qr {base} | R Documentation |
The QR Decomposition of a Matrix
Description
qr
computes the QR decomposition of a matrix.
Usage
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)
Arguments
x |
a numeric or complex matrix whose QR decomposition is to be computed. Logical matrices are coerced to numeric. |
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. |
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
.
(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 any.
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
and inherits
from "qr"
.
It is not possible to coerce objects to mode "qr"
. Objects
either are QR decompositions or they are not.
The LINPACK interface is restricted to matrices x
with less
than 2^{31}
elements.
qr.fitted
and qr.resid
only support the LINPACK interface.
Unsuccessful results from the underlying LAPACK code will result in an error giving a positive error code: these can only be interpreted by detailed study of the FORTRAN code.
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(2)/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
.
*)
dqrdc2
instead of LINPACK's DQRDC
In the (default) LINPACK case (LAPACK = FALSE
), qr()
uses a modified version of LINPACK's DQRDC, called
‘dqrdc2
’. It differs by using the tolerance tol
for a pivoting strategy which moves columns with near-zero 2-norm to
the right-hand edge of the x matrix. This strategy means that
sequential one degree-of-freedom effects can be computed in a natural
way.
Note
To compute the determinant of a matrix (do you really need it?),
the QR decomposition is much more efficient than using eigenvalues
(eigen
). See det
.
Using LAPACK (including in the complex case) uses column pivoting and does not attempt to detect rank-deficient matrices.
Source
For qr
, the LINPACK routine DQRDC
(but modified to
dqrdc2
(*)) and the LAPACK
routines DGEQP3
and ZGEQP3
. Further LINPACK and LAPACK
routines are used for qr.coef
, qr.qy
and qr.aty
.
LAPACK and LINPACK are from https://netlib.org/lapack/ and https://netlib.org/linpack/ and their guides are listed in the references.
References
Anderson. E. and ten others (1999)
LAPACK Users' Guide. Third Edition. SIAM.
Available on-line at
https://netlib.org/lapack/lug/lapack_lug.html.
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.
See Also
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.
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
## 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