747 lines
26 KiB
Python
747 lines
26 KiB
Python
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 (issparse,
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SparseEfficiencyWarning, csc_matrix, csr_matrix)
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from scipy.sparse._sputils import is_pydata_spmatrix, convert_pydata_sparse_to_scipy
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from scipy.linalg import LinAlgError
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import copy
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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 [1]_, [2]_, [3]_, [4]_. over SuperLU. Has effect only
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if ``scikits.umfpack`` is 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
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installed. 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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References
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----------
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.. [1] T. A. Davis, Algorithm 832: UMFPACK - an unsymmetric-pattern
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multifrontal method with a column pre-ordering strategy, ACM
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Trans. on Mathematical Software, 30(2), 2004, pp. 196--199.
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https://dl.acm.org/doi/abs/10.1145/992200.992206
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.. [2] T. A. Davis, A column pre-ordering strategy for the
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unsymmetric-pattern multifrontal method, ACM Trans.
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on Mathematical Software, 30(2), 2004, pp. 165--195.
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https://dl.acm.org/doi/abs/10.1145/992200.992205
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.. [3] T. A. Davis and I. S. Duff, A combined unifrontal/multifrontal
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method for unsymmetric sparse matrices, ACM Trans. on
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Mathematical Software, 25(1), 1999, pp. 1--19.
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https://doi.org/10.1145/305658.287640
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.. [4] T. A. Davis and I. S. Duff, An unsymmetric-pattern multifrontal
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method for sparse LU factorization, SIAM J. Matrix Analysis and
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Computations, 18(1), 1997, pp. 140--158.
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https://doi.org/10.1137/S0895479894246905T.
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Examples
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--------
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>>> import numpy as np
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>>> from scipy.sparse.linalg import use_solver, spsolve
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>>> from scipy.sparse import csc_matrix
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>>> R = np.random.randn(5, 5)
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>>> A = csc_matrix(R)
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>>> b = np.random.randn(5)
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>>> use_solver(useUmfpack=False) # enforce superLU over UMFPACK
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>>> x = spsolve(A, b)
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>>> np.allclose(A.dot(x), b)
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True
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>>> use_solver(useUmfpack=True) # reset umfPack usage to default
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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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# A.dtype.name can only be "float64" or
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# "complex128" in control flow
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f_type = getattr(np, A.dtype.name)
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# control flow may allow for more index
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# types to get through here
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i_type = getattr(np, 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 as e:
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msg = ('only float64 or complex128 matrices with int32 or int64 '
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f'indices are supported! (got: matrix: {f_type}, indices: {i_type})')
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raise ValueError(msg) from e
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# See gh-8278. Considered converting only if
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# A.shape[0]*A.shape[1] > np.iinfo(np.int32).max,
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# but that didn't always fix the issue.
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family = family[0] + "l"
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A_new = copy.copy(A)
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A_new.indptr = np.asarray(A.indptr, dtype=np.int64)
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A_new.indices = np.asarray(A.indices, dtype=np.int64)
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return family, A_new
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def _safe_downcast_indices(A):
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# check for safe downcasting
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max_value = np.iinfo(np.intc).max
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if A.indptr[-1] > max_value: # indptr[-1] is max b/c indptr always sorted
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raise ValueError("indptr values too large for SuperLU")
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if max(*A.shape) > max_value: # only check large enough arrays
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if np.any(A.indices > max_value):
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raise ValueError("indices values too large for SuperLU")
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indices = A.indices.astype(np.intc, copy=False)
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indptr = A.indptr.astype(np.intc, copy=False)
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return indices, indptr
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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 [1]_, [2]_.
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use_umfpack : bool, optional
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if True (default) then use UMFPACK for the solution [3]_, [4]_, [5]_,
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[6]_ . This is only referenced if b is a vector and
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``scikits.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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References
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----------
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.. [1] T. A. Davis, J. R. Gilbert, S. Larimore, E. Ng, Algorithm 836:
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COLAMD, an approximate column minimum degree ordering algorithm,
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ACM Trans. on Mathematical Software, 30(3), 2004, pp. 377--380.
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:doi:`10.1145/1024074.1024080`
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.. [2] T. A. Davis, J. R. Gilbert, S. Larimore, E. Ng, A column approximate
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minimum degree ordering algorithm, ACM Trans. on Mathematical
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Software, 30(3), 2004, pp. 353--376. :doi:`10.1145/1024074.1024079`
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.. [3] T. A. Davis, Algorithm 832: UMFPACK - an unsymmetric-pattern
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multifrontal method with a column pre-ordering strategy, ACM
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Trans. on Mathematical Software, 30(2), 2004, pp. 196--199.
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https://dl.acm.org/doi/abs/10.1145/992200.992206
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.. [4] T. A. Davis, A column pre-ordering strategy for the
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unsymmetric-pattern multifrontal method, ACM Trans.
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on Mathematical Software, 30(2), 2004, pp. 165--195.
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https://dl.acm.org/doi/abs/10.1145/992200.992205
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.. [5] T. A. Davis and I. S. Duff, A combined unifrontal/multifrontal
|
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method for unsymmetric sparse matrices, ACM Trans. on
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Mathematical Software, 25(1), 1999, pp. 1--19.
