scipy.sparse.linalg.

SuperLU#

class scipy.sparse.linalg.SuperLU#

LU factorization of a sparse matrix.

Factorization is represented as:

Pr @ A @ Pc = L @ U

To construct these SuperLU objects, call the splu and spilu functions.

Attributes:

shape: Shape of the original matrix as a tuple of ints.
nnz: Number of nonzero elements in the matrix.
perm_c: Permutation Pc represented as an array of indices.
perm_r: Permutation Pr represented as an array of indices.
L: Lower triangular factor with unit diagonal as a scipy.sparse.csc_matrix.
U: Upper triangular factor as a scipy.sparse.csc_matrix.

Methods

solve(rhs[, trans])

Solves linear system of equations with one or several right-hand sides.

Notes

Added in version 0.14.0.

Examples

The LU decomposition can be used to solve matrix equations. Consider:

>>> import numpy as np
>>> from scipy.sparse import csc_matrix
>>> from scipy.sparse.linalg import splu
>>> A = csc_matrix([[1,2,0,4], [1,0,0,1], [1,0,2,1], [2,2,1,0.]])

This can be solved for a given right-hand side:

>>> lu = splu(A)
>>> b = np.array([1, 2, 3, 4])
>>> x = lu.solve(b)
>>> A.dot(x)
array([ 1.,  2.,  3.,  4.])

The lu object also contains an explicit representation of the decomposition. The permutations are represented as mappings of indices:

>>> lu.perm_r
array([2, 1, 3, 0], dtype=int32)  # may vary
>>> lu.perm_c
array([0, 1, 3, 2], dtype=int32)  # may vary

The L and U factors are sparse matrices in CSC format:

>>> lu.L.toarray()
array([[ 1. ,  0. ,  0. ,  0. ],  # may vary
       [ 0.5,  1. ,  0. ,  0. ],
       [ 0.5, -1. ,  1. ,  0. ],
       [ 0.5,  1. ,  0. ,  1. ]])
>>> lu.U.toarray()
array([[ 2. ,  2. ,  0. ,  1. ],  # may vary
       [ 0. , -1. ,  1. , -0.5],
       [ 0. ,  0. ,  5. , -1. ],
       [ 0. ,  0. ,  0. ,  2. ]])

The permutation matrices can be constructed:

>>> Pr = csc_matrix((np.ones(4), (lu.perm_r, np.arange(4))))
>>> Pc = csc_matrix((np.ones(4), (np.arange(4), lu.perm_c)))

We can reassemble the original matrix:

>>> (Pr.T @ (lu.L @ lu.U) @ Pc.T).toarray()
array([[ 1.,  2.,  0.,  4.],
       [ 1.,  0.,  0.,  1.],
       [ 1.,  0.,  2.,  1.],
       [ 2.,  2.,  1.,  0.]])