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Build a block diagonal sparse matrix from provided matrices.

Parameters ---------- mats : sequence of matrices Input matrices. format : str, optional The sparse format of the result (e.g., 'csr'). If not given, the matrix is returned in 'coo' format. dtype : dtype specifier, optional The data-type of the output matrix. If not given, the dtype is determined from that of `blocks`.

Returns ------- res : sparse matrix

Notes -----

.. versionadded:: 0.11.0

See Also -------- bmat, diags

Examples -------- >>> from scipy.sparse import coo_matrix, block_diag >>> A = coo_matrix([1, 2], [3, 4]) >>> B = coo_matrix([5], [6]) >>> C = coo_matrix([7]) >>> block_diag((A, B, C)).toarray() array([1, 2, 0, 0], [3, 4, 0, 0], [0, 0, 5, 0], [0, 0, 6, 0], [0, 0, 0, 7])

Build a sparse matrix from sparse sub-blocks

Parameters ---------- blocks : array_like Grid of sparse matrices with compatible shapes. An entry of None implies an all-zero matrix. format : 'bsr', 'coo', 'csc', 'csr', 'dia', 'dok', 'lil', optional The sparse format of the result (e.g. 'csr'). By default an appropriate sparse matrix format is returned. This choice is subject to change. dtype : dtype, optional The data-type of the output matrix. If not given, the dtype is determined from that of `blocks`.

Returns ------- bmat : sparse matrix

See Also -------- block_diag, diags

Examples -------- >>> from scipy.sparse import coo_matrix, bmat >>> A = coo_matrix([1, 2], [3, 4]) >>> B = coo_matrix([5], [6]) >>> C = coo_matrix([7]) >>> bmat([A, B], [None, C]).toarray() array([1, 2, 5], [3, 4, 6], [0, 0, 7])

>>> bmat([A, None], [None, C]).toarray() array([1, 2, 0], [3, 4, 0], [0, 0, 7])

Construct a sparse matrix from diagonals.

Parameters ---------- diagonals : sequence of array_like Sequence of arrays containing the matrix diagonals, corresponding to `offsets`. offsets : sequence of int or an int, optional Diagonals to set:

• k = 0 the main diagonal (default)
• k > 0 the kth upper diagonal
• k < 0 the kth lower diagonal shape : tuple of int, optional Shape of the result. If omitted, a square matrix large enough to contain the diagonals is returned. format : 'dia', 'csr', 'csc', 'lil', ..., optional Matrix format of the result. By default (format=None) an appropriate sparse matrix format is returned. This choice is subject to change. dtype : dtype, optional Data type of the matrix.

See Also -------- spdiags : construct matrix from diagonals

Notes ----- This function differs from `spdiags` in the way it handles off-diagonals.

The result from `diags` is the sparse equivalent of::

np.diag(diagonals0, offsets0)

1. ...
2. np.diag(diagonalsk, offsetsk)

Repeated diagonal offsets are disallowed.

.. versionadded:: 0.11

Examples -------- >>> from scipy.sparse import diags >>> diagonals = [1, 2, 3, 4], [1, 2, 3], [1, 2] >>> diags(diagonals, 0, -1, 2).toarray() array([1, 0, 1, 0], [1, 2, 0, 2], [0, 2, 3, 0], [0, 0, 3, 4])

Broadcasting of scalars is supported (but shape needs to be specified):

>>> diags(1, -2, 1, -1, 0, 1, shape=(4, 4)).toarray() array([-2., 1., 0., 0.], [ 1., -2., 1., 0.], [ 0., 1., -2., 1.], [ 0., 0., 1., -2.])

If only one diagonal is wanted (as in `numpy.diag`), the following works as well:

>>> diags(1, 2, 3, 1).toarray() array([ 0., 1., 0., 0.], [ 0., 0., 2., 0.], [ 0., 0., 0., 3.], [ 0., 0., 0., 0.])

Sparse matrix with ones on diagonal

Returns a sparse (m x n) matrix where the kth diagonal is all ones and everything else is zeros.

Parameters ---------- m : int Number of rows in the matrix. n : int, optional Number of columns. Default: `m`. k : int, optional Diagonal to place ones on. Default: 0 (main diagonal). dtype : dtype, optional Data type of the matrix. format : str, optional Sparse format of the result, e.g., format='csr', etc.

