lm(): What is qraux returned by QR decomposition in LINPACK / LAPACK

Question

rich.main3 is a linear model in R. I understand the rest of the elements of the list but I don\'t get what qraux is. The documentation states that

Answer 1

Presumably you don't know how QR factorization is computed. I wrote the following in LaTeX which might help you clarify this. Surely on a programming site I need to show you some code. In the end I offer you a toy R function computing Householder reflection.

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Householder reflection matrix

Householder transformation

Householder QR factorization (without pivoting)

Compact storage of QR and rescaling

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The LAPACK auxiliary routine dlarfg is performing Householder transform. I have also written the following toy R function for demonstration:

dlarfg <- function (x) {
  beta <- -1 * sign(x[1]) * sqrt(as.numeric(crossprod(x)))
  v <- c(1, x[-1] / (x[1] - beta))
  tau <- 1 - x[1] / beta
  y <- c(beta, rep(0, length(x)-1L))
  packed_yv <- c(beta, v[-1])
  oo <- cbind(x, y, v, packed_yv)
  attr(oo, "tau") <- tau
  oo
  }

Suppose we have an input vector

set.seed(0); x <- rnorm(5)

my function gives:

dlarfg(x)
#              x         y           v   packed_yv
#[1,]  1.2629543 -2.293655  1.00000000 -2.29365466
#[2,] -0.3262334  0.000000 -0.09172596 -0.09172596
#[3,]  1.3297993  0.000000  0.37389527  0.37389527
#[4,]  1.2724293  0.000000  0.35776475  0.35776475
#[5,]  0.4146414  0.000000  0.11658336  0.11658336
#attr(,"tau")
#[1] 1.55063