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\name{sparseLU-class}
\docType{class}
\alias{sparseLU-class}
\alias{expand,sparseLU-method}
\title{Sparse LU decomposition of a square sparse matrix}
\description{Objects of this class contain the components of the LU
decomposition of a sparse square matrix.}
\section{Objects from the Class}{
Objects can be created by calls of the form \code{new("sparseLU",
...)} but are more commonly created by function \code{\link{lu}()}
applied to a sparse matrix, such as a matrix of class
\code{\linkS4class{dgCMatrix}}.
}
\section{Slots}{
\describe{
\item{\code{L}:}{Object of class \code{"\linkS4class{dtCMatrix}"}, the lower
triangular factor from the left.}
\item{\code{U}:}{Object of class \code{"\linkS4class{dtCMatrix}"}, the upper
triangular factor from the right.}
\item{\code{p}:}{Object of class \code{"integer"}, permutation
applied from the left. }
\item{\code{q}:}{Object of class \code{"integer"}, permutation
applied from the right.}
\item{\code{Dim}:}{the dimension of the original matrix; inherited
from class \code{\linkS4class{MatrixFactorization}}.}
}
}
\section{Extends}{
Class \code{"\linkS4class{LU}"}, directly.
Class \code{"\linkS4class{MatrixFactorization}"}, by class \code{"LU"}.
}
\section{Methods}{
\describe{
\item{expand}{\code{signature(x = "sparseLU")} Returns a list with
components \code{P}, \code{L}, \code{U}, and \code{Q},
where \eqn{P} and \eqn{Q} represent fill-reducing
permutations, and \eqn{L}, and \eqn{U} the lower and upper
triangular matrices of the decomposition. The original matrix
corresponds to the product \eqn{PLUQ}.}
}
}
%\references{}
\note{
The decomposition is of the form
\deqn{A = P'LUQ}{A = P'LUQ}
where all matrices are sparse and of size \eqn{n\times n}{n by n}.
The matrices \eqn{P}, its transpose \eqn{P'}, and \eqn{Q} are
permutation matrices, \eqn{L} is lower triangular and \eqn{U} is upper
triangular.
}
\seealso{
\code{\link{lu}}, \code{\link[base]{solve}}, \code{\linkS4class{dgCMatrix}}
}
\examples{
## 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))
}
\keyword{classes}
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