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|
https://doi.org/10.1145/305658.287640
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|
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.. [6] T. A. Davis and I. S. Duff, An unsymmetric-pattern multifrontal
|
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method for sparse LU factorization, SIAM J. Matrix Analysis and
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Computations, 18(1), 1997, pp. 140--158.
|
|
https://doi.org/10.1137/S0895479894246905T.
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Examples
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--------
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>>> import numpy as np
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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).toarray(), B.toarray())
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True
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"""
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is_pydata_sparse = is_pydata_spmatrix(b)
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pydata_sparse_cls = b.__class__ if is_pydata_sparse else None
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A = convert_pydata_sparse_to_scipy(A)
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b = convert_pydata_sparse_to_scipy(b)
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if not (issparse(A) and A.format in ("csc", "csr")):
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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, stacklevel=2)
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# b is a vector only if b have shape (n,) or (n, 1)
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b_is_sparse = issparse(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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# sum duplicates for non-canonical format
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A.sum_duplicates()
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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(f"matrix must be square (has shape {(M, N)})")
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if M != b.shape[0]:
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raise ValueError(f"matrix - rhs dimension mismatch ({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_family, A = _get_umf_family(A)
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umf = umfpack.UmfpackContext(umf_family)
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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 A.format == "csc":
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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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indices = A.indices.astype(np.intc, copy=False)
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indptr = A.indptr.astype(np.intc, copy=False)
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options = dict(ColPerm=permc_spec)
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x, info = _superlu.gssv(N, A.nnz, A.data, indices, 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, stacklevel=2)
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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 (b.format == "csc" or is_pydata_spmatrix(b)):
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warn('spsolve is more efficient when sparse b '
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'is in the CSC matrix format',
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SparseEfficiencyWarning, stacklevel=2)
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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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# TODO: replace this with
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# bj = b[:, j].toarray().ravel()
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# once 1D sparse arrays are supported.
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# That is a slightly faster code path.
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bj = b[:, [j]].toarray().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.full(segment_length, j, dtype=int))
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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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if is_pydata_sparse:
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x = pydata_sparse_cls.from_scipy_sparse(x)
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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. Most efficient when provided in CSC
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format. Other formats will be converted to CSC before factorization.
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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 https://portal.nersc.gov/project/sparse/superlu/
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Examples
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--------
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>>> import numpy as np
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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 is_pydata_spmatrix(A):
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def csc_construct_func(*a, cls=type(A)):
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return cls.from_scipy_sparse(csc_matrix(*a))
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A = A.to_scipy_sparse().tocsc()
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else:
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csc_construct_func = csc_matrix
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if not (issparse(A) and A.format == "csc"):
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A = csc_matrix(A)
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warn('splu converted its input to CSC format',
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SparseEfficiencyWarning, stacklevel=2)
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# sum duplicates for non-canonical format
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A.sum_duplicates()
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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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indices, indptr = _safe_downcast_indices(A)
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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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# Ensure that no column permutations are applied
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if (_options["ColPerm"] == "NATURAL"):
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_options["SymmetricMode"] = True
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return _superlu.gstrf(N, A.nnz, A.data, indices, indptr,
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csc_construct_func=csc_construct_func,
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ilu=False, options=_options)
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|
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|
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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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"""
|
|
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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A : (N, N) array_like
|
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Sparse matrix to factorize. Most efficient when provided in CSC format.
|
|
Other formats will be converted to CSC before factorization.
|
|
drop_tol : float, optional
|
|
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
|
|
Specifies the fill ratio upper bound (>= 1.0) for ILU. (default: 10)
|
|
drop_rule : str, optional
|
|
Comma-separated string of drop rules to use.
|
|
Available rules: ``basic``, ``prows``, ``column``, ``area``,
|
|
``secondary``, ``dynamic``, ``interp``. (Default: ``basic,area``)
|
|
|
|
See SuperLU documentation for details.
|
|
|
|
Remaining other options
|
|
Same as for `splu`
|
|
|
|
Returns
|
|
-------
|
|
invA_approx : scipy.sparse.linalg.SuperLU
|
|
Object, which has a ``solve`` method.
|
|
|
|
See also
|
|
--------
|
|
splu : complete LU decomposition
|
|
|
|
Notes
|
|
-----
|
|
To improve the better approximation to the inverse, you may need to
|
|
increase `fill_factor` AND decrease `drop_tol`.
|
|
|
|
This function uses the SuperLU library.
|
|
|
|
Examples
|
|
--------
|
|
>>> import numpy as np
|
|
>>> from scipy.sparse import csc_matrix
|
|
>>> from scipy.sparse.linalg import spilu
|
|
>>> A = csc_matrix([[1., 0., 0.], [5., 0., 2.], [0., -1., 0.]], dtype=float)
|
|
>>> B = spilu(A)
|
|
>>> x = np.array([1., 2., 3.], dtype=float)
|
|
>>> 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 is_pydata_spmatrix(A):
|
|
def csc_construct_func(*a, cls=type(A)):
|
|
return cls.from_scipy_sparse(csc_matrix(*a))
|
|
A = A.to_scipy_sparse().tocsc()
|
|
else:
|
|
csc_construct_func = csc_matrix
|
|
|
|
if not (issparse(A) and A.format == "csc"):
|
|
A = csc_matrix(A)
|
|
warn('spilu converted its input to CSC format',
|
|
SparseEfficiencyWarning, stacklevel=2)
|
|
|
|
# sum duplicates for non-canonical format
|
|
A.sum_duplicates()
|
|
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?
