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 thesplu
andspilu
functions.Notes
New 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, linalg as sla >>> 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 = sla.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([0, 2, 1, 3], dtype=int32) >>> lu.perm_c array([2, 0, 1, 3], dtype=int32)
The L and U factors are sparse matrices in CSC format:
>>> lu.L.A array([[ 1. , 0. , 0. , 0. ], [ 0. , 1. , 0. , 0. ], [ 0. , 0. , 1. , 0. ], [ 1. , 0.5, 0.5, 1. ]]) >>> lu.U.A array([[ 2., 0., 1., 4.], [ 0., 2., 1., 1.], [ 0., 0., 1., 1.], [ 0., 0., 0., -5.]])
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).A array([[ 1., 2., 0., 4.], [ 1., 0., 0., 1.], [ 1., 0., 2., 1.], [ 2., 2., 1., 0.]])
- 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.