`qr`

computes the QR decomposition of a matrix.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 |

`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 |

`qr` |
a QR decomposition of the type computed by |

`y, b` |
a vector or matrix of right-hand sides of equations. |

`a` |
a QR decomposition or ( |

`k` |
effective rank. |

`LAPACK` |
logical. For real |

`...` |
further arguments passed to or from other methods |

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.

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 |

`qraux` |
a vector of length |

`rank` |
the rank of |

`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 DQRDCIn 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.

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.

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 https://www.netlib.org/lapack/ and https://www.netlib.org/linpack/ and their guides are listed in the references.

Anderson. E. and ten others (1999)
*LAPACK Users' Guide*. Third Edition. SIAM.

Available on-line at
https://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.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 | ```
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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