The QR Decomposition of a Matrix (original) (raw)
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 x. Only used if LAPACK is false andx 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 (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.fittedreturn 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 andt(Q) %*% y, where Q is the (complete) \bold{Q} matrix.
All the above functions keep dimnames (and names) ofx and y if there are any.
solve.qr is the method for [solve](../../base/help/solve.html) 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](../../base/help/list.html)and [inherits](../../base/help/inherits.html) 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 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 (LINPACK) and DGEQP3 (LAPACK) differs. |
|---|---|
| qraux | a vector of length ncol(x) in the LINPACK case and min(dim(x)) in the LAPACK case, 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. |
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 tolfor 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](../../base/help/eigen.html)). See [det](../../base/help/det.html).
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 todqrdc2(*)) and the LAPACK routines DGEQP3 and ZGEQP3. Further LINPACK and LAPACK routines are used for qr.coef, qr.qy and qr.qty.
LAPACK and LINPACK are from https://netlib.org/lapack/ andhttps://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 athttps://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](../../base/help/qr.Q.html), [qr.R](../../base/help/qr.R.html), [qr.X](../../base/help/qr.X.html) for reconstruction of the matrices.[lm.fit](../../stats/html/lmfit.html), [lsfit](../../stats/html/lsfit.html),[eigen](../../base/help/eigen.html), [svd](../../base/help/svd.html).
[det](../../base/help/det.html) (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
[Package _base_ version 4.6.0 Index]