[Gretl-devel] Re: qrdecomp with column pivoting?
On Mon, Jun 6, 2022 at 6:47 AM Sven Schreiber <svetosch(a)gmx.net> wrote:
Am 06.06.2022 um 02:46 schrieb Cottrell, Allin:
>
> I tend to think these are different error conditions: (a) the input
> contains exactly collinear data, and (b) one or more of the input
> series/columns are all zero.
But that was just an example, not the encompassing definition of the
problem.
Consider this variation without all-zero columns:
<hansl>
R = {}
A = I(4,1) ~ I(4,2)
print A
Q = qrdecomp(A , &R)
print rank(A) # correct: 2
print R
scalar r = sumc(abs(diag(R)).>1.0e-12)
print r # wrong: 1
</hansl>
Is there any general characterization of what it takes for plain QR
not to reveal rank?
It would appear that the presence of all-zero rows and/or columns is
necessary but not sufficient. With m > n you can construct mostly-zero
matrices where diag(R) does give the right answer. E.g.
<hansl>
function void rank_check (const matrix X)
R = {}
qrdecomp(X, &R)
printf "rank via SVD: %d\n", rank(X)
printf "rank via QR: %d\n\n", sumc(abs(diag(R)) .> 1.0e-12)
end function
matrix X = zeros(8,4)
X[6,1] = 1
X[7,4] = 1
X[8,4] = 1
print X
rank_check(X)
</hansl>
Allin