Source code for sympy.tensor.indexed

r"""Module that defines indexed objects

The classes ``IndexedBase``, ``Indexed``, and ``Idx`` represent a
matrix element ``M[i, j]`` as in the following diagram::

       1) The Indexed class represents the entire indexed object.
                  |
               ___|___
              '       '
               M[i, j]
              /   \__\______
              |             |
              |             |
              |     2) The Idx class represents indices; each Idx can
              |        optionally contain information about its range.
              |
        3) IndexedBase represents the 'stem' of an indexed object, here `M`.
           The stem used by itself is usually taken to represent the entire
           array.

    There can be any number of indices on an Indexed object.  No
    transformation properties are implemented in these Base objects, but
    implicit contraction of repeated indices is supported.

    Note that the support for complicated (i.e. non-atomic) integer
    expressions as indices is limited.  (This should be improved in
    future releases.)

    Examples
    ========

    To express the above matrix element example you would write:

    >>> from sympy import symbols, IndexedBase, Idx
    >>> M = IndexedBase('M')
    >>> i, j = symbols('i j', cls=Idx)
    >>> M[i, j]
    M[i, j]

    Repeated indices in a product implies a summation, so to express a
    matrix-vector product in terms of Indexed objects:

    >>> x = IndexedBase('x')
    >>> M[i, j]*x[j]
    M[i, j]*x[j]

    If the indexed objects will be converted to component based arrays, e.g.
    with the code printers or the autowrap framework, you also need to provide
    (symbolic or numerical) dimensions.  This can be done by passing an
    optional shape parameter to IndexedBase upon construction:

    >>> dim1, dim2 = symbols('dim1 dim2', integer=True)
    >>> A = IndexedBase('A', shape=(dim1, 2*dim1, dim2))
    >>> A.shape
    (dim1, 2*dim1, dim2)
    >>> A[i, j, 3].shape
    (dim1, 2*dim1, dim2)

    If an IndexedBase object has no shape information, it is assumed that the
    array is as large as the ranges of its indices:

    >>> n, m = symbols('n m', integer=True)
    >>> i = Idx('i', m)
    >>> j = Idx('j', n)
    >>> M[i, j].shape
    (m, n)
    >>> M[i, j].ranges
    [(0, m - 1), (0, n - 1)]

    The above can be compared with the following:

    >>> A[i, 2, j].shape
    (dim1, 2*dim1, dim2)
    >>> A[i, 2, j].ranges
    [(0, m - 1), None, (0, n - 1)]

    To analyze the structure of indexed expressions, you can use the methods
    get_indices() and get_contraction_structure():

    >>> from sympy.tensor import get_indices, get_contraction_structure
    >>> get_indices(A[i, j, j])
    ({i}, {})
    >>> get_contraction_structure(A[i, j, j])
    {(j,): {A[i, j, j]}}

    See the appropriate docstrings for a detailed explanation of the output.

"""

#   TODO:  (some ideas for improvement)
#
#   o test and guarantee numpy compatibility
#      - implement full support for broadcasting
#      - strided arrays
#
#   o more functions to analyze indexed expressions
#      - identify standard constructs, e.g matrix-vector product in a subexpression
#
#   o functions to generate component based arrays (numpy and sympy.Matrix)
#      - generate a single array directly from Indexed
#      - convert simple sub-expressions
#
#   o sophisticated indexing (possibly in subclasses to preserve simplicity)
#      - Idx with range smaller than dimension of Indexed
#      - Idx with stepsize != 1
#      - Idx with step determined by function call

from __future__ import print_function, division

import collections

from sympy.core.sympify import _sympify
from sympy.functions.special.tensor_functions import KroneckerDelta
from sympy.core import Expr, Tuple, Symbol, sympify, S
from sympy.core.compatibility import is_sequence, string_types, NotIterable, range