Examples -------- >>> from scipy import sparse >>> sparse.eye(3).toarray() array([ 1., 0., 0.], [ 0., 1., 0.], [ 0., 0., 1.]) >>> sparse.eye(3, dtype=np.int8) <3x3 sparse matrix of type '<class 'numpy.int8'>' with 3 stored elements (1 diagonals) in DIAgonal format>

Return the indices and values of the nonzero elements of a matrix

Parameters ---------- A : dense or sparse matrix Matrix whose nonzero elements are desired.

Returns ------- (I,J,V) : tuple of arrays I,J, and V contain the row indices, column indices, and values of the nonzero matrix entries.

Examples -------- >>> from scipy.sparse import csr_matrix, find >>> A = csr_matrix([7.0, 8.0, 0],[0, 0, 9.0]) >>> find(A) (array(0, 0, 1, dtype=int32), array(0, 1, 2, dtype=int32), array( 7., 8., 9.))

Stack sparse matrices horizontally (column wise)

Parameters ---------- blocks sequence of sparse matrices with compatible shapes format : str sparse format of the result (e.g., 'csr') by default an appropriate sparse matrix format is returned. This choice is subject to change. dtype : dtype, optional The data-type of the output matrix. If not given, the dtype is determined from that of `blocks`.

See Also -------- vstack : stack sparse matrices vertically (row wise)

Examples -------- >>> from scipy.sparse import coo_matrix, hstack >>> A = coo_matrix([1, 2], [3, 4]) >>> B = coo_matrix([5], [6]) >>> hstack(A,B).toarray() array([1, 2, 5], [3, 4, 6])

Identity matrix in sparse format

Returns an identity matrix with shape (n,n) using a given sparse format and dtype.

Parameters ---------- n : int Shape of the identity matrix. dtype : dtype, optional Data type of the matrix format : str, optional Sparse format of the result, e.g., format='csr', etc.

Examples -------- >>> from scipy.sparse import identity >>> identity(3).toarray() array([ 1., 0., 0.], [ 0., 1., 0.], [ 0., 0., 1.]) >>> identity(3, dtype='int8', format='dia') <3x3 sparse matrix of type '<class 'numpy.int8'>' with 3 stored elements (1 diagonals) in DIAgonal format>

Is x of a sparse matrix type?

Parameters ---------- x object to check for being a sparse matrix

Returns ------- bool True if x is a sparse matrix, False otherwise

Notes ----- issparse and isspmatrix are aliases for the same function.

Examples -------- >>> from scipy.sparse import csr_matrix, isspmatrix >>> isspmatrix(csr_matrix([5])) True

>>> from scipy.sparse import isspmatrix >>> isspmatrix(5) False

Is x of a sparse matrix type?

Parameters ---------- x object to check for being a sparse matrix

Returns ------- bool True if x is a sparse matrix, False otherwise

Notes ----- issparse and isspmatrix are aliases for the same function.

Examples -------- >>> from scipy.sparse import csr_matrix, isspmatrix >>> isspmatrix(csr_matrix([5])) True

>>> from scipy.sparse import isspmatrix >>> isspmatrix(5) False

Is x of a bsr_matrix type?

Parameters ---------- x object to check for being a bsr matrix

Returns ------- bool True if x is a bsr matrix, False otherwise

Examples -------- >>> from scipy.sparse import bsr_matrix, isspmatrix_bsr >>> isspmatrix_bsr(bsr_matrix([5])) True

>>> from scipy.sparse import bsr_matrix, csr_matrix, isspmatrix_bsr >>> isspmatrix_bsr(csr_matrix([5])) False

Is x of coo_matrix type?

Parameters ---------- x object to check for being a coo matrix

Returns ------- bool True if x is a coo matrix, False otherwise

Examples -------- >>> from scipy.sparse import coo_matrix, isspmatrix_coo >>> isspmatrix_coo(coo_matrix([5])) True

>>> from scipy.sparse import coo_matrix, csr_matrix, isspmatrix_coo >>> isspmatrix_coo(csr_matrix([5])) False

Is x of csc_matrix type?

Parameters ---------- x object to check for being a csc matrix

Returns ------- bool True if x is a csc matrix, False otherwise

Examples -------- >>> from scipy.sparse import csc_matrix, isspmatrix_csc >>> isspmatrix_csc(csc_matrix([5])) True

>>> from scipy.sparse import csc_matrix, csr_matrix, isspmatrix_csc >>> isspmatrix_csc(csr_matrix([5])) False

Is x of csr_matrix type?

Parameters ---------- x object to check for being a csr matrix

Returns ------- bool True if x is a csr matrix, False otherwise

Examples -------- >>> from scipy.sparse import csr_matrix, isspmatrix_csr >>> isspmatrix_csr(csr_matrix([5])) True

>>> from scipy.sparse import csc_matrix, csr_matrix, isspmatrix_csc >>> isspmatrix_csr(csc_matrix([5])) False

Is x of dia_matrix type?

