Edited the code of xGBMV, xGEMV and xGEMM by removing the test for an entry of
B (or X) being zero to skip a loop so as to enable better NaN (or Inf)
propagation.

Taking DGEMM as an example, if you use notation: 
	C(I,J) = C(I,J) + alpha * A(I,L) * B(L,J), 
the reference BLAS DGEMM is using the JLI version of matrix matrix multiply and
is checking, for each J (from 1 to N) and for each L (from 1 to K), whether
B(L,J) is zero (or not) to save (or not) the 2M following operations.

(See the "IF (B(L,J).NE.ZERO) THEN" in the code below.)

The snippets of code is as follows

              DO 90 J = 1,N
                  DO 80 L = 1,K
                      IF (B(L,J).NE.ZERO) THEN
                          TEMP = ALPHA*B(L,J)
                          DO 70 I = 1,M
                              C(I,J) = C(I,J) + TEMP*A(I,L)
   70                     CONTINUE
                      END IF
   80             CONTINUE
   90         CONTINUE

This induces some non NaN-propagation in a pretty ad-hoc way. For better NaN
propagation, this patch removes the above IF statement.

The snippet of code now becomes

              DO 90 J = 1,N
                  DO 80 L = 1,K
                      TEMP = ALPHA*B(L,J)
                      DO 70 I = 1,M
                          C(I,J) = C(I,J) + TEMP*A(I,L)
   70                 CONTINUE
   80             CONTINUE
   90         CONTINUE

This enables correct NaN propagation for this piece of code.

Rationale: BLAS does not correctly propagate all NaNs (and Infs). We still have
no NaN propagation where for example ALPHA=0, etc. The goal of this commit is
to have correct NaN propagation no matter what the entries of the input
matrices/vectors (A, B, C, X, etc.) are.  BLAS do not correctly propagate NaNs
and Infs based on some values of the scalars (ALPHA, BETA, etc.).

See below the email from Tom Callaway from RedHat, sent on July 9th to
lapack@cs.utk.edu.

Hello LAPACK people,

Martyn & Lejeczek (on CC) reported an issue to Fedora relating to R using our
system copy of BLAS (from LAPACK).

As noted in the R administration and Installation Manual, "R relies on ISO/IEC
60559 compliance of an external BLAS. This can be broken if for example the
code assumes that terms with a zero factor are always zero and do not need to
be computed - whereas x*0 can be NaN. This is checked in the test suite."

In the stock BLAS, DGBMV, DGEMM, and DGEMV fail this. R has been patching their
bundled BLAS to resolve this issue since 2010, but Fedora now uses the system
BLAS.

Attached is a patch (from upstream R) to fix this issue in the LAPACK BLAS.
Please consider applying it.

Thanks,

~tom






git-svn-id: https://icl.cs.utk.edu/svn/lapack-dev/lapack/trunk@1488 8a072113-8704-0410-8d35-dd094bca7971