class IndexException(Exception):
    pass


[docs]class Indexed(Expr): """Represents a mathematical object with indices. >>> from sympy import Indexed, IndexedBase, Idx, symbols >>> i, j = symbols('i j', cls=Idx) >>> Indexed('A', i, j) A[i, j] It is recommended that ``Indexed`` objects be created via ``IndexedBase``: >>> A = IndexedBase('A') >>> Indexed('A', i, j) == A[i, j] True """ is_commutative = True is_Indexed = True is_symbol = True is_Atom = True def __new__(cls, base, *args, **kw_args): from sympy.utilities.misc import filldedent from sympy.tensor.array.ndim_array import NDimArray from sympy.matrices.matrices import MatrixBase if not args: raise IndexException("Indexed needs at least one index.") if isinstance(base, (string_types, Symbol)): base = IndexedBase(base) elif not hasattr(base, '__getitem__') and not isinstance(base, IndexedBase): raise TypeError(filldedent(""" Indexed expects string, Symbol, or IndexedBase as base.""")) args = list(map(sympify, args)) if isinstance(base, (NDimArray, collections.Iterable, Tuple, MatrixBase)) and all([i.is_number for i in args]): if len(args) == 1: return base[args[0]] else: return base[args] return Expr.__new__(cls, base, *args, **kw_args) @property def _diff_wrt(self): """Allow derivatives with respect to an ``Indexed`` object.""" return True def _eval_derivative(self, wrt): from sympy.tensor.array.ndim_array import NDimArray if isinstance(wrt, Indexed) and wrt.base == self.base: if len(self.indices) != len(wrt.indices): msg = "Different # of indices: d({!s})/d({!s})".format(self, wrt) raise IndexException(msg) result = S.One for index1, index2 in zip(self.indices, wrt.indices): result *= KroneckerDelta(index1, index2) return result elif isinstance(self.base, NDimArray): from sympy.tensor.array import derive_by_array return Indexed(derive_by_array(self.base, wrt), *self.args[1:]) else: if Tuple(self.indices).has(wrt): return S.NaN return S.Zero @property def base(self): """Returns the ``IndexedBase`` of the ``Indexed`` object. Examples ======== >>> from sympy import Indexed, IndexedBase, Idx, symbols >>> i, j = symbols('i j', cls=Idx) >>> Indexed('A', i, j).base A >>> B = IndexedBase('B') >>> B == B[i, j].base True """ return self.args[0] @property def indices(self): """ Returns the indices of the ``Indexed`` object. Examples ======== >>> from sympy import Indexed, Idx, symbols >>> i, j = symbols('i j', cls=Idx) >>> Indexed('A', i, j).indices (i, j) """ return self.args[1:] @property def rank(self): """ Returns the rank of the ``Indexed`` object. Examples ======== >>> from sympy import Indexed, Idx, symbols >>> i, j, k, l, m = symbols('i:m', cls=Idx) >>> Indexed('A', i, j).rank 2 >>> q = Indexed('A', i, j, k, l, m) >>> q.rank 5 >>> q.rank == len(q.indices) True """ return len(self.args) - 1 @property def shape(self): """Returns a list with dimensions of each index. Dimensions is a property of the array, not of the indices. Still, if the ``IndexedBase`` does not define a shape attribute, it is assumed that the ranges of the indices correspond to the shape of the array. >>> from sympy import IndexedBase, Idx, symbols >>> n, m = symbols('n m', integer=True) >>> i = Idx('i', m) >>> j = Idx('j', m) >>> A = IndexedBase('A', shape=(n, n)) >>> B = IndexedBase('B') >>> A[i, j].shape (n, n) >>> B[i, j].shape (m, m) """ from sympy.utilities.misc import filldedent if self.base.shape: return self.base.shape try: return Tuple(*[i.upper - i.lower + 1 for i in self.indices]) except AttributeError: raise IndexException(filldedent(""" Range is not defined for all indices in: %s""" % self)) except TypeError: raise IndexException(filldedent(""" Shape cannot be inferred from Idx with undefined range: %s""" % self)) @property def ranges(self): """Returns a list of tuples with lower and upper range of each index. If an index does not define the data members upper and lower, the corresponding slot in the list contains ``None`` instead of a tuple. Examples ======== >>> from sympy import Indexed,Idx, symbols >>> Indexed('A', Idx('i', 2), Idx('j', 4), Idx('k', 8)).ranges [(0, 1), (0, 3), (0, 7)] >>> Indexed('A', Idx('i', 3), Idx('j', 3), Idx('k', 3)).ranges [(0, 2), (0, 2), (0, 2)] >>> x, y, z = symbols('x y z', integer=True) >>> Indexed('A', x, y, z).ranges [None, None, None] """ ranges = [] for i in self.indices: try: ranges.append(Tuple(i.lower, i.upper)) except AttributeError: ranges.append(None) return ranges def _sympystr(self, p): indices = list(map(p.doprint, self.indices)) return "%s[%s]" % (p.doprint(self.base), ", ".join(indices)) @property def free_symbols(self): base_free_symbols = self.base.free_symbols indices_free_symbols = { fs for i in self.indices for fs in i.free_symbols} if base_free_symbols: return {self} | base_free_symbols | indices_free_symbols else: return indices_free_symbols @property def expr_free_symbols(self): return {self}
