# qr

##### The QR Decomposition of a Matrix

`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

##### 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.

##### 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.

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.

a vector of length `ncol(x)`

which contains
additional information on \(\bold{Q}\).

the rank of `x`

as computed by the decomposition(*):
always full rank in the LAPACK case.

information on the pivoting strategy used during the decomposition.

Non-complex 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 rank-deficient matrices.

`*)`

`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.

##### 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

`library(base)`

```
# 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
# }
```

*Documentation reproduced from package base, version 3.6.2, License: Part of R 3.6.2*