Indexed Objects

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.

class sympy.tensor.indexed.Idx(label, range=None, **kw_args)[source]

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)
property 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
property lower

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
property upper

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
class sympy.tensor.indexed.Indexed(base, *args, **kw_args)[source]

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 by indexing IndexedBase: IndexedBase('A')[i, j] instead of Indexed(IndexedBase('A'), i, j).

>>> A = IndexedBase('A')
>>> a_ij = A[i, j]           # Prefer this,
>>> b_ij = Indexed(A, i, j)  # over this.
>>> a_ij == b_ij
True
property base

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
property indices

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)
property 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]
property rank

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
property 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 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)
class sympy.tensor.indexed.IndexedBase(
label,
shape=None,
*,
offset=0,
strides=None,
**kw_args,
)[source]

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)

Assumptions can be specified with keyword arguments the same way as for Symbol:

>>> A_real = IndexedBase('A', real=True)
>>> A_real.is_real
True
>>> A != A_real
True

Assumptions can also be inherited if a Symbol is used to initialize the IndexedBase:

>>> I = symbols('I', integer=True)
>>> C_inherit = IndexedBase(I)
>>> C_explicit = IndexedBase('I', integer=True)
>>> C_inherit == C_explicit
True
property 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
property offset

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]'
property shape

Returns the shape of the IndexedBase object.

Examples

>>> from sympy import IndexedBase, Idx
>>> 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)
property strides

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.