R: The QR Decomposition of a Matrix

qr {base}

R Documentation

The QR Decomposition of a Matrix

Description

qr computes the QR decomposition of a matrix. It provides an interface to the techniques used in the LINPACK routine DQRDC or (for complex matrices) the LAPACK routine ZGEQP3.

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.

k

effective rank.

Details

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.

All the above functions keep dimnames (and names) of x and y if there are.

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

is.qr returns TRUE if x is a list with components named qr, rank and qraux 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 or LAPACK. The components in the returned value correspond directly to the values returned by DQRDC/ZGEQP3.

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: always full rank in the complex case.

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 det.

The complex case uses column pivoting and does not attempt to detect rank-deficient matrices.

References

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

Anderson. E. and ten others (1999) LAPACK Users' Guide. Third Edition. SIAM.
Available on-line at http://www.netlib.org/lapack/lug/lapack_lug.html.

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

[Package base version 1.6.1 ]