Indexed Objects

Module that defines indexed objects

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

1) The Indexed class represents the entire indexed object.
           |
        ___|___
       '       '
        M[i, j]
       /   \__\______
       |             |
       |             |
       |     2) The Idx class represent indices and 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.tensor import IndexedBase, Idx
>>> from sympy import symbols
>>> 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]
x[j]*M[i, 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])
(set([i]), {})
>>> get_contraction_structure(A[i, j, j])
{(j,): set([A[i, j, j]])}

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

class sympy.tensor.indexed.Idx

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: the Idx constructor is rather pedantic in that it only accepts integer arguments. The only exception is that you can use oo and -oo to specify an unbounded range. For all other cases, both label and bounds must be declared as integers, e.g. if n is given as an argument then n.is_integer must return True.

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.tensor import Idx
>>> from sympy import 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)

label

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

lower

Returns the lower bound of the Index.

Examples

>>> from sympy import Idx
>>> Idx('j', 2).lower
0
>>> Idx('j', 5).lower
0
>>> Idx('j').lower is None
True

upper

Returns the upper bound of the Index.

Examples

>>> from sympy import Idx
>>> Idx('j', 2).upper
1
>>> Idx('j', 5).upper
4
>>> Idx('j').upper is None
True

class sympy.tensor.indexed.Indexed

Represents a mathematical object with indices.

>>> from sympy.tensor import Indexed, IndexedBase, Idx
>>> from sympy import symbols
>>> i, j = symbols('i j', cls=Idx)
>>> Indexed('A', i, j)
A[i, j]

It is recommended that Indexed objects are created via IndexedBase:

>>> A = IndexedBase('A')
>>> Indexed('A', i, j) == A[i, j]
True

base

Returns the IndexedBase of the Indexed object.

Examples

>>> from sympy.tensor import Indexed, IndexedBase, Idx
>>> from sympy import symbols
>>> i, j = symbols('i j', cls=Idx)
>>> Indexed('A', i, j).base
A
>>> B = IndexedBase('B')
>>> B == B[i, j].base
True

indices

Returns the indices of the Indexed object.

Examples

>>> from sympy.tensor import Indexed, Idx
>>> from sympy import symbols
>>> i, j = symbols('i j', cls=Idx)
>>> Indexed('A', i, j).indices
(i, j)

ranges

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]

rank

Returns the rank of the Indexed object.

Examples

>>> from sympy.tensor import Indexed, Idx
>>> from sympy import 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

shape

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.tensor.indexed import IndexedBase, Idx
>>> from sympy import 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)

class sympy.tensor.indexed.IndexedBase

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)

label

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

shape

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)