Parameters ---------- x object to check for being a dia matrix

Returns ------- bool True if x is a dia matrix, False otherwise

Examples -------- >>> from scipy.sparse import dia_matrix, isspmatrix_dia >>> isspmatrix_dia(dia_matrix([5])) True

>>> from scipy.sparse import dia_matrix, csr_matrix, isspmatrix_dia >>> isspmatrix_dia(csr_matrix([5])) False

Is x of dok_matrix type?

Parameters ---------- x object to check for being a dok matrix

Returns ------- bool True if x is a dok matrix, False otherwise

Examples -------- >>> from scipy.sparse import dok_matrix, isspmatrix_dok >>> isspmatrix_dok(dok_matrix([5])) True

>>> from scipy.sparse import dok_matrix, csr_matrix, isspmatrix_dok >>> isspmatrix_dok(csr_matrix([5])) False

Is x of lil_matrix type?

Parameters ---------- x object to check for being a lil matrix

Returns ------- bool True if x is a lil matrix, False otherwise

Examples -------- >>> from scipy.sparse import lil_matrix, isspmatrix_lil >>> isspmatrix_lil(lil_matrix([5])) True

>>> from scipy.sparse import lil_matrix, csr_matrix, isspmatrix_lil >>> isspmatrix_lil(csr_matrix([5])) False

kronecker product of sparse matrices A and B

Parameters ---------- A : sparse or dense matrix first matrix of the product B : sparse or dense matrix second matrix of the product format : str, optional format of the result (e.g. 'csr')

Returns ------- kronecker product in a sparse matrix format

Examples -------- >>> from scipy import sparse >>> A = sparse.csr_matrix(np.array([0, 2], [5, 0])) >>> B = sparse.csr_matrix(np.array([1, 2], [3, 4])) >>> sparse.kron(A, B).toarray() array([ 0, 0, 2, 4], [ 0, 0, 6, 8], [ 5, 10, 0, 0], [15, 20, 0, 0])

>>> sparse.kron(A, [1, 2], [3, 4]).toarray() array([ 0, 0, 2, 4], [ 0, 0, 6, 8], [ 5, 10, 0, 0], [15, 20, 0, 0])

kronecker sum of sparse matrices A and B

Kronecker sum of two sparse matrices is a sum of two Kronecker products kron(I_n,A) + kron(B,I_m) where A has shape (m,m) and B has shape (n,n) and I_m and I_n are identity matrices of shape (m,m) and (n,n), respectively.

Parameters ---------- A square matrix B square matrix format : str format of the result (e.g. 'csr')

Returns ------- kronecker sum in a sparse matrix format

Examples --------

Load a sparse matrix from a file using ``.npz`` format.

Parameters ---------- file : str or file-like object Either the file name (string) or an open file (file-like object) where the data will be loaded.

Returns ------- result : csc_matrix, csr_matrix, bsr_matrix, dia_matrix or coo_matrix A sparse matrix containing the loaded data.

Raises ------ IOError If the input file does not exist or cannot be read.

See Also -------- scipy.sparse.save_npz: Save a sparse matrix to a file using ``.npz`` format. numpy.load: Load several arrays from a ``.npz`` archive.

Examples -------- Store sparse matrix to disk, and load it again:

>>> import scipy.sparse >>> sparse_matrix = scipy.sparse.csc_matrix(np.array([0, 0, 3], [4, 0, 0])) >>> sparse_matrix <2x3 sparse matrix of type '<class 'numpy.int64'>' with 2 stored elements in Compressed Sparse Column format> >>> sparse_matrix.todense() matrix([0, 0, 3], [4, 0, 0], dtype=int64)

>>> scipy.sparse.save_npz('/tmp/sparse_matrix.npz', sparse_matrix) >>> sparse_matrix = scipy.sparse.load_npz('/tmp/sparse_matrix.npz')

>>> sparse_matrix <2x3 sparse matrix of type '<class 'numpy.int64'>' with 2 stored elements in Compressed Sparse Column format> >>> sparse_matrix.todense() matrix([0, 0, 3], [4, 0, 0], dtype=int64)

Generate a sparse matrix of the given shape and density with uniformly distributed values.

Parameters ---------- m, n : int shape of the matrix density : real, optional density of the generated matrix: density equal to one means a full matrix, density of 0 means a matrix with no non-zero items. format : str, optional sparse matrix format. dtype : dtype, optional type of the returned matrix values. random_state : numpy.random.RandomState, int, np.random.Generator, optional Random number generator or random seed. If not given, the singleton numpy.random will be used.

