For a sparse \(m \times n\) (“long”: \(m \ge n\))
rectangular matrix \(A\), the sparse QR decomposition is either
of the form \(P A = Q R\) with a (row)
permutation matrix \(P\), (encoded in the p slot of the
result) if the q slot is of length 0,
or of the form \(P A P* = Q R\) with an extra (column) permutation
matrix \(P*\) (encoded in the q slot).
Note that the row permutation \(P A\) in R is simply A[p+1, ]
where p is the p-slot, a 0-based permutation of
1:m applied to the rows of the original matrix.
If the q slot has length n it is a 0-based permutation
of 1:n applied to the columns of the original matrix to reduce
the amount of “fill-in” in the matrix \(R\), and
\(A P*\) in R is simply A[ , q+1].
\(R\) is an \(m\times n\) matrix that is zero below the
main diagonal, i.e., upper triangular (\(m\times m\)) with
\(m-n\) extra zero rows.
The matrix \(Q\) is a "virtual matrix". It is the product of
\(n\) Householder transformations. The information to generate
these Householder transformations is stored in the V and
beta slots.
Note however that qr.Q() returns the row permuted matrix
\(Q* := P^{-1}Q = P'Q\) as permutation matrices are
orthogonal; and \(Q*\) is orthogonal itself because \(Q\) and \(P\) are.
This is useful because then, as in the dense matrix and base R
matrix qr case, we have the mathematical identity
$$P A = Q* R,$$ in R as
A[p+1,] == qr.Q(*) %*% R .
The "sparseQR" methods for the qr.* functions return
objects of class "dgeMatrix" (see
'>dgeMatrix). Results from qr.coef,
qr.resid and qr.fitted (when k == ncol(R)) are
well-defined and should match those from the corresponding dense matrix
calculations. However, because the matrix Q is not uniquely
defined, the results of qr.qy and qr.qty do not
necessarily match those from the corresponding dense matrix
calculations.
Also, the results of qr.qy and qr.qty apply to the
permuted column order when the q slot has length n.