568 lines
19 KiB
Python
568 lines
19 KiB
Python
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from __future__ import division, print_function, absolute_import
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from warnings import warn
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import numpy as np
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from numpy import asarray
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from scipy.sparse import (isspmatrix_csc, isspmatrix_csr, isspmatrix,
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SparseEfficiencyWarning, csc_matrix, csr_matrix)
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from scipy.linalg import LinAlgError
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from . import _superlu
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noScikit = False
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try:
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import scikits.umfpack as umfpack
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except ImportError:
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noScikit = True
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useUmfpack = not noScikit
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__all__ = ['use_solver', 'spsolve', 'splu', 'spilu', 'factorized',
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'MatrixRankWarning', 'spsolve_triangular']
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class MatrixRankWarning(UserWarning):
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pass
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def use_solver(**kwargs):
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"""
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Select default sparse direct solver to be used.
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Parameters
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----------
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useUmfpack : bool, optional
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Use UMFPACK over SuperLU. Has effect only if scikits.umfpack is
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installed. Default: True
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assumeSortedIndices : bool, optional
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Allow UMFPACK to skip the step of sorting indices for a CSR/CSC matrix.
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Has effect only if useUmfpack is True and scikits.umfpack is installed.
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Default: False
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Notes
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-----
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The default sparse solver is umfpack when available
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(scikits.umfpack is installed). This can be changed by passing
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useUmfpack = False, which then causes the always present SuperLU
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based solver to be used.
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Umfpack requires a CSR/CSC matrix to have sorted column/row indices. If
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sure that the matrix fulfills this, pass ``assumeSortedIndices=True``
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to gain some speed.
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"""
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if 'useUmfpack' in kwargs:
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globals()['useUmfpack'] = kwargs['useUmfpack']
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if useUmfpack and 'assumeSortedIndices' in kwargs:
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umfpack.configure(assumeSortedIndices=kwargs['assumeSortedIndices'])
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def _get_umf_family(A):
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"""Get umfpack family string given the sparse matrix dtype."""
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_families = {
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(np.float64, np.int32): 'di',
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(np.complex128, np.int32): 'zi',
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(np.float64, np.int64): 'dl',
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(np.complex128, np.int64): 'zl'
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}
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f_type = np.sctypeDict[A.dtype.name]
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i_type = np.sctypeDict[A.indices.dtype.name]
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try:
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family = _families[(f_type, i_type)]
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except KeyError:
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msg = 'only float64 or complex128 matrices with int32 or int64' \
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' indices are supported! (got: matrix: %s, indices: %s)' \
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% (f_type, i_type)
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raise ValueError(msg)
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return family
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def spsolve(A, b, permc_spec=None, use_umfpack=True):
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"""Solve the sparse linear system Ax=b, where b may be a vector or a matrix.
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Parameters
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----------
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A : ndarray or sparse matrix
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The square matrix A will be converted into CSC or CSR form
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b : ndarray or sparse matrix
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The matrix or vector representing the right hand side of the equation.
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If a vector, b.shape must be (n,) or (n, 1).
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permc_spec : str, optional
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How to permute the columns of the matrix for sparsity preservation.
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(default: 'COLAMD')
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- ``NATURAL``: natural ordering.
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- ``MMD_ATA``: minimum degree ordering on the structure of A^T A.
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- ``MMD_AT_PLUS_A``: minimum degree ordering on the structure of A^T+A.
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- ``COLAMD``: approximate minimum degree column ordering
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use_umfpack : bool, optional
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if True (default) then use umfpack for the solution. This is
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only referenced if b is a vector and ``scikit-umfpack`` is installed.
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Returns
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-------
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x : ndarray or sparse matrix
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the solution of the sparse linear equation.
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If b is a vector, then x is a vector of size A.shape[1]
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If b is a matrix, then x is a matrix of size (A.shape[1], b.shape[1])
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Notes
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-----
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For solving the matrix expression AX = B, this solver assumes the resulting
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matrix X is sparse, as is often the case for very sparse inputs. If the
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resulting X is dense, the construction of this sparse result will be
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relatively expensive. In that case, consider converting A to a dense
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matrix and using scipy.linalg.solve or its variants.
