Occasional NaN returned from qp() with ill-conditioned matrix?

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Occasional NaN returned from qp() with ill-conditioned matrix?

I am having nondeterministic failures with qp(), probably related to
ill-conditioning, and I would love to hear what anyone might have to say about
it (I am so ignorant, no matter what you might say will impress the hell out of
me):

I am using qp() with 86x86 H's that have conditioning numbers like 1.0e+14 to
1.5e+018 (ack!) to do sum of squares minimization.  Most of the time it works
fine, in the sense that the results (1) match the theory and (2) are quite
repeatable.

However, when I run LOTS of them (10000 or so in a script, about an hour), I get
a lot of NaN results results for phat (about 1%); however, if I just rerun qp()
with the same parameters, usually I can get a non-NaN result.

Umm, is there anything I can do to fix this?  Is it a problem with qp() by any
chance?  Is there a transformation I can do on H to make it nicer? Would a 64
bit machine solve my problems?  Anything else I should think about?

Tx