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[Lapack] Bug in LAPACK 3.2 dlarfb.f and likes


I've found a bug in dlarfb.f recently, it can be revealed by dorgql blocked 
algorithm, for instance. I suspect the ordinal tests don't catch it because too 
high crosspoint is set in ilaenv.f NX=128. To catch the bug:
1) set less crosspoint in line 462 of ilaenv.f, for instance, NX=0
2) compile the testcase attached,
3) use the input data attached
You will have non-orthogonal vectors on the output.

dlarfb.f has been changed to reduce the number of reflectors needed to be 
applied, but it turns out sometimes we miss the last reflector to be applied. 
The newly introduced function iladlr detects last non-zero row, but in dlarfb 
we don't pay attention to the fact that the elementary reflectors

v v v
v v v
v v v
1 v v
  1 v

could be kept along with a zero diagonal:

v v v
v v v
v v v
0 v v
  0 v

which in fact reduce the number of reflectors to be applied, that is we miss 
the last one. I find the following logic appropriate in this case: substituting 
line 260 in dlarfb.f

               LASTV = MAX( K, ILADLR( M, K, V, LDV ) )

by the line

<                LASTV = MAX( K, ILADLC( M, K, V, LDV ) )+M-K

The original LASTV initialization is wrong.

Best regards,


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