qr
The QR Decomposition of a Matrix
qr
computes the QR decomposition of a matrix.
Usage
qr(x, ...)
"qr"(x, tol = 1e07 , 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 = 1e7)
"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 ifLAPACK
is false andx
is real.  qr
 a QR decomposition of the type computed by
qr
.  y, b
 a vector or matrix of righthand 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 (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 NewtonRaphson 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 underdetermined
systems, providing a leastsquares 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 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/DGEQP3/ZGEQP3.
 qr
 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.  qraux
 a vector of length
ncol(x)
which contains additional information on $\bold{Q}$.  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. Noncomplex 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 rankdeficient matrices.
Source
For qr
, the LINPACK routine DQRDC
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 http://www.netlib.org/lapack and
http://www.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 online 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)
hilbert < function(n) { i < 1:n; 1 / outer(i  1, i, "+") }
h9 < hilbert(9); h9
qr(h9)$rank #> only 7
qrh9 < qr(h9, tol = 1e10)
qrh9$rank #> 9
## Solve linear equation system H %*% x = y :
y < 1:9/10
x < qr.solve(h9, y, tol = 1e10) # 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 leastsquares 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