\Header{qr}{The QR Decomposition of a Matrix}
\alias{as.qr}{qr}
\alias{is.qr}{qr}
\methalias{qr.coef}{qr}
\methalias{qr.fitted}{qr}
\methalias{qr.qty}{qr}
\methalias{qr.qy}{qr}
\methalias{qr.resid}{qr}
\methalias{qr.solve}{qr}
\keyword{algebra}{qr}
\keyword{array}{qr}
\keyword{regression}{qr}
\begin{Description}\relax
\code{qr} computes the QR decomposition of a matrix.  It provides an
interface to the techniques used in the LINPACK routine DQRDC.\end{Description}
\begin{Usage}
\begin{verbatim}
qr(x, tol=1e-07)
qr.coef(qr, y)
qr.qy(qr, y)
qr.qty(qr, y)
qr.resid(qr, y)
qr.fitted(qr, y, k = qr$rank)
qr.solve(a, b, tol = 1e-7)

is.qr(x)
as.qr(x)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{x}] a matrix whose QR decomposition is to be computed.
\item[\code{tol}] the tolerance for detecting linear dependencies in the
columns of \code{x}.
\item[\code{qr}] a QR decomposition of the type computed by \code{qr}.
\item[\code{y, b}] a vector or matrix of right-hand sides of equations.
\item[\code{a}] A matrix or QR decomposition.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
The QR decomposition plays an important role in many
statistical techniques.  In particular it can be used to solve the
equation \eqn{\bold{Ax} = \bold{b}}{} for given matrix \eqn{\bold{A}}{},
and vector \eqn{\bold{b}}{}.  It is useful for computing regression
coefficients and in applying the Newton-Raphson algorithm.

The functions \code{qr.coef}, \code{qr.resid}, and \code{qr.fitted}
return the coefficients, residuals and fitted values obtained when
fitting \code{y} to the matrix with QR decomposition \code{qr}.
\code{qr.qy} and \code{qr.qty} return \code{Q \%*\% y} and
\code{t(Q) \%*\% y}, where \code{Q} is the \eqn{\bold{Q}}{} matrix.

\code{qr.solve} solves systems of equations via the QR decomposition.

\code{is.qr} returns \code{TRUE} if \code{x} is a \code{\Link{list}}
with components named \code{qr}, \code{rank} and \code{qraux} and
\code{FALSE} otherwise.

It is not possible to coerce objects to mode \code{"qr"}.  Objects
either are QR decompositions or they are not.\end{Details}
\begin{Value}
The QR decomposition of the matrix as computed by LINPACK.
The components in the returned value correspond directly
to the values returned by DQRDC.
\begin{ldescription}
\item[\code{qr}] a matrix with the same dimensions as \code{x}.
The upper triangle contains the \eqn{\bold{R}}{} of the decomposition
and the lower triangle contains information on the \eqn{\bold{Q}}{} of
the decomposition (stored in compact form).
\item[\code{qraux}] a vector of length \code{ncol(x)} which contains
additional information on \eqn{\bold{Q}}{}.
\item[\code{rank}] the rank of \code{x} as computed by the decomposition.
\item[\code{pivot}] information on the pivoting strategy used during
the decomposition.
\end{ldescription}
\end{Value}
\begin{Note}\relax
To compute the determinant of a matrix (do you \emph{really} need it?),
the QR decomposition is much more efficient than using Eigen values
(\code{\Link{eigen}}).  See \code{det2} in the examples below.\end{Note}
\begin{References}\relax
Dongarra, J. J., Bunch, J. R., Moler, C. B. and Stewart, G. W. (1978)
\emph{LINPACK Users Guide.}  Philadelphia: SIAM Publications.\end{References}
\begin{SeeAlso}\relax
\code{\Link{qr.Q}},  \code{\Link{qr.R}},  \code{\Link{qr.X}} for
reconstruction of the matrices.
\code{\Link{solve.qr}},  \code{\Link{lsfit}},
\code{\Link{eigen}}, \code{\Link{svd}}.

\code{\Link{det}} (using \code{qr}) to compute the determinant of a matrix.\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
hilbert <- function(n) { i <- 1:n; 1 / outer(i - 1, i, "+") }
h9 <- hilbert(9); h9
qr(h9)$rank           #--> only 7
qrh9 <- qr(h9, tol = 1e-10)
qrh9$rank             #--> 9
##-- Solve linear equation system  H %*% x = y :
y <- 1:9/10
x <- qr.solve(h9, y, tol = 1e-10) # or equivalently :
x <- qr.coef(qrh9, y) #-- is == but much better than
                      #-- solve(h9) %*% y
h9 %*% x              # = y
\end{ExampleCode}
\end{Examples}

