The QR Decomposition of a Matrix

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Description

qr computes the QR decomposition of a matrix.

Usage

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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 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 (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 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(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 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.

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 Eigen values (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 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 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

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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

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