## The QR Decomposition of a Matrix

## Usage

    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)

## Arguments

`x` | a matrix whose QR decomposition is to be computed.
`tol` | the tolerance for detecting linear dependencies in the columns of `x`.
`qr` | a QR decomposition of the type computed by `qr`.
`y, b` | a vector or matrix of right-hand sides of equations.
`a` | A matrix or QR decomposition.

## Description

`qr` provides an interface to the techniques used in the LINPACK routine DQRDC. The QR decomposition plays an important role in many statistical techniques. In particular it can be used to solve the equation Ax = b for given matrix A, and vector b. It is useful for computing regression coefficients and in applying the Newton-Raphson algorithm.

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

`qr.solve` solves systems of equations via the QR decomposition.

`is.qr` returns `TRUE` if `x` is a list with a component named `qr` and `FALSE` otherwise.

It is not possible to coerce objects to mode `"qr"`. Objects either are QR decompositions or they are not.

## 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.

`qr` | a matrix with the same dimensions as `x`. The upper triangle contains the R of the decomposition and the lower triangle contains information on the Q of the decomposition (stored in compact form).
`qraux` | a vector of length `ncol(x)` which contains additional information on Q.
`rank` | the rank of `x` as computed by the decomposition.
`pivot` | information on the pivoting strategy used during the decomposition.

## References

Dongarra, J. J., J. R. Bunch, C. B. Moler and G. W. Stewart (1978). LINPACK Users Guide. Philadelphia, PA: SIAM Publications.

## See Also

`qr.Q`, `qr.R`, `qr.X` for reconstruction of the matrices. `solve.qr`, `lsfit`, `eigen`, `svd`.

## Examples

    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

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