299 lines
8.4 KiB
Python
299 lines
8.4 KiB
Python
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"""
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=====================================
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Sparse matrices (:mod:`scipy.sparse`)
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=====================================
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.. currentmodule:: scipy.sparse
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SciPy 2-D sparse array package for numeric data.
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.. note::
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This package is switching to an array interface, compatible with
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NumPy arrays, from the older matrix interface. We recommend that
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you use the array objects (`bsr_array`, `coo_array`, etc.) for
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all new work.
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When using the array interface, please note that:
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- ``x * y`` no longer performs matrix multiplication, but
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element-wise multiplication (just like with NumPy arrays). To
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make code work with both arrays and matrices, use ``x @ y`` for
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matrix multiplication.
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- Operations such as `sum`, that used to produce dense matrices, now
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produce arrays, whose multiplication behavior differs similarly.
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- Sparse arrays currently must be two-dimensional. This also means
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that all *slicing* operations on these objects must produce
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two-dimensional results, or they will result in an error. This
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will be addressed in a future version.
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The construction utilities (`eye`, `kron`, `random`, `diags`, etc.)
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have not yet been ported, but their results can be wrapped into arrays::
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A = csr_array(eye(3))
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Contents
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========
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Sparse array classes
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--------------------
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.. autosummary::
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:toctree: generated/
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bsr_array - Block Sparse Row array
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coo_array - A sparse array in COOrdinate format
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csc_array - Compressed Sparse Column array
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csr_array - Compressed Sparse Row array
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dia_array - Sparse array with DIAgonal storage
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dok_array - Dictionary Of Keys based sparse array
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lil_array - Row-based list of lists sparse array
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Sparse matrix classes
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---------------------
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.. autosummary::
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:toctree: generated/
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bsr_matrix - Block Sparse Row matrix
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coo_matrix - A sparse matrix in COOrdinate format
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csc_matrix - Compressed Sparse Column matrix
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csr_matrix - Compressed Sparse Row matrix
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dia_matrix - Sparse matrix with DIAgonal storage
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dok_matrix - Dictionary Of Keys based sparse matrix
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lil_matrix - Row-based list of lists sparse matrix
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spmatrix - Sparse matrix base class
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Functions
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---------
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Building sparse matrices:
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.. autosummary::
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:toctree: generated/
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eye - Sparse MxN matrix whose k-th diagonal is all ones
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identity - Identity matrix in sparse format
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kron - kronecker product of two sparse matrices
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kronsum - kronecker sum of sparse matrices
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diags - Return a sparse matrix from diagonals
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spdiags - Return a sparse matrix from diagonals
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block_diag - Build a block diagonal sparse matrix
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tril - Lower triangular portion of a matrix in sparse format
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triu - Upper triangular portion of a matrix in sparse format
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bmat - Build a sparse matrix from sparse sub-blocks
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hstack - Stack sparse matrices horizontally (column wise)
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vstack - Stack sparse matrices vertically (row wise)
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rand - Random values in a given shape
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random - Random values in a given shape
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Save and load sparse matrices:
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.. autosummary::
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:toctree: generated/
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save_npz - Save a sparse matrix to a file using ``.npz`` format.
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load_npz - Load a sparse matrix from a file using ``.npz`` format.
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Sparse matrix tools:
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.. autosummary::
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:toctree: generated/
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find
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Identifying sparse matrices:
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.. autosummary::
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:toctree: generated/
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issparse
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isspmatrix
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isspmatrix_csc
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isspmatrix_csr
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isspmatrix_bsr
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isspmatrix_lil
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isspmatrix_dok
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isspmatrix_coo
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isspmatrix_dia
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Submodules
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----------
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.. autosummary::
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csgraph - Compressed sparse graph routines
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linalg - sparse linear algebra routines
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Exceptions
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----------
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.. autosummary::
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:toctree: generated/
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SparseEfficiencyWarning
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SparseWarning
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Usage information
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=================
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There are seven available sparse matrix types:
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1. csc_matrix: Compressed Sparse Column format
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2. csr_matrix: Compressed Sparse Row format
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3. bsr_matrix: Block Sparse Row format
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4. lil_matrix: List of Lists format
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5. dok_matrix: Dictionary of Keys format
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6. coo_matrix: COOrdinate format (aka IJV, triplet format)
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7. dia_matrix: DIAgonal format
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To construct a matrix efficiently, use either dok_matrix or lil_matrix.