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Examples
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--------
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>>> from scipy.sparse import csc_matrix
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>>> from scipy.sparse.linalg import spsolve
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>>> A = csc_matrix([[3, 2, 0], [1, -1, 0], [0, 5, 1]], dtype=float)
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>>> B = csc_matrix([[2, 0], [-1, 0], [2, 0]], dtype=float)
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>>> x = spsolve(A, B)
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>>> np.allclose(A.dot(x).todense(), B.todense())
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True
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"""
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if not (isspmatrix_csc(A) or isspmatrix_csr(A)):
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A = csc_matrix(A)
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warn('spsolve requires A be CSC or CSR matrix format',
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SparseEfficiencyWarning)
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# b is a vector only if b have shape (n,) or (n, 1)
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b_is_sparse = isspmatrix(b)
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if not b_is_sparse:
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b = asarray(b)
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b_is_vector = ((b.ndim == 1) or (b.ndim == 2 and b.shape[1] == 1))
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A.sort_indices()
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A = A.asfptype() # upcast to a floating point format
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result_dtype = np.promote_types(A.dtype, b.dtype)
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if A.dtype != result_dtype:
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A = A.astype(result_dtype)
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if b.dtype != result_dtype:
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b = b.astype(result_dtype)
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# validate input shapes
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M, N = A.shape
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if (M != N):
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raise ValueError("matrix must be square (has shape %s)" % ((M, N),))
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if M != b.shape[0]:
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raise ValueError("matrix - rhs dimension mismatch (%s - %s)"
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% (A.shape, b.shape[0]))
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use_umfpack = use_umfpack and useUmfpack
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if b_is_vector and use_umfpack:
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if b_is_sparse:
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b_vec = b.toarray()
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else:
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b_vec = b
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b_vec = asarray(b_vec, dtype=A.dtype).ravel()
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if noScikit:
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raise RuntimeError('Scikits.umfpack not installed.')
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if A.dtype.char not in 'dD':
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raise ValueError("convert matrix data to double, please, using"
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" .astype(), or set linsolve.useUmfpack = False")
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umf = umfpack.UmfpackContext(_get_umf_family(A))
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x = umf.linsolve(umfpack.UMFPACK_A, A, b_vec,
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autoTranspose=True)
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else:
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if b_is_vector and b_is_sparse:
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b = b.toarray()
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b_is_sparse = False
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if not b_is_sparse:
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if isspmatrix_csc(A):
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flag = 1 # CSC format
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else:
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flag = 0 # CSR format
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options = dict(ColPerm=permc_spec)
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x, info = _superlu.gssv(N, A.nnz, A.data, A.indices, A.indptr,
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b, flag, options=options)
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if info != 0:
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warn("Matrix is exactly singular", MatrixRankWarning)
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x.fill(np.nan)
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if b_is_vector:
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x = x.ravel()
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else:
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# b is sparse
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Afactsolve = factorized(A)
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if not isspmatrix_csc(b):
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warn('spsolve is more efficient when sparse b '
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'is in the CSC matrix format', SparseEfficiencyWarning)
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b = csc_matrix(b)
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# Create a sparse output matrix by repeatedly applying
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# the sparse factorization to solve columns of b.
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data_segs = []
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row_segs = []
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col_segs = []
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for j in range(b.shape[1]):
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bj = b[:, j].A.ravel()
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xj = Afactsolve(bj)
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w = np.flatnonzero(xj)
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segment_length = w.shape[0]
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row_segs.append(w)
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col_segs.append(np.ones(segment_length, dtype=int)*j)
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data_segs.append(np.asarray(xj[w], dtype=A.dtype))
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sparse_data = np.concatenate(data_segs)
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sparse_row = np.concatenate(row_segs)
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sparse_col = np.concatenate(col_segs)
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x = A.__class__((sparse_data, (sparse_row, sparse_col)),
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shape=b.shape, dtype=A.dtype)
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return x
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def splu(A, permc_spec=None, diag_pivot_thresh=None,
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relax=None, panel_size=None, options=dict()):
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"""
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Compute the LU decomposition of a sparse, square matrix.
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Parameters
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----------
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A : sparse matrix
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Sparse matrix to factorize. Should be in CSR or CSC format.
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permc_spec : str, optional
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How to permute the columns of the matrix for sparsity preservation.
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(default: 'COLAMD')
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- ``NATURAL``: natural ordering.
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- ``MMD_ATA``: minimum degree ordering on the structure of A^T A.
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- ``MMD_AT_PLUS_A``: minimum degree ordering on the structure of A^T+A.
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- ``COLAMD``: approximate minimum degree column ordering
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diag_pivot_thresh : float, optional
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Threshold used for a diagonal entry to be an acceptable pivot.
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See SuperLU user's guide for details [1]_
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relax : int, optional
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Expert option for customizing the degree of relaxing supernodes.
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See SuperLU user's guide for details [1]_
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panel_size : int, optional
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Expert option for customizing the panel size.