Returns ------- res : sparse matrix

Notes ----- Only float types are supported for now.

See Also -------- scipy.sparse.random : Similar function that allows a user-specified random data source.

Examples -------- >>> from scipy.sparse import rand >>> matrix = rand(3, 4, density=0.25, format='csr', random_state=42) >>> matrix <3x4 sparse matrix of type '<class 'numpy.float64'>' with 3 stored elements in Compressed Sparse Row format> >>> matrix.todense() matrix([0.05641158, 0. , 0. , 0.65088847], [0. , 0. , 0. , 0.14286682], [0. , 0. , 0. , 0. ])

Generate a sparse matrix of the given shape and density with randomly distributed values.

Parameters ---------- m, n : int shape of the matrix density : real, optional density of the generated matrix: density equal to one means a full matrix, density of 0 means a matrix with no non-zero items. format : str, optional sparse matrix format. dtype : dtype, optional type of the returned matrix values. random_state : numpy.random.RandomState, int, optional Random number generator or random seed. If not given, the singleton numpy.random will be used. This random state will be used for sampling the sparsity structure, but not necessarily for sampling the values of the structurally nonzero entries of the matrix. data_rvs : callable, optional Samples a requested number of random values. This function should take a single argument specifying the length of the ndarray that it will return. The structurally nonzero entries of the sparse random matrix will be taken from the array sampled by this function. By default, uniform 0, 1) random values will be sampled using the same random state as is used for sampling the sparsity structure. Returns ------- res : sparse matrix Notes ----- Only float types are supported for now. Examples -------- >>> from scipy.sparse import random >>> from scipy import stats >>> class CustomRandomState(np.random.RandomState): ... def randint(self, k): ... i = np.random.randint(k) ... return i - i % 2 >>> np.random.seed(12345) >>> rs = CustomRandomState() >>> rvs = stats.poisson(25, loc=10).rvs >>> S = random(3, 4, density=0.25, random_state=rs, data_rvs=rvs) >>> S.A array([[ 36., 0., 33., 0.], # random [ 0., 0., 0., 0.], [ 0., 0., 36., 0.]]) >>> from scipy.sparse import random >>> from scipy.stats import rv_continuous >>> class CustomDistribution(rv_continuous): ... def _rvs(self, size=None, random_state=None): ... return random_state.randn( *size) >>> X = CustomDistribution(seed=2906) >>> Y = X() # get a frozen version of the distribution >>> S = random(3, 4, density=0.25, random_state=2906, data_rvs=Y.rvs) >>> S.A array([[ 0. , 0. , 0. , 0. ], [ 0.13569738, 1.9467163 , -0.81205367, 0. ], [ 0. , 0. , 0. , 0. ]])

Save a sparse matrix to a file using ``.npz`` format.

Parameters ---------- file : str or file-like object Either the file name (string) or an open file (file-like object) where the data will be saved. If file is a string, the ``.npz`` extension will be appended to the file name if it is not already there. matrix: spmatrix (format: ``csc``, ``csr``, ``bsr``, ``dia`` or coo``) The sparse matrix to save. compressed : bool, optional Allow compressing the file. Default: True

See Also -------- scipy.sparse.load_npz: Load a sparse matrix from a file using ``.npz`` format. numpy.savez: Save several arrays into a ``.npz`` archive. numpy.savez_compressed : Save several arrays into a compressed ``.npz`` archive.

Examples -------- Store sparse matrix to disk, and load it again:

>>> import scipy.sparse >>> sparse_matrix = scipy.sparse.csc_matrix(np.array([0, 0, 3], [4, 0, 0])) >>> sparse_matrix <2x3 sparse matrix of type '<class 'numpy.int64'>' with 2 stored elements in Compressed Sparse Column format> >>> sparse_matrix.todense() matrix([0, 0, 3], [4, 0, 0], dtype=int64)

>>> scipy.sparse.save_npz('/tmp/sparse_matrix.npz', sparse_matrix) >>> sparse_matrix = scipy.sparse.load_npz('/tmp/sparse_matrix.npz')

>>> sparse_matrix <2x3 sparse matrix of type '<class 'numpy.int64'>' with 2 stored elements in Compressed Sparse Column format> >>> sparse_matrix.todense() matrix([0, 0, 3], [4, 0, 0], dtype=int64)

Return a sparse matrix from diagonals.

