the deep spike or more erratic behavior of the LINPACK QRCP is likely
to happen when the scaled condition of A (cond(A with equilibrated columns))
exceeds 1/sqrt(machine precision). In other words, if scaled condition
is taken as a realistic sensitivity measure (and for the QRF it is),
then LINPACK-style QRCP causes
problems even if we could get half of the digits in the answer.
Zvonimir and myself did not go into fixing this in the ScaLAPACK, we
that ScaLAPACK developers will do that. (How far this bag can go, see
in the attachment. That report contains some discussion related to our
ACM TOMS paper
I should point out here that the fix we provided for LAPACK is not the best
we can -- a truly reliable implementation requires extra N scalars in
and that we could not afford because of backward compatibility. Best fix
should get different name, because of this extra N locations in the
Please let me know if you are interested in having this in (Sca)LAPACK.
On 20.10.2011. 15:26, James Demmel wrote:
On 10/20/11 6:01 AM, Sven Hammarling wrote:
Unless I misunderstand (quite likely!), I think that Zlatko and
Zvonimir showed a similar example in Figure 1 of LAWN 176 - it
certainly has a deep spike.
Once ScaLAPACK gets the LAPACK fix it will be interesting to see the
result on David's example.
In that figure, the "true" answer is around macheps * norm(A), and
down way below that. So everything is tiny. I think this is the only
way you can
get such completely out-of-order R(i,i), when everything is
and so possibly totally losing all leading digits.
In the example from David Wang below, R(3,3) is O(norm(A)) = O(1), and
R(2,2) = O(macheps), which can't be explained by the bug Zlatko and
R(0,0) -0.707107 + 0i
R(1,1) -0.707107 + 0i
R(2,2) 8.16273e-17 + 0i
R(3,3) -0.243173 + 0i
R(4,4) 4.82605e-17 + 0i
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