qr {base}

R Documentation

The QR Decomposition of a Matrix

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 \bold{Ax} = \bold{b} for given matrix \bold{A}, and vector \bold{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 \bold{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.

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.

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 \bold{R} of the decomposition and the lower triangle contains information on the \bold{Q} of the decomposition (stored in compact form).
qraux	a vector of length ncol(x) which contains additional information on \bold{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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