Matrix (version 1.5-3)

# qr-methods: QR Decomposition -- S4 Methods and Generic

## Description

The Matrix package provides methods for the QR decomposition of special classes of matrices. There is a generic function which uses `qr` as default, but methods defined in this package can take extra arguments. In particular there is an option for determining a fill-reducing permutation of the columns of a sparse, rectangular matrix.

## Usage

```qr(x, ...)
qrR(qr, complete=FALSE, backPermute=TRUE, row.names = TRUE)```

## Arguments

x

a numeric or complex matrix whose QR decomposition is to be computed. Logical matrices are coerced to numeric.

qr

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

complete

logical indicating whether the \(\bold{R}\) matrix is to be completed by binding zero-value rows beneath the square upper triangle.

backPermute

logical indicating if the rows of the \(\bold{R}\) matrix should be back permuted such that `qrR()`'s result can be used directly to reconstruct the original matrix \(\bold{X}\).

row.names

logical indicating if `rownames` should propagated to the result.

...

further arguments passed to or from other methods

## Methods

x = "dgCMatrix"

QR decomposition of a general sparse double-precision matrix with `nrow(x) >= ncol(x)`. Returns an object of class `"sparseQR"`.

x = "sparseMatrix"

works via `"dgCMatrix"`.

`qr`; then, the class documentations, mainly `sparseQR`, and also `dgCMatrix`.

## Examples

Run this code
``````
##------------- example of pivoting -- from base'  qraux.Rd -------------
X <- cbind(int = 1,
b1=rep(1:0, each=3), b2=rep(0:1, each=3),
c1=rep(c(1,0,0), 2), c2=rep(c(0,1,0), 2), c3=rep(c(0,0,1),2))
rownames(X) <- paste0("r", seq_len(nrow(X)))
dnX <- dimnames(X)
bX <- X # [b]ase version of X
X <- as(bX, "sparseMatrix")
X # is singular, columns "b2" and "c3" are "extra"
stopifnot(identical(dimnames(X), dnX))# some versions changed X's dimnames!
c(rankMatrix(X)) # = 4 (not 6)
m <- function(.) as(., "matrix")

##----- regular case ------------------------------------------
Xr <- X[ , -c(3,6)] # the "regular" (non-singular) version of X
stopifnot(rankMatrix(Xr) == ncol(Xr))
Y <- cbind(y <- setNames(1:6, paste0("y", 1:6)))
## regular case:
qXr   <- qr(  Xr)
qxr   <- qr(m(Xr))
qxrLA <- qr(m(Xr), LAPACK=TRUE) # => qr.fitted(), qr.resid() not supported
qcfXy <- qr.coef (qXr, y) # vector
qcfXY <- qr.coef (qXr, Y) # 4x1 dgeMatrix
cf <- c(int=6, b1=-3, c1=-2, c2=-1)
doExtras <- interactive() || nzchar(Sys.getenv("R_MATRIX_CHECK_EXTRA"))
tolE <- if(doExtras) 1e-15 else 1e-13
stopifnot(exprs = {
all.equal(qr.coef(qxr,  y),   cf,  tol=tolE)
all.equal(qr.coef(qxrLA,y),   cf,  tol=tolE)
all.equal(qr.coef(qxr,  Y), m(cf), tol=tolE)
all.equal(  qcfXy,    cf, tol=tolE)
all.equal(m(qcfXY), m(cf), tol=tolE)
all.equal(y, qr.fitted(qxr, y), tol=2*tolE)
all.equal(y, qr.fitted(qXr, y), tol=2*tolE)
all.equal(m(qr.fitted(qXr, Y)), qr.fitted(qxr, Y), tol=tolE)
all.equal(  qr.resid (qXr, y),  qr.resid (qxr, y), tol=tolE)
all.equal(m(qr.resid (qXr, Y)), qr.resid (qxr, Y), tol=tolE)
})

##----- rank-deficient ("singular") case ------------------------------------

(qX <- qr(  X))           # both @p and @q are non-trivial permutations
qx <- qr(m(X)) ; str(qx) # \$pivot is non-trivial, too

drop0(R. <- qr.R(qX), tol=tolE) # columns *permuted*: c3 b1 ..
Q. <- qr.Q(qX)
qI <- sort.list(qX@q) # the inverse 'q' permutation
(X. <- drop0(Q. %*% R.[, qI], tol=tolE))## just = X, incl. correct colnames
stopifnot(all(X - X.) < 8*.Machine\$double.eps,
## qrR(.) returns R already "back permuted" (as with qI):
identical(R.[, qI], qrR(qX)) )
##
## In this sense, classical qr.coef() is fine:
cfqx <- qr.coef(qx, y) # quite different from
nna <- !is.na(cfqx)
stopifnot(all.equal(unname(qr.fitted(qx,y)),
as.numeric(X[,nna] %*% cfqx[nna])))
## FIXME: do these make *any* sense? --- should give warnings !
qr.coef(qX, y)
qr.coef(qX, Y)
rm(m)
``````

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