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The lil_matrix class supports basic slicing and fancy indexing with a
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similar syntax to NumPy arrays. As illustrated below, the COO format
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may also be used to efficiently construct matrices. Despite their
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similarity to NumPy arrays, it is **strongly discouraged** to use NumPy
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functions directly on these matrices because NumPy may not properly convert
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them for computations, leading to unexpected (and incorrect) results. If you
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do want to apply a NumPy function to these matrices, first check if SciPy has
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its own implementation for the given sparse matrix class, or **convert the
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sparse matrix to a NumPy array** (e.g., using the `toarray()` method of the
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class) first before applying the method.
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To perform manipulations such as multiplication or inversion, first
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convert the matrix to either CSC or CSR format. The lil_matrix format is
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row-based, so conversion to CSR is efficient, whereas conversion to CSC
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is less so.
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All conversions among the CSR, CSC, and COO formats are efficient,
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linear-time operations.
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Matrix vector product
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---------------------
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To do a vector product between a sparse matrix and a vector simply use
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the matrix `dot` method, as described in its docstring:
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>>> import numpy as np
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>>> from scipy.sparse import csr_matrix
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>>> A = csr_matrix([[1, 2, 0], [0, 0, 3], [4, 0, 5]])
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>>> v = np.array([1, 0, -1])
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>>> A.dot(v)
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array([ 1, -3, -1], dtype=int64)
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.. warning:: As of NumPy 1.7, `np.dot` is not aware of sparse matrices,
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therefore using it will result on unexpected results or errors.
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The corresponding dense array should be obtained first instead:
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>>> np.dot(A.toarray(), v)
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array([ 1, -3, -1], dtype=int64)
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but then all the performance advantages would be lost.
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The CSR format is specially suitable for fast matrix vector products.
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Example 1
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---------
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Construct a 1000x1000 lil_matrix and add some values to it:
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>>> from scipy.sparse import lil_matrix
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>>> from scipy.sparse.linalg import spsolve
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>>> from numpy.linalg import solve, norm
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>>> from numpy.random import rand
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>>> A = lil_matrix((1000, 1000))
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>>> A[0, :100] = rand(100)
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>>> A[1, 100:200] = A[0, :100]
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>>> A.setdiag(rand(1000))
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Now convert it to CSR format and solve A x = b for x:
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>>> A = A.tocsr()
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>>> b = rand(1000)
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>>> x = spsolve(A, b)
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Convert it to a dense matrix and solve, and check that the result
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is the same:
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>>> x_ = solve(A.toarray(), b)
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Now we can compute norm of the error with:
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>>> err = norm(x-x_)
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>>> err < 1e-10
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True
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It should be small :)
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Example 2
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---------
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Construct a matrix in COO format:
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>>> from scipy import sparse
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>>> from numpy import array
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>>> I = array([0,3,1,0])
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>>> J = array([0,3,1,2])
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>>> V = array([4,5,7,9])
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>>> A = sparse.coo_matrix((V,(I,J)),shape=(4,4))
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Notice that the indices do not need to be sorted.
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Duplicate (i,j) entries are summed when converting to CSR or CSC.
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>>> I = array([0,0,1,3,1,0,0])
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>>> J = array([0,2,1,3,1,0,0])
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>>> V = array([1,1,1,1,1,1,1])
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>>> B = sparse.coo_matrix((V,(I,J)),shape=(4,4)).tocsr()
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This is useful for constructing finite-element stiffness and mass matrices.
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Further details
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---------------
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CSR column indices are not necessarily sorted. Likewise for CSC row
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indices. Use the .sorted_indices() and .sort_indices() methods when
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sorted indices are required (e.g., when passing data to other libraries).
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"""
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# Original code by Travis Oliphant.
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# Modified and extended by Ed Schofield, Robert Cimrman,
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# Nathan Bell, and Jake Vanderplas.
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import warnings as _warnings
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from ._base import *
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from ._csr import *
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from ._csc import *
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from ._lil import *
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from ._dok import *
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from ._coo import *
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from ._dia import *
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from ._bsr import *
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from ._construct import *
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from ._extract import *
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from ._matrix_io import *
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from ._arrays import (
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csr_array, csc_array, lil_array, dok_array, coo_array, dia_array, bsr_array
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)
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# For backward compatibility with v0.19.
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from . import csgraph
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# Deprecated namespaces, to be removed in v2.0.0
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from . import (
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base, bsr, compressed, construct, coo, csc, csr, data, dia, dok, extract,
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lil, sparsetools, sputils
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)
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__all__ = [s for s in dir() if not s.startswith('_')]
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# Filter PendingDeprecationWarning for np.matrix introduced with numpy 1.15
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_warnings.filterwarnings('ignore', message='the matrix subclass is not the recommended way')
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from scipy._lib._testutils import PytestTester
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test = PytestTester(__name__)
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del PytestTester
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