| 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, 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 x. Only used if LAPACK is false and
x is 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.solve only) a rectangular matrix. |
k |
effective rank. |
LAPACK |
logical. For real x, if true use LAPACK
otherwise use LINPACK. |
... |
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 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 (complete) 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 minimal-length solution or 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 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). Note that the storage
used by DQRDC and DGEQP3 differs. |
qraux |
a vector of length ncol(x) which contains
additional information on Q. |
rank |
the rank of x as computed by the decomposition:
always full rank in the LAPACK case. |
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