QR Decomposition – S4 Methods and Generic

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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

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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".

See Also

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

Examples

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##------------- example of pivoting -- from base'  qraux.Rd -------------
X <- Matrix(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)),
            sparse=TRUE)
rownames(X) <- paste0("r", seq_len(nrow(X)))
dnX <- dimnames(X)
X # is singular, columns "b2" and "c3" are "extra"
c(rankMatrix(X)) # = 4 (not 6)
##----- 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:
m <- as.matrix
qXr <- qr(  Xr)
qxr <- qr(m(Xr))
qcfXy <- qr.coef (qXr, y)
qcfXY <- qr.coef (qXr, Y)
stopifnot(
          all.equal(qr.coef(qxr, y), cf <- c(int=6, b1=-3, c1=-2, c2=-1), tol=1e-15)
,
          all.equal(qr.coef(qxr, Y), as.matrix(cf), tol=1e-15)
,
          all.equal(unname(qcfXy), unname(cf), tol=1e-15) || # FAIL names: ## FIXME_______
          all.equal(qcfXy, cf, tol=1e-15)
,
          all.equal(unname(m(qcfXY)), unname(m(cf)), tol=1e-15) || # FAIL dimnames: ## FIXME_______
          all.equal(m(qcfXY), m(cf), tol=1e-15)
,
          all.equal(y, qr.fitted(qxr, y), tol=2e-15)
,
          all.equal(y, qr.fitted(qXr, y), tol=2e-15)
,
          all.equal(m(qr.fitted(qXr, Y)), qr.fitted(qxr, Y), tol=1e-15)
,
          all.equal(  qr.resid (qXr, y),  qr.resid (qxr, y), tol=1e-15)
,
          all.equal(m(qr.resid (qXr, Y)), qr.resid (qxr, Y), tol=1e-15)
)

##----- singular case -----------------------------------------------
(qX <- qr(  X))
 qx <- qr(m(X))
# both @p and @q are non-trivial permutations
stopifnot(identical(dimnames(X), dnX))# some versions changed X's dimnames!

drop0(R. <- qr.R(qX), tol=1e-15) # columns *permuted*: c3 b1 ..
Q. <- qr.Q(qX)
qI <- sort.list(qX@q) # the inverse 'q' permutation
(X. <- drop0(Q. %*% R.[, qI], tol=1e-15))## 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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