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See SuperLU user's guide for details [1]_
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options : dict, optional
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Dictionary containing additional expert options to SuperLU.
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See SuperLU user guide [1]_ (section 2.4 on the 'Options' argument)
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for more details. For example, you can specify
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``options=dict(Equil=False, IterRefine='SINGLE'))``
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to turn equilibration off and perform a single iterative refinement.
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Returns
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-------
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invA : scipy.sparse.linalg.SuperLU
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Object, which has a ``solve`` method.
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See also
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--------
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spilu : incomplete LU decomposition
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Notes
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-----
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This function uses the SuperLU library.
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References
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----------
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.. [1] SuperLU http://crd.lbl.gov/~xiaoye/SuperLU/
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Examples
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--------
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>>> from scipy.sparse import csc_matrix
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>>> from scipy.sparse.linalg import splu
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>>> A = csc_matrix([[1., 0., 0.], [5., 0., 2.], [0., -1., 0.]], dtype=float)
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>>> B = splu(A)
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>>> x = np.array([1., 2., 3.], dtype=float)
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>>> B.solve(x)
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array([ 1. , -3. , -1.5])
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>>> A.dot(B.solve(x))
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array([ 1., 2., 3.])
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>>> B.solve(A.dot(x))
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array([ 1., 2., 3.])
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"""
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if not isspmatrix_csc(A):
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A = csc_matrix(A)
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warn('splu requires CSC matrix format', SparseEfficiencyWarning)
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A.sort_indices()
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A = A.asfptype() # upcast to a floating point format
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M, N = A.shape
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if (M != N):
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raise ValueError("can only factor square matrices") # is this true?
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_options = dict(DiagPivotThresh=diag_pivot_thresh, ColPerm=permc_spec,
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PanelSize=panel_size, Relax=relax)
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if options is not None:
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_options.update(options)
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return _superlu.gstrf(N, A.nnz, A.data, A.indices, A.indptr,
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ilu=False, options=_options)
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def spilu(A, drop_tol=None, fill_factor=None, drop_rule=None, permc_spec=None,
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diag_pivot_thresh=None, relax=None, panel_size=None, options=None):
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"""
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Compute an incomplete LU decomposition for a sparse, square matrix.
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The resulting object is an approximation to the inverse of `A`.
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Parameters
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----------
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A : (N, N) array_like
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Sparse matrix to factorize
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drop_tol : float, optional
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Drop tolerance (0 <= tol <= 1) for an incomplete LU decomposition.
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(default: 1e-4)
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fill_factor : float, optional
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Specifies the fill ratio upper bound (>= 1.0) for ILU. (default: 10)
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drop_rule : str, optional
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Comma-separated string of drop rules to use.
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Available rules: ``basic``, ``prows``, ``column``, ``area``,
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``secondary``, ``dynamic``, ``interp``. (Default: ``basic,area``)
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See SuperLU documentation for details.
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Remaining other options
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Same as for `splu`
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Returns
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-------
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invA_approx : scipy.sparse.linalg.SuperLU
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Object, which has a ``solve`` method.
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See also
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--------
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splu : complete LU decomposition
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Notes
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-----
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To improve the better approximation to the inverse, you may need to
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increase `fill_factor` AND decrease `drop_tol`.
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This function uses the SuperLU library.
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Examples
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--------
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>>> from scipy.sparse import csc_matrix
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>>> from scipy.sparse.linalg import spilu
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>>> A = csc_matrix([[1., 0., 0.], [5., 0., 2.], [0., -1., 0.]], dtype=float)
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>>> B = spilu(A)
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>>> x = np.array([1., 2., 3.], dtype=float)
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>>> B.solve(x)
|
||
|
|
array([ 1. , -3. , -1.5])
|
||
|
|
>>> A.dot(B.solve(x))
|
||
|
|
array([ 1., 2., 3.])
|
||
|
|
>>> B.solve(A.dot(x))
|
||
|
|
array([ 1., 2., 3.])
|
||
|
|
"""
|
||
|
|
if not isspmatrix_csc(A):
|
||
|
|
A = csc_matrix(A)
|
||
|
|
warn('splu requires CSC matrix format', SparseEfficiencyWarning)
|
||
|
|
|
||
|
|
A.sort_indices()
|
||
|
|
A = A.asfptype() # upcast to a floating point format
|
||
|
|
|
||
|
|
M, N = A.shape
|
||
|
|
if (M != N):
|
||
|
|
raise ValueError("can only factor square matrices") # is this true?
