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
or the LAPACK routines DGEQP3 and (for complex matrices) ZGEQP3.
qr(x, ...)
## Default S3 method:
qr(x, tol = 1e-07 , LAPACK = FALSE, ...)
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)
## S3 method for class 'qr'
solve(a, b, ...)
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 |
qr |
a QR decomposition of the type computed by |
y , b |
a vector or matrix of right-hand sides of equations. |
a |
a QR decomposition or ( |
k |
effective rank. |
LAPACK |
logical. For real |
... |
further arguments passed to or from other methods |
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
.
(If pivoting is used, some of the coefficients will be NA
.)
qr.qy
and qr.qty
return Q %*% y
and
t(Q) %*% y
, where Q
is the (complete) \bold{Q}
matrix.
All the above functions keep dimnames
(and names
) of
x
and y
if there are.
solve.qr
is the method for solve
for qr
objects.
qr.solve
solves systems of equations via the QR decomposition:
if a
is a QR decomposition it is the same as solve.qr
,
but if a
is a rectangular matrix the QR decomposition is
computed first. Either will handle over- and under-determined
systems, providing a least-squares fit if appropriate.
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.
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/DGEQP3/ZGEQP3.
qr |
a matrix with the same dimensions as |
qraux |
a vector of length |
rank |
the rank of |
pivot |
information on the pivoting strategy used during the decomposition. |
Non-complex QR objects computed by LAPACK have the attribute
"useLAPACK"
with value TRUE
.
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
.
Using LAPACK (including in the complex case) uses column pivoting and does not attempt to detect rank-deficient matrices.
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.
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.
qr.Q
, qr.R
, qr.X
for
reconstruction of the matrices.
lm.fit
, lsfit
,
eigen
, svd
.
det
(using qr
) to compute the determinant of a matrix.
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
## overdetermined system
A <- matrix(runif(12), 4)
b <- 1:4
qr.solve(A, b) # or solve(qr(A), b)
solve(qr(A, LAPACK=TRUE), b)
# this is a least-squares solution, cf. lm(b ~ 0 + A)
## underdetermined system
A <- matrix(runif(12), 3)
b <- 1:3
qr.solve(A, b)
solve(qr(A, LAPACK=TRUE), b)
# solutions will have one zero, not necessarily the same one