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error estimates (was: GSL K0/K1)

On the question of error estimates. I have addressed
some of this before on this list, although in more
general contexts. Here is what I have to say about
error estimates in particular.

Warning: This is probably going to be a long one,
like many of my cries of despair on this list...

First, some agreement on the behavior of function evaluations.
The best possible behavior for an evaluation is to conform to
the following specification:
  The function f(x) must return a result, for a machine
  represented number x, which is the closest possible
  machine representable value to the mathematically
  correct value, correctly rounded.

For something like the platform-provided sin(x) and cos(x),
this is definitely the specification. Of course, some
implementations fall short. There are examples, and
associated discussions to be googled...
I have not checked this one myself. But you get the idea.
I hope this one is not really true... anybody want to check?

If a function conforms to the best-possible specification,
then there is no need for an error estimate. The estimate
is implicit in the contract.

But this is hard enough to achieve, even for sin(x).
If we stuck to that, there might no functions at all.
So we sacrifice consistency and quality for coverage.

The overriding problem with the special functions
is the wide range of quality. Some of the functions are pretty
good (although maybe not as good as hoped, i.e. K0/K1). Some
are kind of difficult and lose in corner cases. And some are
just plain nuts. Amongst the "just plain nuts" examples, the
confluent hypergeometrics stick out.

The error estimates are a kind of apology for this state
of affairs, but they are obviously not very desirable

Here is a quote from a message that I sent to this
list several years ago:
  "This turns out to be a poor apology, since it tends to gum-up the
   works for the whole sublibrary, eating performance and occupying
   a large piece of intellectual real estate.

   The error estimation code must, at the least, be factored out. More
   to the point, it should likely be discarded in the main line of
   development. Other notions of error control should be investigated."

So I think we are in agreement about what is desirable.
But I have only the vaguest notions of how to get there.

Here is a further quote from the same message:
  "Similar to the notion of sublibrary dependence discussed elsewhere,
   the special functions should be hierarchically organized. The
   dependencies should be made clear, and foundational levels of
   ironclad functions should be made explicit. Such ironclad
   functions should occupy the same place in the users mind as
   platform-native implementations of sin(x); they should be
   beyond question for daily use.

   Other functions, which suffer from implementation problems
   or are simply too complicated to guarantee the same level
   of correctness should be explicitly identified, as part
   of the design (not just the documentation!)."

For example, functions which are ironclad should not bother
to do error estimates. Functions which are harder to control
should do something to indicate how they are behaving,
whether this means returning an error estimate, or just
returning some kind of "loss of accuracy" flag. Of course,
the heuristics needed to identify loss of accuracy might be
just as expensive as or equivalent to an actual error estimate.

Functions which are "just plain nuts" might be pushed
into an "experimental" sublibrary.

Along with the problems involving variable quality, there
is some actual intellectual confusion in this area as well.
Examples of this are the versions of trig functions which return
error estimates. Not only should these be among the ironclad functions
(and probably not even provided by the library... what is the point),
but the notion of "error estimate" expressed there is just confused.
What those functions are trying to do is detect and report some
form of ill-conditioning in the arguments, related to argument
reduction. This is completely different from the theology
expressed in the best-possible specification above. In that
specification, the argument x is a known machine represented
number, so there is no question of its conditioning.

I remember writing these trig functions and puzzling over
what I was doing. I was trying to fix some instability
problem in the testing of functions which depended on
trig evaluation; the ill-conditioning for large arguments
was making it hard to create stable tests. But this was
simply a problem with the tests and not with the functions.
That is one day that I should have just sat on my hands and
typed nothing; we would have been better off in the long run.

I think the problem of argument conditioning is fundamentally
a client-side issue. We should probably do nothing about it.
If the client wants to evaluate sin(1.0e300), then it is
probably not our problem. We should just do our best to
return f(x) for the given machine-represented x.

Ok, back to the real problem of error estimation and variable
quality. What constitutes an ironclad function? In essence
any function of a single variable (with no other parameters)
must be ironclad. At the very least, it can be expressed
with some form of Chebyshev expansion, argument transformation,
etc, which should meet the best-possible specification. In other
cases, there is an efficient high-quality algorithm which makes
the evaluation easy, like the Lanczos algorithm for gamma(z).

This includes functions of a single complex variable which
it should be possible to treat in the same manner.

What about functions with one "parameter", like besselK(nu, x)?
These are harder and tend to have problems with corner cases
or large arguments. But many of these should also be ironclad.
Most of them have well-known stable methods, and the
implementation only needs to take some care to get
everything right.

What about besselJ(nu, x)? This should also be ironclad. Although
it may have some problems with argument conditioning, which can
arise for many oscillatory functions, we have decided that
argument-conditioning is not a problem we can address.
So it should be no harder than sin(x), in principle.

Stable iterative methods are applicable in many cases.
Continued fraction methods are good, and I use these
in many places. But they can also fail mysteriously.
Gautschi has written some classic papers on this subject.
Asymptotic methods are needed to fill the gaps. The
tradeoff between iteration and asymptotic expansion
has not been studied well enough. Certainly not by
me. It would be best if the tradeoff point were
detected automatically in some way, by the
implementation. Otherwise, it becomes harder to
understand what the code is doing, with many
hard-wired constants, etc, that control the
switch on methods. These matching problems
are really tedious to think about explicitly;
the machine should be tasked to figure that out.

What about functions with more parameters? First the good
news. All the elliptic functions and related elementary
functions should be ironclad. A great deal is known
about the evaluation of these. At the very least,
you can always fall back on the arithmetic-geometric
mean, which is stable and correct, though maybe not
always the most efficient.

Now the bad news: hypergeometric functions, including
the regular and confluent functions. These are terrible.
Probably all must go to the "experimental" ghetto.

Functions with more parameters are harder to test as well.
Maybe even impossible to test. Certainly, blind testing
seems useless. You have to know what errors to look for.

Another area of doubtful intellectual content: the
"mode" argument for some of the evaluations. At the
time, I wanted to be able to decrease accuracy at
runtime, for performance gain in return. The
obvious thing to do was to control the order
of Chebyshev expansions. Fewer coefficients
should be faster right? Well, who knows. It was
never really studied properly. And it was only
applicable for functions which depend on
underlying Chebyshev expansions. Probably
just a bad idea. The main library type is 'double';
the functions should probably just run as close to
double precision as possible. Architectures have
changes a lot since then as well, so spending
extra cycles on some small local computations
is not a big deal.

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