qr {base} | R Documentation |
qr
computes the QR decomposition of a matrix. It provides an
interface to the techniques used in the LINPACK routine DQRDC.
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$rank) qr.solve(a, b, tol = 1e-7) is.qr(x) as.qr(x)
x |
a matrix whose QR decomposition is to be computed. |
tol |
the tolerance for detecting linear dependencies in the
columns of x . |
qr |
a QR decomposition of the type computed by qr . |
y, b |
a vector or matrix of right-hand sides of equations. |
a |
A matrix or QR decomposition. |
The QR decomposition plays an important role in many statistical techniques. In particular it can be used to solve the equation Ax = b for given matrix A, and vector b. It is useful for computing regression coefficients and in applying the Newton-Raphson algorithm.
The functions qr.coef
, qr.resid
, and qr.fitted
return the coefficients, residuals and fitted values obtained when
fitting y
to the matrix with QR decomposition qr
.
qr.qy
and qr.qty
return Q %*% y
and
t(Q) %*% y
, where Q
is the Q matrix.
qr.solve
solves systems of equations via the QR decomposition.
is.qr
returns TRUE
if x
is a list with a
component named qr
and FALSE
otherwise.
It is not possible to coerce objects to mode "qr"
. Objects
either are QR decompositions or they are not.
qr |
a matrix with the same dimensions as x .
The upper triangle contains the R of the decomposition
and the lower triangle contains information on the Q of
the decomposition (stored in compact form). |
qraux |
a vector of length ncol(x) which contains
additional information on Q. |
rank |
the rank of x as computed by the decomposition. |
pivot |
information on the pivoting strategy used during the decomposition. |
To compute the determinant of a matrix (do you really need it?),
the QR decomposition is much more efficient than using Eigen values
(eigen
). See det2
in the examples below.
Dongarra, J. J., Bunch, J. R., Moler, C. B. and Stewart, G. W. (1978) LINPACK Users Guide. Philadelphia: SIAM Publications.
qr.Q
, qr.R
, qr.X
for
reconstruction of the matrices.
solve.qr
, lsfit
,
eigen
, svd
.
## The determinant of a matrix -- if you really must have it det2 <- function(x) prod(diag(qr(x)$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