R: Reconstruct the Q, R, or X Matrices from a QR Object

QR.Auxiliaries {base}

R Documentation

Reconstruct the Q, R, or X Matrices from a QR Object

Description

Returns the original matrix from which the object was constructed or the components of the decomposition.

Usage

qr.X(qr, complete = FALSE, ncol =)
qr.Q(qr, complete = FALSE, Dvec =)
qr.R(qr, complete = FALSE)

Arguments

`qr`	object representing a QR decomposition. This will typically have come from a previous call to `qr` or `lsfit`.
`complete`	logical expression of length 1. Indicates whether an arbitrary orthogonal completion of the \bold{Q} or \bold{X} matrices is to be made, or whether the \bold{R} matrix is to be completed by binding zero-value rows beneath the square upper triangle.
`ncol`	integer in the range `1:nrow(qr$qr)`. The number of columns to be in the reconstructed \bold{X}. The default when `complete` is `FALSE` is the first `min(ncol(X), nrow(X))` columns of the original \bold{X} from which the qr object was constructed. The default when `complete` is `TRUE` is a square matrix with the original \bold{X} in the first `ncol(X)` columns and an arbitrary orthogonal completion (unitary completion in the complex case) in the remaining columns.
`Dvec`	vector (not matrix) of diagonal values. Each column of the returned \bold{Q} will be multiplied by the corresponding diagonal value. Defaults to all `1`s.

Value

qr.X returns \bold{X}, the original matrix from which the qr object was constructed, provided ncol(X) <= nrow(X). If complete is TRUE or the argument ncol is greater than ncol(X), additional columns from an arbitrary orthogonal (unitary) completion of X are returned.

qr.Q returns part or all of Q, the order-nrow(X) orthogonal (unitary) transformation represented by qr. If complete is TRUE, Q has nrow(X) columns. If complete is FALSE, Q has ncol(X) columns. When Dvec is specified, each column of Q is multiplied by the corresponding value in Dvec.

Note that qr.Q(qr, *) is a special case of qr.qy(qr, y) (with a “diagonal” y), and qr.X(qr, *) is basically qr.qy(qr, R) (apart from pivoting and dimnames setting).

qr.R returns R. This may be pivoted, e.g., if a <- qr(x) then x[, a$pivot] = QR. The number of rows of R is either nrow(X) or ncol(X) (and may depend on whether complete is TRUE or FALSE).

Examples

p <- ncol(x <- LifeCycleSavings[, -1]) # not the 'sr'
qrstr <- qr(x)   # dim(x) == c(n,p)
qrstr $ rank # = 4 = p
Q <- qr.Q(qrstr) # dim(Q) == dim(x)
R <- qr.R(qrstr) # dim(R) == ncol(x)
X <- qr.X(qrstr) # X == x
range(X - as.matrix(x))  # ~ < 6e-12
## X == Q %*% R if there has been no pivoting, as here:
all.equal(unname(X),
          unname(Q %*% R))

# example of pivoting
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))
x # is singular, columns "b2" and "c3" are "extra"
a <- qr(x)
zapsmall(qr.R(a)) # columns are int b1 c1 c2 b2 c3
a$pivot
pivI <- sort.list(a$pivot) # the inverse permutation
all.equal (x,            qr.Q(a) %*% qr.R(a)) # no, no
stopifnot(
 all.equal(x[, a$pivot], qr.Q(a) %*% qr.R(a)),          # TRUE
 all.equal(x           , qr.Q(a) %*% qr.R(a)[, pivI]))  # TRUE too!