scipy.sparse

Sparse arrays (scipy.sparse)

• sparse.csgraph

• sparse.linalg

• sparse.migration_to_sparray

• sparse.spmatrix_api

SciPy 2-D sparse array package for numeric data.

Submodules

.. autosummary:: 
    csgraph - Compressed sparse graph routines
    linalg - Sparse linear algebra routines

Sparse array classes

.. autosummary:: 
    :toctree:generated/
    bsr_array - Block Sparse Row array
    coo_array - A sparse array in COOrdinate format
    csc_array - Compressed Sparse Column array
    csr_array - Compressed Sparse Row array
    dia_array - Sparse array with DIAgonal storage
    dok_array - Dictionary Of Keys based sparse array
    lil_array - Row-based list of lists sparse array
    sparray - Sparse array base class

Building sparse arrays

.. autosummary:: 
    :toctree:generated/
    diags_array - Return a sparse array from diagonals
    eye_array - Sparse MxN array whose k-th diagonal is all ones
    random_array - Random values in a given shape array
    block_array - Build a sparse array from sub-blocks

Combining arrays

.. autosummary:: 
    :toctree:generated/
    kron - Kronecker product of two sparse arrays
    kronsum - Kronecker sum of sparse arrays
    block_diag - Build a block diagonal sparse array
    tril - Lower triangular portion of a sparse array
    triu - Upper triangular portion of a sparse array
    hstack - Stack sparse arrays horizontally (column wise)
    vstack - Stack sparse arrays vertically (row wise)

Manipulating arrays

.. autosummary:: 
    :toctree:generated/
    matrix_transpose - Transpose a matrix (or a batch of matrices)
    swapaxes - swap two axes of a sparse array
    expand_dims - add a new (trivial) axis to a sparse array
    permute_dims - reorder the axes of a sparse array

Sparse tools

.. autosummary:: 
    :toctree:generated/
    save_npz - Save a sparse array to a file using ``.npz`` format.
    load_npz - Load a sparse array from a file using ``.npz`` format.
    find - Return the indices and values of the nonzero elements
    get_index_dtype - determine a good dtype for index arrays.
    safely_cast_index_arrays - cast index array dtype or raise if shape too big

Identifying sparse arrays

.. autosummary:: 
    :toctree:generated/
    issparse
    isspmatrix

Warnings

.. autosummary:: 
    :toctree:generated/
    SparseEfficiencyWarning
    SparseWarning

Usage information

There are seven available sparse array types:

• csc_array: Compressed Sparse Column format

• csr_array: Compressed Sparse Row format

• bsr_array: Block Sparse Row format

• lil_array: List of Lists format

• dok_array: Dictionary of Keys format

• coo_array: COOrdinate format (aka IJV, triplet format)

• dia_array: DIAgonal format

To construct an array efficiently, use any of coo_array, dok_array or lil_array. They each support basic slicing and fancy indexing with a similar syntax to NumPy arrays. The COO format is recommended and other formats use it under the hood to allow efficient construction using data values and coordinate arrays.

Despite their similarity to NumPy arrays, it is strongly discouraged to use NumPy functions directly on these arrays because NumPy typically treats them as generic Python objects rather than arrays, leading to unexpected (and incorrect) results. If you are tempted to apply a NumPy function to these arrays, check if SciPy has its own implementation for the given sparse array class and, if not, convert the sparse array to a NumPy array (e.g., using the toarray method of the class) before applying the method.

All conversions among the CSR, CSC, and COO formats are efficient, linear-time operations.

To perform manipulations such as multiplication or inversion, first convert the array to either CSC or CSR format. The lil_array format is row-based, so conversion to CSR is efficient, but conversion to CSC is less so.

Matrix vector product

To do a vector product between a 2D sparse array and a vector use the matmul operator (i.e., @) which performs a dot product (like the dot method):

>>> import numpy as np
>>> from scipy.sparse import csr_array
>>> A = csr_array([[1, 2, 0], [0, 0, 3], [4, 0, 5]])
>>> v = np.array([1, 0, -1])
>>> A @ v
array([ 1, -3, -1], dtype=int64)

The CSR format is especially suitable for fast matrix vector products.

Example 1

Construct a 1000x1000 lil_array and add some values to it:

>>> from scipy.sparse import lil_array
>>> from scipy.sparse.linalg import spsolve
>>> from numpy.linalg import solve, norm
>>> from numpy.random import rand

>>> A = lil_array((1000, 1000))
>>> A[0, :100] = rand(100)
>>> A.setdiag(rand(1000))

Now convert it to CSR format and solve A x = b for x:

>>> A = A.tocsr()
>>> b = rand(1000)
>>> x = spsolve(A, b)

Convert it to a dense array and solve, and check that the result is the same:

>>> x_ = solve(A.toarray(), b)

Now we can compute norm of the error with:

>>> err = norm(x-x_)
>>> err < 1e-9
True

It should be small :)

Example 2

Construct an array in COO format:

>>> from scipy import sparse
>>> from numpy import array
>>> I = array([0,3,1,0])
>>> J = array([0,3,1,2])
>>> V = array([4,5,7,9])
>>> A = sparse.coo_array((V,(I,J)),shape=(4,4))

Notice that the indices do not need to be sorted.

Duplicate (i,j) entries are summed when converting to CSR or CSC.

>>> I = array([0,0,1,3,1,0,0])
>>> J = array([0,2,1,3,1,0,0])
>>> V = array([1,1,1,1,1,1,1])
>>> B = sparse.coo_array((V,(I,J)),shape=(4,4)).tocsr()

This is useful for constructing finite-element stiffness and mass matrices.

Further details

CSR column indices are not necessarily sorted. Likewise for CSC row indices. And similarly for COO coordinates. Use the .sorted_indices() and .sort_indices() methods when sorted indices are required (e.g., when passing data to other libraries).