Matrix (version 1.6-1.1)

# qr-methods: Methods for QR Factorization

## Description

Computes the pivoted QR factorization of an $$m \times n$$ real matrix $$A$$, which has the general form $$P_{1} A P_{2} = Q R$$ or (equivalently) $$A = P_{1}' Q R P_{2}'$$ where $$P_{1}$$ and $$P_{2}$$ are permutation matrices, $$Q = \prod_{j = 1}^{n} H_{j}$$ is an $$m \times m$$ orthogonal matrix equal to the product of $$n$$ Householder matrices $$H_{j}$$, and $$R$$ is an $$m \times n$$ upper trapezoidal matrix.

denseMatrix use the default method implemented in base, namely qr.default. It is built on LINPACK routine dqrdc and LAPACK routine dgeqp3, which do not pivot rows, so that $$P_{1}$$ is an identity matrix.

Methods for sparseMatrix are built on CSparse routines cs_sqr and cs_qr, which require $$m \ge n$$.

## Usage

qr(x, ...)
# S4 method for dgCMatrix
qr(x, order = 3L, ...)

## Value

An object representing the factorization, inheriting from virtual S4 class QR or S3 class

qr. The specific class is qr

unless x inherits from virtual class

sparseMatrix, in which case it is

sparseQR.

## Arguments

x

a finite matrix or Matrix to be factorized, satisfying nrow(x) >= ncol(x) if sparse.

order

an integer in 0:3 passed to CSparse routine cs_sqr, indicating a strategy for choosing the column permutation $$P_{2}$$. 0 means no column permutation. 1, 2, and 3 indicate a fill-reducing ordering of $$A + A'$$, $$\tilde{A}' \tilde{A}$$, and $$A' A$$, where $$\tilde{A}$$ is $$A$$ with “dense” rows removed. Do not set to 0 unless you know that the column order of $$A$$ is already sensible.

...

further arguments passed to or from methods.

## Details

If x is sparse and structurally rank deficient, having structural rank $$r < n$$, then x is augmented with $$(n-r)$$ rows of (partly non-structural) zeros, such that the augmented matrix has structural rank $$n$$. This augmented matrix is factorized as described above: $$P_1 A P_2 = P_1 \begin{bmatrix} A_{0} \\ 0 \end{bmatrix} P_2 = Q R$$ where $$A_0$$ denotes the original, user-supplied $$(m-(n-r)) \times n$$ matrix.

## References

Davis, T. A. (2006). Direct methods for sparse linear systems. Society for Industrial and Applied Mathematics. tools:::Rd_expr_doi("10.1137/1.9780898718881")

Golub, G. H., & Van Loan, C. F. (2013). Matrix computations (4th ed.). Johns Hopkins University Press. tools:::Rd_expr_doi("10.56021/9781421407944")

Class sparseQR and its methods.

Class dgCMatrix.

Generic function qr from base, whose default method qr.default “defines” the S3 class qr of dense QR factorizations.

Generic functions expand1 and expand2, for constructing matrix factors from the result.

Generic functions Cholesky, BunchKaufman, Schur, and lu, for computing other factorizations.

## Examples

Run this code
showMethods("qr", inherited = FALSE)

## Rank deficient: columns 3 {b2} and 6 {c3} are "extra"
M <- as(cbind(a1 = 1,
b1 = rep(c(1, 0), each = 3L),
b2 = rep(c(0, 1), each = 3L),
c1 = rep(c(1, 0, 0), 2L),
c2 = rep(c(0, 1, 0), 2L),
c3 = rep(c(0, 0, 1), 2L)),
"CsparseMatrix")
rownames(M) <- paste0("r", seq_len(nrow(M)))
b <- 1:6
eps <- .Machine\$double.eps

## .... [1] full rank ..................................................
## ===> a least squares solution of A x = b exists
##      and is unique _in exact arithmetic_

(A1 <- M[, -c(3L, 6L)])
(qr.A1 <- qr(A1))

stopifnot(exprs = {
rankMatrix(A1) == ncol(A1)
{ d1 <- diag(qr.A1@R); sum(d1 < max(d1) * eps) == 0L }
rcond(crossprod(A1)) >= eps
all.equal(qr.coef(qr.A1, b), drop(solve(crossprod(A1), crossprod(A1, b))))
all.equal(qr.fitted(qr.A1, b) + qr.resid(qr.A1, b), b)
})

## .... [2] numerically rank deficient with full structural rank .......
## ===> a least squares solution of A x = b does not
##      exist or is not unique _in exact arithmetic_

(A2 <- M)
(qr.A2 <- qr(A2))

stopifnot(exprs = {
rankMatrix(A2) == ncol(A2) - 2L
{ d2 <- diag(qr.A2@R); sum(d2 < max(d2) * eps) == 2L }
rcond(crossprod(A2)) < eps

## 'qr.coef' computes unique least squares solution of "nearby" problem
## Z x = b for some full rank Z ~ A, currently without warning {FIXME} !
tryCatch({ qr.coef(qr.A2, b); TRUE }, condition = function(x) FALSE)

all.equal(qr.fitted(qr.A2, b) + qr.resid(qr.A2, b), b)
})

## .... [3] numerically and structurally rank deficient ................
## ===> factorization of _augmented_ matrix with
##      full structural rank proceeds as in [2]

##  NB: implementation details are subject to change; see (*) below

A3 <- M
A3[, c(3L, 6L)] <- 0
A3
(qr.A3 <- qr(A3)) # with a warning ... "additional 2 row(s) of zeros"

stopifnot(exprs = {
## sparseQR object preserves the unaugmented dimensions (*)
dim(qr.A3  ) == dim(A3)
dim(qr.A3@V) == dim(A3) + c(2L, 0L)
dim(qr.A3@R) == dim(A3) + c(2L, 0L)

## The augmented matrix remains numerically rank deficient
rankMatrix(A3) == ncol(A3) - 2L
{ d3 <- diag(qr.A3@R); sum(d3 < max(d3) * eps) == 2L }
rcond(crossprod(A3)) < eps
})

## Auxiliary functions accept and return a vector or matrix
## with dimensions corresponding to the unaugmented matrix (*),
## in all cases with a warning
qr.coef  (qr.A3, b)
qr.fitted(qr.A3, b)
qr.resid (qr.A3, b)

## .... [4] yet more examples ..........................................

## By disabling column pivoting, one gets the "vanilla" factorization
## A = Q~ R, where Q~ := P1' Q is orthogonal because P1 and Q are

(qr.A1.pp <- qr(A1, order = 0L)) # partial pivoting

ae1 <- function(a, b, ...) all.equal(as(a, "matrix"), as(b, "matrix"), ...)
ae2 <- function(a, b, ...) ae1(unname(a), unname(b), ...)

stopifnot(exprs = {
length(qr.A1   @q) == ncol(A1)
length(qr.A1.pp@q) == 0L # indicating no column pivoting
ae2(A1[, qr.A1@q + 1L], qr.Q(qr.A1   ) %*% qr.R(qr.A1   ))
ae2(A1                , qr.Q(qr.A1.pp) %*% qr.R(qr.A1.pp))
})


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