\Header{QR.Auxiliaries}{Reconstruct the Q, R, or X Matrices from a QR Object}
\alias{qr.Q}{QR.Auxiliaries}
\alias{qr.R}{QR.Auxiliaries}
\alias{qr.X}{QR.Auxiliaries}
\keyword{algebra}{QR.Auxiliaries}
\keyword{array}{QR.Auxiliaries}
\begin{Description}\relax
Returns the original matrix from which the object was constructed or
the components of the decomposition.\end{Description}
\begin{Usage}
\begin{verbatim}
qr.X(qr, complete = FALSE, ncol =)
qr.Q(qr, complete = FALSE, Dvec = 1)
qr.R(qr, complete = FALSE)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{qr}] object representing a QR decomposition.  This will
typically have come from a previous call to \code{\Link{qr}} or
\code{\Link{lsfit}}.
\item[\code{complete}] logical expression of length 1.  Indicates whether an
arbitrary  orthogonal completion of the \eqn{\bold{Q}}{} or
\eqn{\bold{X}}{} matrices is to be made, or whether the \eqn{\bold{R}}{}
matrix is to be completed  by binding zero-value rows beneath the
square upper triangle.
\item[\code{ncol}] integer in the range \code{1:nrow(qr\$qr)}.  The number
of columns to be in the reconstructed \eqn{\bold{X}}{}.  The default
when \code{complete} is \code{FALSE} is the first
\code{min(ncol(X), nrow(X))} columns of the original \eqn{\bold{X}}{}
from which the qr object was constructed.  The default when
\code{complete} is \code{TRUE} is a square matrix with the original
\eqn{\bold{X}}{} in the first \code{ncol(X)} columns and an arbitrary
orthogonal completion (unitary completion in the complex case) in
the remaining columns.
\item[\code{Dvec}] vector (not matrix) of diagonal values.  Each column of
the returned \eqn{\bold{Q}}{} will be multiplied by the corresponding
diagonal value.
\end{ldescription}
\end{Arguments}
\begin{Value}
\code{qr.X} returns \eqn{\bold{X}}{}, the original matrix from
which the qr object was constructed, provided \code{ncol(X) <= nrow(X)}.
If \code{complete} is \code{TRUE} or the argument \code{ncol} is greater than
\code{ncol(X)}, additional columns from an arbitrary orthogonal
(unitary) completion of \code{X} are returned.

\code{qr.Q} returns \bold{Q}, the order-nrow(X) orthogonal (unitary)
transformation represented by \code{qr}.  If \code{complete} is
\code{TRUE}, \bold{Q} has \code{nrow(X)} columns.  If \code{complete}
is \code{FALSE}, \bold{Q} has \code{ncol(X)} columns.  When
\code{Dvec} is specified, each column of \bold{Q} is multiplied by the
corresponding value in \code{Dvec}.

\code{qr.R} returns \bold{R}, the upper triangular matrix such that
\code{X == Q \%*\% R}.  The number of rows of \bold{R} is
\code{nrow(X)} or \code{ncol(X)}, depending on whether \code{complete}
is \code{TRUE} or \code{FALSE}.\end{Value}
\begin{SeeAlso}\relax
\code{\Link{qr}},
\code{\Link{qr.qy}}.\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
data(LifeCycleSavings)
p <- ncol(x <- LifeCycleSavings[,-1]) # not the `sr'
qrstr <- qr(x)   # dim(x) == c(n,p)
qrstr $ rank # = 4 = p
Q <- qr.Q(qrstr) # dim(Q) == dim(x)
R <- qr.R(qrstr) # dim(R) == ncol(x)
X <- qr.X(qrstr) # X == x
range(X - as.matrix(x))# ~ < 6e-12

## X == Q %*% R :
all((1 - X /( Q %*% R))< 100*.Machine$double.eps)#TRUE

dim(Qc <- qr.Q(qrstr, complete=TRUE)) # Square: dim(Qc) == rep(nrow(x),2)
all((crossprod(Qc) - diag(nrow(x))) < 10*.Machine $double.eps)

QD <- qr.Q(qrstr, D=1:p)      # QD == Q %*% diag(1:p)
all(QD - Q %*% diag(1:p)  < 8* .Machine$double.eps)

dim(Rc <- qr.R(qrstr, complete=TRUE)) # == dim(x)
dim(Xc <- qr.X(qrstr, complete=TRUE)) # square: nrow(x) ^ 2
all(Xc[,1:p] == X)
\end{ExampleCode}
\end{Examples}

