?test_rfp: error in the test

[AlexanderZotkevich]

Description

Test can fail for DIAG = 'U' when diagonal elements of test matrix are supposed to be unit.

in ?drvrf3 one can read:

 *
 *                             Generate A our NA--by--NA triangular
 *                             matrix.
 *                             Our test is based on forward error so we
 *                             do want A to be well conditioned! To get
 *                             a well-conditioned triangular matrix, we
 *                             take the R factor of the QR/LQ factorization
 *                             of a random matrix.

And that's right, the 1/(condition number) of the test matrix calculated with ?LANGE + ?GETRF + ?GECON is about 1E-5 for single precision when DIAG is not 'U'.

But when DIAG is 'U', setting diagonal elements to be unit spoils the test matrix: 1/(condition number) becomes about 1E-10.

This can lead to test fail for different combinations of hardware/BLAS/compiler.

Checklist

• I'd suggest to fix this dividing rows of the matrix by 2 * corresponding diagonal elements for DIAG ='U' case:

IUPLO = 1:

                                 IF ( LSAME( DIAG, 'U' ) ) THEN
                                    DO J = 1, NA
                                       DO I = 1, J
                                          A(I, J) = A(I, J) /
     +                                          ( 2.0 * A(J, J) )
                                       END DO
                                    END DO
                                 END IF

IUPLO != 1:

                                 IF ( LSAME( DIAG, 'U' ) ) THEN
                                    DO I = 1, NA
                                       DO J = 1, I
                                          A(I, J) = A(I, J) /
     +                                          ( 2.0 * A(I, I) )
                                       END DO
                                    END DO
                                 END IF

[langou]

Thanks @AlexanderZotkevich. Nice. Indeed a random matrix is almost always well-conditioned, and consequently so is an R factor of one of its QR factorization. But, if we change the diagonal entries of R to ones, well, that changes everything! Nice catch. Cute bug. And nice fix.

So when the diagonal entries are set to ONE, you suggest to consider this as a diagonal scaling of the matrix R. (So divide rows or cols by diagonal element.) And then you help a little the condition number by dividing by 2 the off-diagonal entries. Fun. Question in my head right now: what is the expected condition number of the so-obtained R matrix? :)

As a side comment, I do not understand why this test was not written with a backward error check. I am re-reading it now and I do not get it. It would have been easy enough to take a "b", solve "Ax=b", and then check the backward error.

That being said, thanks @AlexanderZotkevich. Would you mind contributing a PR with the change as you suggest? Your proposed changes sounds great to me.

Thanks a lot for submitting an issue.

[AlexanderZotkevich]

Q. Question in my head right now: what is the expected condition number of the so-obtained R matrix? :)
A. I believe it's not too high. :)

Q. Would you mind contributing a PR with the change as you suggest?
A. OK. But please give me some time for this.