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from __future__ import division, print_function, absolute_import
from warnings import warn
import numpy as np
from numpy import asarray, empty, ravel, nonzero
from scipy.sparse import (isspmatrix_csc, isspmatrix_csr, isspmatrix,
SparseEfficiencyWarning, csc_matrix, csr_matrix)
from scipy.linalg import LinAlgError
from . import _superlu
noScikit = False
try:
import scikits.umfpack as umfpack
except ImportError:
noScikit = True
useUmfpack = not noScikit
__all__ = ['use_solver', 'spsolve', 'splu', 'spilu', 'factorized',
'MatrixRankWarning', 'spsolve_triangular']
class MatrixRankWarning(UserWarning):
pass
def use_solver(**kwargs):
"""
Select default sparse direct solver to be used.
Parameters
----------
useUmfpack : bool, optional
Use UMFPACK over SuperLU. Has effect only if scikits.umfpack is
installed. Default: True
Notes
-----
The default sparse solver is umfpack when available
(scikits.umfpack is installed). This can be changed by passing
useUmfpack = False, which then causes the always present SuperLU
based solver to be used.
Umfpack requires a CSR/CSC matrix to have sorted column/row indices. If
sure that the matrix fulfills this, pass ``assumeSortedIndices=True``
to gain some speed.
"""
if 'useUmfpack' in kwargs:
globals()['useUmfpack'] = kwargs['useUmfpack']
#TODO: pass other options to scikit
def _get_umf_family(A):
"""Get umfpack family string given the sparse matrix dtype."""
family = {'di': 'di', 'Di': 'zi', 'dl': 'dl', 'Dl': 'zl'}
dt = A.dtype.char + A.indices.dtype.char
return family[dt]
def spsolve(A, b, permc_spec=None, use_umfpack=True):
"""Solve the sparse linear system Ax=b, where b may be a vector or a matrix.
Parameters
----------
A : ndarray or sparse matrix
The square matrix A will be converted into CSC or CSR form
b : ndarray or sparse matrix
The matrix or vector representing the right hand side of the equation.
If a vector, b.shape must be (n,) or (n, 1).
permc_spec : str, optional
How to permute the columns of the matrix for sparsity preservation.
(default: 'COLAMD')
- ``NATURAL``: natural ordering.
- ``MMD_ATA``: minimum degree ordering on the structure of A^T A.
- ``MMD_AT_PLUS_A``: minimum degree ordering on the structure of A^T+A.
- ``COLAMD``: approximate minimum degree column ordering
use_umfpack : bool, optional
if True (default) then use umfpack for the solution. This is
only referenced if b is a vector and ``scikit-umfpack`` is installed.
Returns
-------
x : ndarray or sparse matrix
the solution of the sparse linear equation.
If b is a vector, then x is a vector of size A.shape[1]
If b is a matrix, then x is a matrix of size (A.shape[1], b.shape[1])
Notes
-----
For solving the matrix expression AX = B, this solver assumes the resulting
matrix X is sparse, as is often the case for very sparse inputs. If the
resulting X is dense, the construction of this sparse result will be
relatively expensive. In that case, consider converting A to a dense
matrix and using scipy.linalg.solve or its variants.
Examples
--------
>>> from scipy.sparse import csc_matrix
>>> from scipy.sparse.linalg import spsolve
>>> A = csc_matrix([[3, 2, 0], [1, -1, 0], [0, 5, 1]], dtype=float)
>>> B = csc_matrix([[2, 0], [-1, 0], [2, 0]], dtype=float)
>>> x = spsolve(A, B)
>>> np.allclose(A.dot(x).todense(), B.todense())
True
"""
if not (isspmatrix_csc(A) or isspmatrix_csr(A)):
A = csc_matrix(A)
warn('spsolve requires A be CSC or CSR matrix format',
SparseEfficiencyWarning)
# b is a vector only if b have shape (n,) or (n, 1)
b_is_sparse = isspmatrix(b)
if not b_is_sparse:
b = asarray(b)
b_is_vector = ((b.ndim == 1) or (b.ndim == 2 and b.shape[1] == 1))
A.sort_indices()
A = A.asfptype() # upcast to a floating point format
result_dtype = np.promote_types(A.dtype, b.dtype)
if A.dtype != result_dtype:
A = A.astype(result_dtype)
if b.dtype != result_dtype:
b = b.astype(result_dtype)
# validate input shapes
M, N = A.shape
if (M != N):
raise ValueError("matrix must be square (has shape %s)" % ((M, N),))
if M != b.shape[0]:
raise ValueError("matrix - rhs dimension mismatch (%s - %s)"
% (A.shape, b.shape[0]))
use_umfpack = use_umfpack and useUmfpack
if b_is_vector and use_umfpack:
if b_is_sparse:
b_vec = b.toarray()
else:
b_vec = b
b_vec = asarray(b_vec, dtype=A.dtype).ravel()
if noScikit:
raise RuntimeError('Scikits.umfpack not installed.')
