qr
The QR Decomposition of a Matrix
qr computes the QR decomposition of a matrix.
Usage
qr(x, …)
# S3 method for default
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 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
x. Only used ifLAPACKis false andxis real.- qr
a QR decomposition of the type computed by
qr.- y, b
a vector or matrix of right-hand sides of equations.
- a
a QR decomposition or (
qr.solveonly) a rectangular matrix.- k
effective rank.
- LAPACK
logical. For real
x, if true use LAPACK otherwise use LINPACK (the default).- …
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.
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). Note that the storage
used by DQRDC and DGEQP3 differs.
a vector of length ncol(x) which contains
additional information on \(\bold{Q}\).
the rank of x as computed by the decomposition(*):
always full rank in the LAPACK case.
information on the pivoting strategy used during the decomposition.
Non-complex QR objects computed by LAPACK have the attribute "useLAPACK" with value TRUE.
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.
Using LAPACK (including in the complex case) uses column pivoting and does not attempt to detect rank-deficient matrices.
*) 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.
References
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.
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
library(base)
# NOT RUN {
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
# }