R: The QR Decomposition of a Matrix

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 computes the QR decomposition of a matrix. It provides an interface to the techniques used in the LINPACK routine DQRDC.

Details

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.

Note

To compute the determinant of a matrix (do you really need it?), the QR decomposition is much more efficient than using Eigen values (eigen). See det2 in the examples below.

References

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

Examples

## The determinant of a matrix  -- if you really must have it
det2 <- function(x) prod(diag(qr(x)$qr))*(-1)^(ncol(x)-1)
det2(print(cbind(1,1:3,c(2,0,1))))

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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