On Tuesday, 15 September 2015 at 05:16:53 UTC, deadalnix wrote:
On Saturday, 11 July 2015 at 03:02:24 UTC, Nick B wrote:
On Thursday, 20 February 2014 at 10:10:13 UTC, Nick B wrote:
...
If you are at all interested in computer arithmetic or
numerical methods, read this book. It is destined to be a
classic.
To be honest, that sound like snake oil salesman speech to me
rather than science. It's all hand waving and nothing concrete
is provided, the whole thing wrapped in way too much
superlatives.
The guy seems to have good credential. Why should I read that
book ?
I read the whole book and did not regret it at all, but I was
already looking for good interval arithmetic implementations. I
found that the techniques are not too different (though improved
in important ways) from what is mainstream in verified computing.
I found signs that techniques like this are standard in beam
physics. (Caveat, I am not a beam physicist, but my friend at
CERN is.) And the bibliography for MSU's COSY INFINITY, a
verified computing tool from the beam physics community, provided
a lot of interesting background information:
http://www.bt.pa.msu.edu/pub/
What is perhaps most surprising is not that the techniques work
well but that they can be implemented efficiently in a sensible
amount of hardware, little more expensive than bare floating
point hardware. Even maybe the most surprising results about
search-based global optimization with unums have precedent, see
e.g. "Rigorous Global Search using Taylor Models," M. Berz, K.
Makino, Symbolic Numeric Computation 2009, (2009) 11-19,
http://bt.pa.msu.edu/cgi-bin/display.pl?name=GOSNC09 (Taylor
model techniques are similar to direct computation with intervals
but add a bit more sophistication.)
So, from my perspective, I think a unum library would at least be
an interesting and useful library, and roughly in the style of
the Mathematica / Python libraries could reduce unum interval
computations to floating point computations with a modest amount
of overhead. There is a sense in which we might expect the small
overhead up front to be well worth it in the overall system: less
haste, more speed. Hacks to try to compensate for incorrect
behavior after the fact may end up being more costly overall,
certainly to the programmer but perhaps also to the machine. For
example, the common need to stick a loop over an entire vector or
matrix into the inner loop of an iterative method to renormalize
to prevent floating point rounding errors from accumulating.
Whether this library should be part of the standard library, I
don't know. It would seem to depend on how much people want the
standard library to support verified numerical computing. If it
is clear that verified numerical computing needs good support in
the standard library, something like unums should be there, maybe
even with some other techniques built on top of them (Taylor
model or Levi-Civita for example).
Anthony
