qr {base} | R Documentation |

`qr`

computes the QR decomposition of a matrix.

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)

`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`

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.

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/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`

.

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`

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.

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.

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 3.3.0 Index]