|
|
|
|
indices, indptr = _safe_downcast_indices(A)
|
|
|
|
_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)
|
|
|
|
# Ensure that no column permutations are applied
|
|
if (_options["ColPerm"] == "NATURAL"):
|
|
_options["SymmetricMode"] = True
|
|
|
|
return _superlu.gstrf(N, A.nnz, A.data, indices, indptr,
|
|
csc_construct_func=csc_construct_func,
|
|
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. A in CSC format is most efficient. A CSR format matrix will
|
|
be converted to CSC before factorization.
|
|
|
|
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
|
|
--------
|
|
>>> import numpy as np
|
|
>>> from scipy.sparse.linalg import factorized
|
|
>>> from scipy.sparse import csc_matrix
|
|
>>> A = np.array([[ 3. , 2. , -1. ],
|
|
... [ 2. , -2. , 4. ],
|
|
... [-1. , 0.5, -1. ]])
|
|
>>> solve = factorized(csc_matrix(A)) # Makes LU decomposition.
|
|
>>> rhs1 = np.array([1, -2, 0])
|
|
>>> solve(rhs1) # Uses the LU factors.
|
|
array([ 1., -2., -2.])
|
|
|
|
"""
|
|
if is_pydata_spmatrix(A):
|
|
A = A.to_scipy_sparse().tocsc()
|
|
|
|
if useUmfpack:
|
|
if noScikit:
|
|
raise RuntimeError('Scikits.umfpack not installed.')
|
|
|
|
if not (issparse(A) and A.format == "csc"):
|
|
A = csc_matrix(A)
|
|
warn('splu converted its input to CSC format',
|
|
SparseEfficiencyWarning, stacklevel=2)
|
|
|
|
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_family, A = _get_umf_family(A)
|
|
umf = umfpack.UmfpackContext(umf_family)
|
|
|
|
# Make LU decomposition.
|
|
umf.numeric(A)
|
|
|
|
def solve(b):
|
|
with np.errstate(divide="ignore", invalid="ignore"):
|
|
# Ignoring warnings with numpy >= 1.23.0, see gh-16523
|
|
result = umf.solve(umfpack.UMFPACK_A, A, b, autoTranspose=True)
|
|
|
|
return result
|
|
|
|
return solve
|
|
else:
|
|
return splu(A).solve
|
|
|
|
|
|
def spsolve_triangular(A, b, lower=True, overwrite_A=False, overwrite_b=False,
|
|
unit_diagonal=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.
|
|
unit_diagonal : bool, optional
|
|
If True, diagonal elements of `a` are assumed to be 1 and will not be
|
|
referenced.
|
|
|
|
.. versionadded:: 1.4.0
|
|
|
|
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
|
|
--------
|
|
>>> import numpy as np
|
|
>>> 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
|
|
"""
|
|
|
|
if is_pydata_spmatrix(A):
|
|
A = A.to_scipy_sparse().tocsr()
|
|
|
|
# Check the input for correct type and format.
|
|
if not (issparse(A) and A.format == "csr"):
|
|
warn('CSR matrix format is required. Converting to CSR matrix.',
|
|
SparseEfficiencyWarning, stacklevel=2)
|
|
A = csr_matrix(A)
|
|
elif not overwrite_A:
|
|
A = A.copy()
|
|
|
|
if A.shape[0] != A.shape[1]:
|
|
raise ValueError(
|
|
f'A must be a square matrix but its shape is {A.shape}.')
|
|
|
|
# sum duplicates for non-canonical format
|
|
A.sum_duplicates()
|
|
|
|
b = np.asanyarray(b)
|
|
|
|
if b.ndim not in [1, 2]:
|
|
raise ValueError(
|
|
f'b must have 1 or 2 dims but its shape is {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 '
|
|
f'{A.shape} and the shape of b is {b.shape}.'
|
|
)
|
|
|
|
# Init x as (a copy of) b.
|
|
x_dtype = np.result_type(A.data, b, np.float64)
|
|
if overwrite_b:
|
|
if np.can_cast(b.dtype, x_dtype, casting='same_kind'):
|
|
x = b
|
|
else:
|
|
raise ValueError(
|
|
f'Cannot overwrite b (dtype {b.dtype}) with result '
|
|
f'of type {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 not unit_diagonal and (indptr_stop <= indptr_start
|
|
or A.indices[A_diagonal_index_row_i] < i):
|
|
raise LinAlgError(
|
|
f'A is singular: diagonal {i} is zero.')
|
|
if not unit_diagonal and 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.
|
|
if not unit_diagonal:
|
|
x[i] /= A.data[A_diagonal_index_row_i]
|
|
|
|
return x
|