[docs]class IndexedBase(Expr, NotIterable): """Represent the base or stem of an indexed object The IndexedBase class represent an array that contains elements. The main purpose of this class is to allow the convenient creation of objects of the Indexed class. The __getitem__ method of IndexedBase returns an instance of Indexed. Alone, without indices, the IndexedBase class can be used as a notation for e.g. matrix equations, resembling what you could do with the Symbol class. But, the IndexedBase class adds functionality that is not available for Symbol instances: - An IndexedBase object can optionally store shape information. This can be used in to check array conformance and conditions for numpy broadcasting. (TODO) - An IndexedBase object implements syntactic sugar that allows easy symbolic representation of array operations, using implicit summation of repeated indices. - The IndexedBase object symbolizes a mathematical structure equivalent to arrays, and is recognized as such for code generation and automatic compilation and wrapping. >>> from sympy.tensor import IndexedBase, Idx >>> from sympy import symbols >>> A = IndexedBase('A'); A A >>> type(A) <class 'sympy.tensor.indexed.IndexedBase'> When an IndexedBase object receives indices, it returns an array with named axes, represented by an Indexed object: >>> i, j = symbols('i j', integer=True) >>> A[i, j, 2] A[i, j, 2] >>> type(A[i, j, 2]) <class 'sympy.tensor.indexed.Indexed'> The IndexedBase constructor takes an optional shape argument. If given, it overrides any shape information in the indices. (But not the index ranges!) >>> m, n, o, p = symbols('m n o p', integer=True) >>> i = Idx('i', m) >>> j = Idx('j', n) >>> A[i, j].shape (m, n) >>> B = IndexedBase('B', shape=(o, p)) >>> B[i, j].shape (o, p) """ is_commutative = True is_symbol = True is_Atom = True def __new__(cls, label, shape=None, **kw_args): from sympy import MatrixBase, NDimArray if isinstance(label, string_types): label = Symbol(label) elif isinstance(label, Symbol): pass elif isinstance(label, (MatrixBase, NDimArray)): return label elif isinstance(label, collections.Iterable): return _sympify(label) else: label = _sympify(label) if is_sequence(shape): shape = Tuple(*shape) elif shape is not None: shape = Tuple(shape) offset = kw_args.pop('offset', S.Zero) strides = kw_args.pop('strides', None) if shape is not None: obj = Expr.__new__(cls, label, shape, **kw_args) else: obj = Expr.__new__(cls, label, **kw_args) obj._shape = shape obj._offset = offset obj._strides = strides return obj def __getitem__(self, indices, **kw_args): if is_sequence(indices): # Special case needed because M[*my_tuple] is a syntax error. if self.shape and len(self.shape) != len(indices): raise IndexException("Rank mismatch.") return Indexed(self, *indices, **kw_args) else: if self.shape and len(self.shape) != 1: raise IndexException("Rank mismatch.") return Indexed(self, indices, **kw_args) @property def shape(self): """Returns the shape of the ``IndexedBase`` object. Examples ======== >>> from sympy import IndexedBase, Idx, Symbol >>> from sympy.abc import x, y >>> IndexedBase('A', shape=(x, y)).shape (x, y) Note: If the shape of the ``IndexedBase`` is specified, it will override any shape information given by the indices. >>> A = IndexedBase('A', shape=(x, y)) >>> B = IndexedBase('B') >>> i = Idx('i', 2) >>> j = Idx('j', 1) >>> A[i, j].shape (x, y) >>> B[i, j].shape (2, 1) """ 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)
[docs]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 IndexedBase, 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): from sympy.utilities.misc import filldedent if isinstance(label, string_types): 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: raise TypeError("Idx object requires integer bounds.") args = label, Tuple(*range) elif isinstance(range, Expr): if not (range.is_integer or range is S.Infinity): 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 free_symbols(self): return {self} def __le__(self, other): if isinstance(other, Idx): other_upper = other if other.upper is None else other.upper other_lower = other if other.lower is None else other.lower else: other_upper = other other_lower = other if self.upper is not None and (self.upper <= other_lower) == True: return True if self.lower is not None and (self.lower > other_upper) == True: return False return super(Idx, self).__le__(other) def __ge__(self, other): if isinstance(other, Idx): other_upper = other if other.upper is None else other.upper other_lower = other if other.lower is None else other.lower else: other_upper = other other_lower = other if self.lower is not None and (self.lower >= other_upper) == True: return True if self.upper is not None and (self.upper < other_lower) == True: return False return super(Idx, self).__ge__(other) def __lt__(self, other): if isinstance(other, Idx): other_upper = other if other.upper is None else other.upper other_lower = other if other.lower is None else other.lower else: other_upper = other other_lower = other if self.upper is not None and (self.upper < other_lower) == True: return True if self.lower is not None and (self.lower >= other_upper) == True: return False return super(Idx, self).__lt__(other) def __gt__(self, other): if isinstance(other, Idx): other_upper = other if other.upper is None else other.upper other_lower = other if other.lower is None else other.lower else: other_upper = other other_lower = other if self.lower is not None and (self.lower > other_upper) == True: return True if self.upper is not None and (self.upper <= other_lower) == True: return False return super(Idx, self).__gt__(other)