qr {base} R Documentation

## The QR Decomposition of a Matrix

### Description

`qr` computes the QR decomposition of a matrix. It provides an interface to the techniques used in the LINPACK routine DQRDC or the LAPACK routines DGEQP3 and (for complex matrices) ZGEQP3.

### Usage

```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 matrix whose QR decomposition is to be computed. `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. `...` 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 Ax = b for given matrix A, and vector 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`. `qr.qy` and `qr.qty` return `Q %*% y` and `t(Q) %*% y`, where `Q` is the (complete) Q matrix.

All the above functions keep `dimnames` (and `names`) of `x` and `y` if there are.

`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 minimal-length solution or 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.

### 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/DGEQP3/ZGEQP3.

 `qr` a matrix with the same dimensions as `x`. The upper triangle contains the R of the decomposition and the lower triangle contains information on the 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 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`.

### 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.

### References

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.

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.

`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

```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
```

[Package base version 2.5.0 Index]