base (version 3.5.1)

# qr: The QR Decomposition of a Matrix

## Description

`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 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

## 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.

## <code>*)</code> <code>dqrdc2</code> 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.

## 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.

## 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.

`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

Run this code
```# 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
# }
```

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