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\name{sparseQR-class}
\docType{class}
\alias{sparseQR-class}
\alias{qr.coef,sparseQR,dgeMatrix-method}
\alias{qr.coef,sparseQR,matrix-method}
\alias{qr.coef,sparseQR,numeric-method}
\alias{qr.fitted,sparseQR,dgeMatrix-method}
\alias{qr.fitted,sparseQR,matrix-method}
\alias{qr.fitted,sparseQR,numeric-method}
\alias{qr.qty,sparseQR,dgeMatrix-method}
\alias{qr.qty,sparseQR,matrix-method}
\alias{qr.qty,sparseQR,numeric-method}
\alias{qr.qy,sparseQR,dgeMatrix-method}
\alias{qr.qy,sparseQR,matrix-method}
\alias{qr.qy,sparseQR,numeric-method}
\alias{qr.resid,sparseQR,dgeMatrix-method}
\alias{qr.resid,sparseQR,matrix-method}
\alias{qr.resid,sparseQR,numeric-method}
\title{Sparse QR decomposition of a sparse matrix}
\description{Objects of this class represent a QR decomposition of a
sparse rectangular matrix.}
\section{Objects from the Class}{
Objects can be created by calls of the form \code{new("sparseQR", ...)}
but are more commonly created by function \code{\link[base]{qr}} applied
to a sparse matrix such as a matrix of class
\code{\linkS4class{dgCMatrix}}.}
}
\section{Slots}{
\describe{
\item{\code{V}:}{Object of class \code{"dgCMatrix"} The columns of
\code{V} are the vectors that generate the Householder
transformations of which the matrix Q is composed.}
\item{\code{beta}:}{Object of class \code{"numeric"} The normalizing
factors for the Householder transformations.}
\item{\code{p}:}{Object of class \code{"integer"} Permutation
applied to the rows of the original matrix.}
\item{\code{R}:}{Object of class \code{"dgCMatrix"} An upper
triangular matrix of dimension \ }
\item{\code{q}:}{Object of class \code{"integer"} Permutation
applied from the right. Can be of length 0 which implies no
permutation.}
}
}
\section{Methods}{
\describe{
\item{qr.coef}{\code{signature(qr = "sparseQR", y = "dgeMatrix")}: ... }
\item{qr.coef}{\code{signature(qr = "sparseQR", y = "matrix")}: ... }
\item{qr.coef}{\code{signature(qr = "sparseQR", y = "numeric")}: ... }
\item{qr.fitted}{\code{signature(qr = "sparseQR", y = "dgeMatrix")}: ... }
\item{qr.fitted}{\code{signature(qr = "sparseQR", y = "matrix")}: ... }
\item{qr.fitted}{\code{signature(qr = "sparseQR", y = "numeric")}: ... }
\item{qr.qty}{\code{signature(qr = "sparseQR", y = "dgeMatrix")}: ... }
\item{qr.qty}{\code{signature(qr = "sparseQR", y = "matrix")}: ... }
\item{qr.qty}{\code{signature(qr = "sparseQR", y = "numeric")}: ... }
\item{qr.qy}{\code{signature(qr = "sparseQR", y = "dgeMatrix")}: ... }
\item{qr.qy}{\code{signature(qr = "sparseQR", y = "matrix")}: ... }
\item{qr.qy}{\code{signature(qr = "sparseQR", y = "numeric")}: ... }
\item{qr.resid}{\code{signature(qr = "sparseQR", y = "dgeMatrix")}: ... }
\item{qr.resid}{\code{signature(qr = "sparseQR", y = "matrix")}: ... }
\item{qr.resid}{\code{signature(qr = "sparseQR", y = "numeric")}: ... }
}
}
%\references{}
%\author{}
\note{The decomposition is of the form \code{A[p+1:1,] == Q \%*\% R}, if
the \code{q} slot if of length 0 or \code{A[p+1:1,q+1:1] == Q \%*\% R}
where A is a sparse \eqn{m\times n}{m by n} matrix (\eqn{m\ge n}{m >= n}),
\eqn{R} is an \eqn{m\times n}{m by n} matrix that is zero below the
main diagonal. The \code{p} slot is a 0-based permutation of
\code{1:m} applied to the rows of the original matrix. If the \code{q}
slot has length \code{n} it is a 0-based permutation of \code{1:n}
applied to the columns of the original matrix to reduce the amount
of "fill-in" in the matrix \eqn{R}.
The matrix \eqn{Q} is a "virtual matrix". It is the product of
\eqn{n} Householder transformations. The information to generate
these Householder transformations is stored in the \code{V} and
\code{beta} slots.
The \code{"sparseQR"} methods for the \code{qr.*} functions return
objects of class \code{"dgeMatrix"} (see
\code{\linkS4class{dgeMatrix}}). Results from \code{qr.coef},
\code{qr.resid} and \code{qr.fitted} (when \code{k == ncol(R)}) are
well-defined and should match those from the corresponding dense matrix
calculations. However, because the matrix \code{Q} is not uniquely
defined, the results of \code{qr.qy} and \code{qr.qty} do not
necessarily match those from the corresponding dense matrix
calculations.
Also, the results of \code{qr.qy} and \code{qr.qty} apply to the
permuted column order when the \code{q} slot has length \code{n}.
}
\seealso{
\code{\link[base]{qr}}, \code{\link[base]{qr.Q}},
\code{\link[base]{qr.R}}, \code{\link[base]{qr.fitted}},
\code{\link[base]{qr.resid}}, \code{\link[base]{qr.coef}},
\code{\link[base]{qr.qty}}, \code{\link[base]{qr.qy}},
\code{\linkS4class{dgCMatrix}}, \code{\linkS4class{dgeMatrix}}.
}
\examples{
data(KNex); mm <- KNex$mm
str(mmQR <- qr(mm))
}
\keyword{classes}
\keyword{algebra}
\keyword{array}
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