Sorry to bother again. I understand that for a test this criterion is sufficient. In the worst case it rejects a factorization even if it is numerically correct.

But just out of curiosity: I was looking up the error analysis in "Applied Numerical Linear Algebra" (Demmel). There it was shown that

- Screen Shot 2015-12-30 at 23.11.50.png (10.54 KiB) Viewed 10583 times

So this couldn't be used to check if the factorization is correct? If I understand it correctly, the criterion

- Screen Shot 2015-12-30 at 21.27.33.png (11.92 KiB) Viewed 10583 times

shows that the numerical method itself is backward stable for this particular matrix A. So it justifies using this method in this case and shows that the implementation is correct. So this is a good thing.

However, I wonder if I got that right: Using the above criterion would merely check if the factorization was computed "as correct as numerically possible" (even in cases dgetrf should not be used)?

So for pathological matrices like

- Screen Shot 2015-12-30 at 22.15.46.png (11.92 KiB) Viewed 10583 times

were dgetrf is not "the right" choice the first criterion would accept the factorization and the second might not (for large n).