Parameters ---------- data : array_like matrix diagonals stored row-wise diags : diagonals to set

• k = 0 the main diagonal
• k > 0 the k-th upper diagonal
• k < 0 the k-th lower diagonal m, n : int shape of the result format : str, optional Format of the result. By default (format=None) an appropriate sparse matrix format is returned. This choice is subject to change.

See Also -------- diags : more convenient form of this function dia_matrix : the sparse DIAgonal format.

Examples -------- >>> from scipy.sparse import spdiags >>> data = np.array([1, 2, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4]) >>> diags = np.array(0, -1, 2) >>> spdiags(data, diags, 4, 4).toarray() array([1, 0, 3, 0], [1, 2, 0, 4], [0, 2, 3, 0], [0, 0, 3, 4])

Return the lower triangular portion of a matrix in sparse format

Returns the elements on or below the k-th diagonal of the matrix A.

• k = 0 corresponds to the main diagonal
• k > 0 is above the main diagonal
• k < 0 is below the main diagonal

Parameters ---------- A : dense or sparse matrix Matrix whose lower trianglar portion is desired. k : integer : optional The top-most diagonal of the lower triangle. format : string Sparse format of the result, e.g. format='csr', etc.

Returns ------- L : sparse matrix Lower triangular portion of A in sparse format.

See Also -------- triu : upper triangle in sparse format

Examples -------- >>> from scipy.sparse import csr_matrix, tril >>> A = csr_matrix([1, 2, 0, 0, 3], [4, 5, 0, 6, 7], [0, 0, 8, 9, 0], ... dtype='int32') >>> A.toarray() array([1, 2, 0, 0, 3], [4, 5, 0, 6, 7], [0, 0, 8, 9, 0]) >>> tril(A).toarray() array([1, 0, 0, 0, 0], [4, 5, 0, 0, 0], [0, 0, 8, 0, 0]) >>> tril(A).nnz 4 >>> tril(A, k=1).toarray() array([1, 2, 0, 0, 0], [4, 5, 0, 0, 0], [0, 0, 8, 9, 0]) >>> tril(A, k=-1).toarray() array([0, 0, 0, 0, 0], [4, 0, 0, 0, 0], [0, 0, 0, 0, 0]) >>> tril(A, format='csc') <3x5 sparse matrix of type '<class 'numpy.int32'>' with 4 stored elements in Compressed Sparse Column format>

Return the upper triangular portion of a matrix in sparse format

Returns the elements on or above the k-th diagonal of the matrix A.

• k = 0 corresponds to the main diagonal
• k > 0 is above the main diagonal
• k < 0 is below the main diagonal

Parameters ---------- A : dense or sparse matrix Matrix whose upper trianglar portion is desired. k : integer : optional The bottom-most diagonal of the upper triangle. format : string Sparse format of the result, e.g. format='csr', etc.

Returns ------- L : sparse matrix Upper triangular portion of A in sparse format.

See Also -------- tril : lower triangle in sparse format

Examples -------- >>> from scipy.sparse import csr_matrix, triu >>> A = csr_matrix([1, 2, 0, 0, 3], [4, 5, 0, 6, 7], [0, 0, 8, 9, 0], ... dtype='int32') >>> A.toarray() array([1, 2, 0, 0, 3], [4, 5, 0, 6, 7], [0, 0, 8, 9, 0]) >>> triu(A).toarray() array([1, 2, 0, 0, 3], [0, 5, 0, 6, 7], [0, 0, 8, 9, 0]) >>> triu(A).nnz 8 >>> triu(A, k=1).toarray() array([0, 2, 0, 0, 3], [0, 0, 0, 6, 7], [0, 0, 0, 9, 0]) >>> triu(A, k=-1).toarray() array([1, 2, 0, 0, 3], [4, 5, 0, 6, 7], [0, 0, 8, 9, 0]) >>> triu(A, format='csc') <3x5 sparse matrix of type '<class 'numpy.int32'>' with 8 stored elements in Compressed Sparse Column format>

Stack sparse matrices vertically (row wise)

Parameters ---------- blocks sequence of sparse matrices with compatible shapes format : str, optional sparse format of the result (e.g., 'csr') by default an appropriate sparse matrix format is returned. This choice is subject to change. dtype : dtype, optional The data-type of the output matrix. If not given, the dtype is determined from that of `blocks`.

See Also -------- hstack : stack sparse matrices horizontally (column wise)

Examples -------- >>> from scipy.sparse import coo_matrix, vstack >>> A = coo_matrix([1, 2], [3, 4]) >>> B = coo_matrix([5, 6]) >>> vstack(A, B).toarray() array([1, 2], [3, 4], [5, 6])