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))
x = umf.linsolve(umfpack.UMFPACK_A, A, b_vec,
autoTranspose=True)
else:
if b_is_vector and b_is_sparse:
b = b.toarray()
b_is_sparse = False
if not b_is_sparse:
if isspmatrix_csc(A):
flag = 1 # CSC format
else:
flag = 0 # CSR format
options = dict(ColPerm=permc_spec)
x, info = _superlu.gssv(N, A.nnz, A.data, A.indices, A.indptr,
b, flag, options=options)
if info != 0:
warn("Matrix is exactly singular", MatrixRankWarning)
x.fill(np.nan)
if b_is_vector:
x = x.ravel()
else:
# b is sparse
Afactsolve = factorized(A)
if not isspmatrix_csc(b):
warn('spsolve is more efficient when sparse b '
'is in the CSC matrix format', SparseEfficiencyWarning)
b = csc_matrix(b)
# Create a sparse output matrix by repeatedly applying
# the sparse factorization to solve columns of b.
data_segs = []
row_segs = []
col_segs = []
for j in range(b.shape[1]):
bj = b[:, j].A.ravel()
xj = Afactsolve(bj)
w = np.flatnonzero(xj)
segment_length = w.shape[0]
row_segs.append(w)
col_segs.append(np.ones(segment_length, dtype=int)*j)
data_segs.append(np.asarray(xj[w], dtype=A.dtype))
sparse_data = np.concatenate(data_segs)
sparse_row = np.concatenate(row_segs)
sparse_col = np.concatenate(col_segs)
x = A.__class__((sparse_data, (sparse_row, sparse_col)),
shape=b.shape, dtype=A.dtype)
return x
def splu(A, permc_spec=None, diag_pivot_thresh=None,
drop_tol=None, relax=None, panel_size=None, options=dict()):
"""
Compute the LU decomposition of a sparse, square matrix.
Parameters
----------
A : sparse matrix
Sparse matrix to factorize. Should be in CSR or CSC format.
permc_spec : str, optional
How to permute the columns of the matrix for sparsity preservation.
(default: 'COLAMD')
- ``NATURAL``: natural ordering.
- ``MMD_ATA``: minimum degree ordering on the structure of A^T A.
- ``MMD_AT_PLUS_A``: minimum degree ordering on the structure of A^T+A.
- ``COLAMD``: approximate minimum degree column ordering
diag_pivot_thresh : float, optional
Threshold used for a diagonal entry to be an acceptable pivot.
See SuperLU user's guide for details [1]_
drop_tol : float, optional
(deprecated) No effect.
relax : int, optional
Expert option for customizing the degree of relaxing supernodes.
See SuperLU user's guide for details [1]_
panel_size : int, optional
Expert option for customizing the panel size.
See SuperLU user's guide for details [1]_
options : dict, optional
Dictionary containing additional expert options to SuperLU.
See SuperLU user guide [1]_ (section 2.4 on the 'Options' argument)
for more details. For example, you can specify
``options=dict(Equil=False, IterRefine='SINGLE'))``
to turn equilibration off and perform a single iterative refinement.
Returns
-------
invA : scipy.sparse.linalg.SuperLU
Object, which has a ``solve`` method.
See also
--------
spilu : incomplete LU decomposition
Notes
-----
This function uses the SuperLU library.
References
----------
.. [1] SuperLU http://crd.lbl.gov/~xiaoye/SuperLU/
Examples
--------
>>> from scipy.sparse import csc_matrix
>>> from scipy.sparse.linalg import splu
>>> A = csc_matrix([[1., 0., 0.], [5., 0., 2.], [0., -1., 0.]], dtype=float)
>>> B = splu(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 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(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=False, options=_options)
def spilu(A, drop_tol=None, fill_factor=None, drop_rule=None, permc_spec=None,
diag_pivot_thresh=None, relax=None, panel_size=None, options=None):
"""
Compute an incomplete LU decomposition for a sparse, square matrix.
The resulting object is an approximation to the inverse of `A`.
Parameters
----------
A : (N, N) array_like
Sparse matrix to factorize
drop_tol : float, optional
Drop tolerance (0 <= tol <= 1) for an incomplete LU decomposition.
(default: 1e-4)
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
--------
>>> 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 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.sort_indices()
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.
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 copy of b.
if overwrite_b:
x = b
else:
x = b.copy()
# 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: '
'{}th diagonal is zero!'.format(i))
if A.indices[A_diagonal_index_row_i] > i:
raise LinAlgError('A is no triangular matrix: entry '
'[{},{}] is not zero!'.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