Re: Hyperterminants

[Ilya Zakharevich <nospam-abuse@ilyaz.org>]

On Mon, Nov 20, 2023 at 07:59:45PM -0800, Ilya Zakharevich wrote:
> Are there scripts to calculate hyperterminants (beyond erfc) in Pari?
> (Googling is becoming more and more useless every decade…)
> 
>   (Just in case: in last 30 years, these seem to become one of the
>    principal tools when dealing with asymptotic expansions…)

It is hard to explain quickly, since it seems to be a part of “the new
revolution in mathematical bedrock” of analysis.  (A third revolution,
if you count Newton’s fluents/fluxions, then Cauchy’s ε/δ, and now
these resurgence/multi-summation/transseries/etc. approaches.)

  Very briefly: these are new “universal” transcendental functions
  which (practically) appear in ALL problems of asymptotic analysis
  (when one goes close to or beyond the obvious “sum to the smallest
  term of the asymptotic series).

  So THEIR usability is very similar to the statement “(practically)
  it is useful to be able to calculate INTEGRALS of functions”.
  Probably this was quite surprising at Newton/Leibniz time too…

⁜⁜⁜⁜⁜⁜⁜⁜⁜⁜⁜

Just one of the examples I’m staring at in the last few days: suppose
you want to calculate numerically Σ k! tᵏ.  If t=-23, then near n=23
the terms stop to decrease, so the Cauchy’s ε/δ approach fails to find
the exact value¹⁾ of the sum.

  ¹⁾ The whole point of
     resurgence/multi-summation/transseries/etc. approaches is that
     there are (absolutely non-obvious) isomorphisms between (“widely
     extended”²⁾) algebras of power series and algebras of functions.
     Hence the notion of “exact value” makes perfect sense — and for
     this series, it can be found using a suitable generalization of
     the (original) Borel transform.

     (Well, to be more precise, I should better say “PARTIAL
     isomorphisms”.)  

  ²⁾ Both in their size as the sets, AND in the collection of
     operations one can make, like composition (hence log, exp etc.),
     compositional-inversion, convolution etc. 

Stopping at n=23 gives³⁾ a precision of about 13 decimal places.
However, if instead of “abruptly stopping at n=23” you stop slowly,
choosing the cutoff coefficients “smartly”, you can sum more terms and
get more precision.

  ³⁾ Well, IIRC I was using (2n-1)!!/2ⁿ instead of n!, though.  But
     the difference should be minimal.

I did some experiments with using ½erfc((n-N₀)/w) as the cutoff
coefficients (this is WAY beyond Cesàro summation).  With a
particular N₀, the result depends on w — but there is a clearly seen
critical value.  Take it.

With N₀=23, the error is down to 24 decimal places.  With N=23·4.5, it
is down to 58 decimal places.  However, increasing N further leads to
growing errors: with N₀=23*5, the error is up to 45 places.

  (This does not seem to be investigated, and so far I cannot convince
   myself I understand the details of what is happening.  For example,
   the extremum in w is very sharp — about 1%.(

⁜⁜⁜⁜⁜⁜⁜⁜⁜⁜⁜

Here come hyperterminants: for the series above, one considers the
Riemann surface of lambertw().  Calculating hyperterminants with
parameters being the branch points of this Riemann surface

  SHOULD GIVE ALL THE INFORMATION ABOUT THE CONVERGENCE OF THIS SERIES.

In particular, (I expect that in my problem) one should be able to
describe what is w at the critical points, what is the error etc.

I cannot be more specific.  (But formally, hyperterminants are
repeated Cauchy–Heine transforms of exp.)  One of the first references
is

  Berry–Howls ’90 “Hyperasymptotics …”

Hope this helps,
Ilya