|
||
|
|
|
||
|
|
_options = dict(ILU_DropRule=drop_rule, ILU_DropTol=drop_tol,
|
||
|
|
ILU_FillFactor=fill_factor,
|
||
|
|
DiagPivotThresh=diag_pivot_thresh, ColPerm=permc_spec,
|
||
|
|
PanelSize=panel_size, Relax=relax)
|
||
|
|
if options is not None:
|
||
|
|
_options.update(options)
|
||
|
|
return _superlu.gstrf(N, A.nnz, A.data, A.indices, A.indptr,
|
||
|
|
ilu=True, options=_options)
|
||
|
|
|
||
|
|
|
||
|
|
def factorized(A):
|
||
|
|
"""
|
||
|
|
Return a function for solving a sparse linear system, with A pre-factorized.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
A : (N, N) array_like
|
||
|
|
Input.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
solve : callable
|
||
|
|
To solve the linear system of equations given in `A`, the `solve`
|
||
|
|
callable should be passed an ndarray of shape (N,).
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> from scipy.sparse.linalg import factorized
|
||
|
|
>>> A = np.array([[ 3. , 2. , -1. ],
|
||
|
|
... [ 2. , -2. , 4. ],
|
||
|
|
... [-1. , 0.5, -1. ]])
|
||
|
|
>>> solve = factorized(A) # Makes LU decomposition.
|
||
|
|
>>> rhs1 = np.array([1, -2, 0])
|
||
|
|
>>> solve(rhs1) # Uses the LU factors.
|
||
|
|
array([ 1., -2., -2.])
|
||
|
|
|
||
|
|
"""
|
||
|
|
if useUmfpack:
|
||
|
|
if noScikit:
|
||
|
|
raise RuntimeError('Scikits.umfpack not installed.')
|
||
|
|
|
||
|
|
if not isspmatrix_csc(A):
|
||
|
|
A = csc_matrix(A)
|
||
|
|
warn('splu requires CSC matrix format', SparseEfficiencyWarning)
|
||
|
|
|
||
|
|
A = A.asfptype() # upcast to a floating point format
|
||
|
|
|
||
|
|
if A.dtype.char not in 'dD':
|
||
|
|
raise ValueError("convert matrix data to double, please, using"
|
||
|
|
" .astype(), or set linsolve.useUmfpack = False")
|
||
|
|
|
||
|
|
umf = umfpack.UmfpackContext(_get_umf_family(A))
|
||
|
|
|
||
|
|
# Make LU decomposition.
|
||
|
|
umf.numeric(A)
|
||
|
|
|
||
|
|
def solve(b):
|
||
|
|
return umf.solve(umfpack.UMFPACK_A, A, b, autoTranspose=True)
|
||
|
|
|
||
|
|
return solve
|
||
|
|
else:
|
||
|
|
return splu(A).solve
|
||
|
|
|
||
|
|
|
||
|
|
def spsolve_triangular(A, b, lower=True, overwrite_A=False, overwrite_b=False):
|
||
|
|
"""
|
||
|
|
Solve the equation `A x = b` for `x`, assuming A is a triangular matrix.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
A : (M, M) sparse matrix
|
||
|
|
A sparse square triangular matrix. Should be in CSR format.
|
||
|
|
b : (M,) or (M, N) array_like
|
||
|
|
Right-hand side matrix in `A x = b`
|
||
|
|
lower : bool, optional
|
||
|
|
Whether `A` is a lower or upper triangular matrix.
|
||
|
|
Default is lower triangular matrix.
|
||
|
|
overwrite_A : bool, optional
|
||
|
|
Allow changing `A`. The indices of `A` are going to be sorted and zero
|
||
|
|
entries are going to be removed.
|
||
|
|
Enabling gives a performance gain. Default is False.
|
||
|
|
overwrite_b : bool, optional
|
||
|
|
Allow overwriting data in `b`.
|
||
|
|
Enabling gives a performance gain. Default is False.
|
||
|
|
If `overwrite_b` is True, it should be ensured that
|
||
|
|
`b` has an appropriate dtype to be able to store the result.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
x : (M,) or (M, N) ndarray
|
||
|
|
Solution to the system `A x = b`. Shape of return matches shape of `b`.
|
||
|
|
|
||
|
|
Raises
|
||
|
|
------
|
||
|
|
LinAlgError
|
||
|
|
If `A` is singular or not triangular.
