\HeaderA{QR.Auxiliaries}{Reconstruct the Q, R, or X Matrices from a QR Object}{QR.Auxiliaries}
\aliasA{qr.Q}{QR.Auxiliaries}{qr.Q}
\aliasA{qr.R}{QR.Auxiliaries}{qr.R}
\aliasA{qr.X}{QR.Auxiliaries}{qr.X}
\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 =)
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{\LinkA{qr}{qr}} or
\code{\LinkA{lsfit}{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.  Defaults to all \code{1}s.
\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 part or all of \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 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{\LinkA{qr}{qr}},
\code{\LinkA{qr.qy}{qr.qy}}.
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
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 if there has been no pivoting, as here.
Q %*% R
\end{ExampleCode}
\end{Examples}

