added to the resulting index when the
|
|
2D Indexed object is unrolled to a 1D form. Used in code
|
|
generation.
|
|
|
|
Examples
|
|
==========
|
|
>>> from sympy.printing import ccode
|
|
>>> from sympy.tensor import IndexedBase, Idx
|
|
>>> from sympy import symbols
|
|
>>> l, m, n, o = symbols('l m n o', integer=True)
|
|
>>> A = IndexedBase('A', strides=(l, m, n), offset=o)
|
|
>>> i, j, k = map(Idx, 'ijk')
|
|
>>> ccode(A[i, j, k])
|
|
'A[l*i + m*j + n*k + o]'
|
|
|
|
"""
|
|
return self._offset
|
|
|
|
@property
|
|
def label(self):
|
|
"""Returns the label of the ``IndexedBase`` object.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy import IndexedBase
|
|
>>> from sympy.abc import x, y
|
|
>>> IndexedBase('A', shape=(x, y)).label
|
|
A
|
|
|
|
"""
|
|
return self.args[0]
|
|
|
|
def _sympystr(self, p):
|
|
return p.doprint(self.label)
|
|
|
|
|
|
class Idx(Expr):
|
|
"""Represents an integer index as an ``Integer`` or integer expression.
|
|
|
|
There are a number of ways to create an ``Idx`` object. The constructor
|
|
takes two arguments:
|
|
|
|
``label``
|
|
An integer or a symbol that labels the index.
|
|
``range``
|
|
Optionally you can specify a range as either
|
|
|
|
* ``Symbol`` or integer: This is interpreted as a dimension. Lower and
|
|
upper bounds are set to ``0`` and ``range - 1``, respectively.
|
|
* ``tuple``: The two elements are interpreted as the lower and upper
|
|
bounds of the range, respectively.
|
|
|
|
Note: bounds of the range are assumed to be either integer or infinite (oo
|
|
and -oo are allowed to specify an unbounded range). If ``n`` is given as a
|
|
bound, then ``n.is_integer`` must not return false.
|
|
|
|
For convenience, if the label is given as a string it is automatically
|
|
converted to an integer symbol. (Note: this conversion is not done for
|
|
range or dimension arguments.)
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy import Idx, symbols, oo
|
|
>>> n, i, L, U = symbols('n i L U', integer=True)
|
|
|
|
If a string is given for the label an integer ``Symbol`` is created and the
|
|
bounds are both ``None``:
|
|
|
|
>>> idx = Idx('qwerty'); idx
|
|
qwerty
|
|
>>> idx.lower, idx.upper
|
|
(None, None)
|
|
|
|
Both upper and lower bounds can be specified:
|
|
|
|
>>> idx = Idx(i, (L, U)); idx
|
|
i
|
|
>>> idx.lower, idx.upper
|
|
(L, U)
|
|
|
|
When only a single bound is given it is interpreted as the dimension
|
|
and the lower bound defaults to 0:
|
|
|
|
>>> idx = Idx(i, n); idx.lower, idx.upper
|
|
(0, n - 1)
|
|
>>> idx = Idx(i, 4); idx.lower, idx.upper
|
|
(0, 3)
|
|
>>> idx = Idx(i, oo); idx.lower, idx.upper
|
|
(0, oo)
|
|
|
|
"""
|
|
|
|
is_integer = True
|
|
is_finite = True
|
|
is_real = True
|
|
is_symbol = True
|
|
is_Atom = True
|
|
_diff_wrt = True
|
|
|
|
def __new__(cls, label, range=None, **kw_args):
|
|
|
|
if isinstance(label, str):
|
|
label = Symbol(label, integer=True)
|
|
label, range = list(map(sympify, (label, range)))
|
|
|
|
if label.is_Number:
|
|
if not label.is_integer:
|
|
raise TypeError("Index is not an integer number.")
|
|
return label
|
|
|
|
if not label.is_integer:
|
|
raise TypeError("Idx object requires an integer label.")
