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Re: [Axiom-developer] Bug 215: sin asin(7.0::COMPLEX FLOAT)

From: Gabriel Dos Reis
Subject: Re: [Axiom-developer] Bug 215: sin asin(7.0::COMPLEX FLOAT)
Date: Wed, 17 Jan 2007 10:51:16 -0600 (CST)

On Wed, 17 Jan 2007, Waldek Hebisch wrote:

| Gabriel Dos Reis wrote:
| > Waldek Hebisch <address@hidden> writes:
| > | However, the problem with asin is just one special case.  We
| > | would like to support many multivalued special functions (which in
| > | numerical version require branch cuts).  Tracking that we get
| > | "correct" values on cuts may well take significant portion of
| > | effert to implements those functions.  OTOH computing with
| > | values on branch cuts does not seem very useful.  So I am tempted
| > | to declare that arguments branch cuts are errors (like divison
| > | by 0).
| >
| > I'm concerned with that approach.
| >
| > After all, this is a mathematical computational platform.  If we go
| > that way, how else can we expect other people to take branch cuts
| > seriously?
| >
| 1)
| What can be more serious than signaling error?
| 2)
| I am mathematician and I do not "take branch cuts seriously".

I'm a computational mathematician -- even though I ended up corrupted by
computer scientists -- and I do take branch cuts seriously.

[ In my PhD thesis work on Constant Mean Curvature surfaces, where
  I did lot of numerical simulation and "construction", Riemann
  surfaces were my benchwork. ]

| Serious math works with arbitrary branches, multivalued functions
| or Riemennian surfaces.

Which is why just signaling an error is not an option, from my perspective.
We must do better.  See links below.

|  Branch cuts are an artifical convention
| which pretends that multivalued functions can be used naively in
| numerical computations.  In some sense it is an ideal field for
| standarisation: many choices are arbitrary, but for effective
| shortcut communication everybody should use the same choices.
| However, standarizing branch cuts produces a formal structure
| which has little to do with original functions.  Once such structure
| is available there is good chance that somebody will abuse it to
| archive some good effect.  But in most cases it would be better to
| use another mechanizm.  More specifically, serious complex numerical
| computation can not depend on values on branch cuts

But, they do.  And that is a matter of life Axiom has to take into
account if it ever has to be serious about "computational platform"

Somewhere we should record relevant literature on Axiom website.

-- Gaby

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