# Sparse Tools¶

sympy.matrices.sparsetools._doktocsr()[source]

Converts a sparse matrix to Compressed Sparse Row (CSR) format.

Parameters

A : contains non-zero elements sorted by key (row, column)

JA : JA[i] is the column corresponding to A[i]

IA : IA[i] contains the index in A for the first non-zero element

of row[i]. Thus IA[i+1] - IA[i] gives number of non-zero elements row[i]. The length of IA is always 1 more than the number of rows in the matrix.

Examples

>>> from sympy.matrices.sparsetools import _doktocsr
>>> from sympy import SparseMatrix, diag
>>> m = SparseMatrix(diag(1, 2, 3))
>>> m[2, 0] = -1
>>> _doktocsr(m)
[[1, 2, -1, 3], [0, 1, 0, 2], [0, 1, 2, 4], [3, 3]]

sympy.matrices.sparsetools._csrtodok()[source]

Converts a CSR representation to DOK representation.

Examples

>>> from sympy.matrices.sparsetools import _csrtodok
>>> _csrtodok([[5, 8, 3, 6], [0, 1, 2, 1], [0, 0, 2, 3, 4], [4, 3]])
Matrix([
[0, 0, 0],
[5, 8, 0],
[0, 0, 3],
[0, 6, 0]])

sympy.matrices.sparsetools.banded(**kwargs)[source]

Returns a SparseMatrix from the given dictionary describing the diagonals of the matrix. The keys are positive for upper diagonals and negative for those below the main diagonal. The values may be:

• expressions or single-argument functions,

• lists or tuples of values,

• matrices

Unless dimensions are given, the size of the returned matrix will be large enough to contain the largest non-zero value provided.

Kwargs

rowsrows of the resulting matrix; computed if

not given.

colscolumns of the resulting matrix; computed if

not given.

Examples

>>> from sympy import banded, ones, Matrix
>>> from sympy.abc import x


If explicit values are given in tuples, the matrix will autosize to contain all values, otherwise a single value is filled onto the entire diagonal:

>>> banded({1: (1, 2, 3), -1: (4, 5, 6), 0: x})
Matrix([
[x, 1, 0, 0],
[4, x, 2, 0],
[0, 5, x, 3],
[0, 0, 6, x]])


A function accepting a single argument can be used to fill the diagonal as a function of diagonal index (which starts at 0). The size (or shape) of the matrix must be given to obtain more than a 1x1 matrix:

>>> s = lambda d: (1 + d)**2
>>> banded(5, {0: s, 2: s, -2: 2})
Matrix([
[1, 0, 1,  0,  0],
[0, 4, 0,  4,  0],
[2, 0, 9,  0,  9],
[0, 2, 0, 16,  0],
[0, 0, 2,  0, 25]])


The diagonal of matrices placed on a diagonal will coincide with the indicated diagonal:

>>> vert = Matrix([1, 2, 3])
>>> banded({0: vert}, cols=3)
Matrix([
[1, 0, 0],
[2, 1, 0],
[3, 2, 1],
[0, 3, 2],
[0, 0, 3]])

>>> banded(4, {0: ones(2)})
Matrix([
[1, 1, 0, 0],
[1, 1, 0, 0],
[0, 0, 1, 1],
[0, 0, 1, 1]])


Errors are raised if the designated size will not hold all values an integral number of times. Here, the rows are designated as odd (but an even number is required to hold the off-diagonal 2x2 ones):

>>> banded({0: 2, 1: ones(2)}, rows=5)
Traceback (most recent call last):
...
ValueError:
sequence does not fit an integral number of times in the matrix


And here, an even number of rows is given…but the square matrix has an even number of columns, too. As we saw in the previous example, an odd number is required:

>>> banded(4, {0: 2, 1: ones(2)})  # trying to make 4x4 and cols must be odd
Traceback (most recent call last):
...
ValueError:
sequence does not fit an integral number of times in the matrix


A way around having to count rows is to enclosing matrix elements in a tuple and indicate the desired number of them to the right:

>>> banded({0: 2, 2: (ones(2),)*3})
Matrix([
[2, 0, 1, 1, 0, 0, 0, 0],
[0, 2, 1, 1, 0, 0, 0, 0],
[0, 0, 2, 0, 1, 1, 0, 0],
[0, 0, 0, 2, 1, 1, 0, 0],
[0, 0, 0, 0, 2, 0, 1, 1],
[0, 0, 0, 0, 0, 2, 1, 1]])


An error will be raised if more than one value is written to a given entry. Here, the ones overlap with the main diagonal if they are placed on the first diagonal:

>>> banded({0: (2,)*5, 1: (ones(2),)*3})
Traceback (most recent call last):
...
ValueError: collision at (1, 1)


By placing a 0 at the bottom left of the 2x2 matrix of ones, the collision is avoided:

>>> u2 = Matrix([
... [1, 1],
... [0, 1]])
>>> banded({0: *5, 1: [u2]*3})
Matrix([
[2, 1, 1, 0, 0, 0, 0],
[0, 2, 1, 0, 0, 0, 0],
[0, 0, 2, 1, 1, 0, 0],
[0, 0, 0, 2, 1, 0, 0],
[0, 0, 0, 0, 2, 1, 1],
[0, 0, 0, 0, 0, 0, 1]])