|
|
|
|
elif is_sequence(range):
|
|
if len(range) != 2:
|
|
raise ValueError(filldedent("""
|
|
Idx range tuple must have length 2, but got %s""" % len(range)))
|
|
for bound in range:
|
|
if (bound.is_integer is False and bound is not S.Infinity
|
|
and bound is not S.NegativeInfinity):
|
|
raise TypeError("Idx object requires integer bounds.")
|
|
args = label, Tuple(*range)
|
|
elif isinstance(range, Expr):
|
|
if range is not S.Infinity and fuzzy_not(range.is_integer):
|
|
raise TypeError("Idx object requires an integer dimension.")
|
|
args = label, Tuple(0, range - 1)
|
|
elif range:
|
|
raise TypeError(filldedent("""
|
|
The range must be an ordered iterable or
|
|
integer SymPy expression."""))
|
|
else:
|
|
args = label,
|
|
|
|
obj = Expr.__new__(cls, *args, **kw_args)
|
|
obj._assumptions["finite"] = True
|
|
obj._assumptions["real"] = True
|
|
return obj
|
|
|
|
@property
|
|
def label(self):
|
|
"""Returns the label (Integer or integer expression) of the Idx object.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy import Idx, Symbol
|
|
>>> x = Symbol('x', integer=True)
|
|
>>> Idx(x).label
|
|
x
|
|
>>> j = Symbol('j', integer=True)
|
|
>>> Idx(j).label
|
|
j
|
|
>>> Idx(j + 1).label
|
|
j + 1
|
|
|
|
"""
|
|
return self.args[0]
|
|
|
|
@property
|
|
def lower(self):
|
|
"""Returns the lower bound of the ``Idx``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy import Idx
|
|
>>> Idx('j', 2).lower
|
|
0
|
|
>>> Idx('j', 5).lower
|
|
0
|
|
>>> Idx('j').lower is None
|
|
True
|
|
|
|
"""
|
|
try:
|
|
return self.args[1][0]
|
|
except IndexError:
|
|
return
|
|
|
|
@property
|
|
def upper(self):
|
|
"""Returns the upper bound of the ``Idx``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy import Idx
|
|
>>> Idx('j', 2).upper
|
|
1
|
|
>>> Idx('j', 5).upper
|
|
4
|
|
>>> Idx('j').upper is None
|
|
True
|
|
|
|
"""
|
|
try:
|
|
return self.args[1][1]
|
|
except IndexError:
|
|
return
|
|
|
|
def _sympystr(self, p):
|
|
return p.doprint(self.label)
|
|
|
|
@property
|
|
def name(self):
|
|
return self.label.name if self.label.is_Symbol else str(self.label)
|
|
|
|
@property
|
|
def free_symbols(self):
|
|
return {self}
|
|
|
|
|
|
@dispatch(Idx, Idx)
|
|
def _eval_is_ge(lhs, rhs): # noqa:F811
|
|
|
|
other_upper = rhs if rhs.upper is None else rhs.upper
|
|
other_lower = rhs if rhs.lower is None else rhs.lower
|
|
|
|
if lhs.lower is not None and (lhs.lower >= other_upper) == True:
|
|
return True
|
|
if lhs.upper is not None and (lhs.upper < other_lower) == True:
|
|
return False
|
|
return None
|
|
|
|
|
|
@dispatch(Idx, Number) # type:ignore
|
|
def _eval_is_ge(lhs, rhs): # noqa:F811
|
|
|
|
other_upper = rhs
|
|
other_lower = rhs
|
|
|
|
if lhs.lower is not None and (lhs.lower >= other_upper) == True:
|
|
return True
|
|
if lhs.upper is not None and (lhs.upper < other_lower) == True:
|
|
return False
|
|
return None
|
|
|
|
|
|
@dispatch(Number, Idx) # type:ignore
|
|
def _eval_is_ge(lhs, rhs): # noqa:F811
|
|
|
|
other_upper = lhs
|
|
other_lower = lhs
|
|
|
|
if rhs.upper is not None and (rhs.upper <= other_lower) == True:
|
|
return True
|
|
if rhs.lower is not None and (rhs.lower > other_upper) == True:
|
|
return False
|
|
return None
|