|
||
|
|
ValueError
|
||
|
|
If shape of `A` or shape of `b` do not match the requirements.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
.. versionadded:: 0.19.0
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> from scipy.sparse import csr_matrix
|
||
|
|
>>> from scipy.sparse.linalg import spsolve_triangular
|
||
|
|
>>> A = csr_matrix([[3, 0, 0], [1, -1, 0], [2, 0, 1]], dtype=float)
|
||
|
|
>>> B = np.array([[2, 0], [-1, 0], [2, 0]], dtype=float)
|
||
|
|
>>> x = spsolve_triangular(A, B)
|
||
|
|
>>> np.allclose(A.dot(x), B)
|
||
|
|
True
|
||
|
|
"""
|
||
|
|
|
||
|
|
# Check the input for correct type and format.
|
||
|
|
if not isspmatrix_csr(A):
|
||
|
|
warn('CSR matrix format is required. Converting to CSR matrix.',
|
||
|
|
SparseEfficiencyWarning)
|
||
|
|
A = csr_matrix(A)
|
||
|
|
elif not overwrite_A:
|
||
|
|
A = A.copy()
|
||
|
|
|
||
|
|
if A.shape[0] != A.shape[1]:
|
||
|
|
raise ValueError(
|
||
|
|
'A must be a square matrix but its shape is {}.'.format(A.shape))
|
||
|
|
|
||
|
|
A.eliminate_zeros()
|
||
|
|
A.sort_indices()
|
||
|
|
|
||
|
|
b = np.asanyarray(b)
|
||
|
|
|
||
|
|
if b.ndim not in [1, 2]:
|
||
|
|
raise ValueError(
|
||
|
|
'b must have 1 or 2 dims but its shape is {}.'.format(b.shape))
|
||
|
|
if A.shape[0] != b.shape[0]:
|
||
|
|
raise ValueError(
|
||
|
|
'The size of the dimensions of A must be equal to '
|
||
|
|
'the size of the first dimension of b but the shape of A is '
|
||
|
|
'{} and the shape of b is {}.'.format(A.shape, b.shape))
|
||
|
|
|
||
|
|
# Init x as (a copy of) b.
|
||
|
|
x_dtype = np.result_type(A.data, b, np.float)
|
||
|
|
if overwrite_b:
|
||
|
|
if np.can_cast(b.dtype, x_dtype, casting='same_kind'):
|
||
|
|
x = b
|
||
|
|
else:
|
||
|
|
raise ValueError(
|
||
|
|
'Cannot overwrite b (dtype {}) with result '
|
||
|
|
'of type {}.'.format(b.dtype, x_dtype))
|
||
|
|
else:
|
||
|
|
x = b.astype(x_dtype, copy=True)
|
||
|
|
|
||
|
|
# Choose forward or backward order.
|
||
|
|
if lower:
|
||
|
|
row_indices = range(len(b))
|
||
|
|
else:
|
||
|
|
row_indices = range(len(b) - 1, -1, -1)
|
||
|
|
|
||
|
|
# Fill x iteratively.
|
||
|
|
for i in row_indices:
|
||
|
|
|
||
|
|
# Get indices for i-th row.
|
||
|
|
indptr_start = A.indptr[i]
|
||
|
|
indptr_stop = A.indptr[i + 1]
|
||
|
|
if lower:
|
||
|
|
A_diagonal_index_row_i = indptr_stop - 1
|
||
|
|
A_off_diagonal_indices_row_i = slice(indptr_start, indptr_stop - 1)
|
||
|
|
else:
|
||
|
|
A_diagonal_index_row_i = indptr_start
|
||
|
|
A_off_diagonal_indices_row_i = slice(indptr_start + 1, indptr_stop)
|
||
|
|
|
||
|
|
# Check regularity and triangularity of A.
|
||
|
|
if indptr_stop <= indptr_start or A.indices[A_diagonal_index_row_i] < i:
|
||
|
|
raise LinAlgError(
|
||
|
|
'A is singular: diagonal {} is zero.'.format(i))
|
||
|
|
if A.indices[A_diagonal_index_row_i] > i:
|
||
|
|
raise LinAlgError(
|
||
|
|
'A is not triangular: A[{}, {}] is nonzero.'
|
||
|
|
''.format(i, A.indices[A_diagonal_index_row_i]))
|
||
|
|
|
||
|
|
# Incorporate off-diagonal entries.
|
||
|
|
A_column_indices_in_row_i = A.indices[A_off_diagonal_indices_row_i]
|
||
|
|
A_values_in_row_i = A.data[A_off_diagonal_indices_row_i]
|
||
|
|
x[i] -= np.dot(x[A_column_indices_in_row_i].T, A_values_in_row_i)
|
||
|
|
|
||
|
|
# Compute i-th entry of x.
|
||
|
|
x[i] /= A.data[A_diagonal_index_row_i]
|
||
|
|
|
||
|
|
return x
|