| sparseLU-class {Matrix} | R Documentation |
Objects of this class contain the components of the LU decomposition of a sparse square matrix.
Objects can be created by calls of the form new("sparseLU",
...) but are more commonly created by function lu()
applied to a sparse matrix, such as a matrix of class
dgCMatrix.
L:Object of class "dtCMatrix", the lower
triangular factor from the left.
U:Object of class "dtCMatrix", the upper
triangular factor from the right.
p:Object of class "integer", permutation
applied from the left.
q:Object of class "integer", permutation
applied from the right.
Dim:the dimension of the original matrix; inherited
from class MatrixFactorization.
Class "LU", directly.
Class "MatrixFactorization", by class "LU".
signature(x = "sparseLU") Returns a list with
components P, L, U, and Q,
where P and Q represent fill-reducing
permutations, and L, and U the lower and upper
triangular matrices of the decomposition. The original matrix
corresponds to the product PLUQ.
The decomposition is of the form
A = P'LUQ
where all matrices are sparse and of size n by n. The matrices P, its transpose P', and Q are permutation matrices, L is lower triangular and U is upper triangular.
## Extending the one in examples(lu), calling the matrix A,
## and confirming the factorization identities :
A <- as(readMM(system.file("external/pores_1.mtx",
package = "Matrix")),
"CsparseMatrix")
str(luA <- lu(A)) # p is a 0-based permutation of the rows
# q is a 0-based permutation of the columns
xA <- expand(luA)
## which is simply doing
stopifnot(identical(xA$ L, luA@L),
identical(xA$ U, luA@U),
identical(xA$ P, as(luA@p +1L, "pMatrix")),
identical(xA$ Q, as(luA@q +1L, "pMatrix")))
P.LUQ <- with(xA, t(P) %*% L %*% U %*% Q)
stopifnot(all.equal(A, P.LUQ, tol = 1e-12))
## permute rows and columns of original matrix
pA <- A[luA@p + 1L, luA@q + 1L]
stopifnot(identical(pA, with(xA, P %*% A %*% t(Q))))
pLU <- drop0(luA@L %*% luA@U) # L %*% U -- dropping extra zeros
stopifnot(all.equal(pA, pLU))