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 nnz perm_c perm_r L U
Methods ------- solve
Notes -----
.. versionadded:: 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.])