From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Subject: Re: Smoothing function
Date: 26 Jan 2000 21:33:34 GMT
Newsgroups: sci.math.num-analysis
Summary: [missing]

Laurent Guerby <guerby@acm.org> writes:
> I need to to find a function [0,1]->[0,1] that has the following
> properties:
> 
> * f(0)=0, f(1)=1
> * all derivatives (or at least up to some order n) are 0 in 0 and 1
> * as fast as possible to compute (64 bits IEEE FP)

Art Werschulz <agw@shekel.mcl.cs.columbia.edu> writes:
> The function f(x) = exp(-1/(x^2(1-x)^2)) does the trick.  See Gelbaum
> and Olmstead, "Counterexamples in Analysis", pg 40 (item 11).

In article <86wvow3a32.fsf@ppp-223-101.villette.club-internet.fr>,
Laurent Guerby  <guerby@acm.org> wrote:
>
>Some people mailed me about solutions of this form (thanks Dave!), but
>this does not solve my problem, since f(1) must be 1, and it looks
>like it's where the difficulty is.
>
>The mathematical solution is just to integrate and rescale the f you
>described, but unfortunately raw numerical integration will lead to
>poor performance and/or poor higher order derivative behaviour.

Well, I suppose you'll still get poor numerical performance, but you
can get functions of the sort you want, all in terms of  exp  .
The reason that things like  exp(-1/x)  work near zero is that  exp(1/x)
blows up so rapidly at x=0. Your problem is that it doesn't approach  1
fast enough as  x-> 1. Well, keep the  exp() but replace the  1/x  by
something which goes to oo as x -> 0 and yet decreases very rapidly
to  0  as  x->1, say  (1/x)* exp(-1/(1-x))  . So a suggestion is
    f(x) = exp( - exp(-1/(1-x))/x )
But I guess I prefer something more symmetrical, say
    g(x) = arctan(x*exp(1/(1-x^2)))
suitably normalized.

The ability to construct C^\infty functions with compact support is of
crucial importance in the development of differential geometry, where
examples like these, and their variants, are used freely -- with the 
usual complete disregard of issues of efficiency or numerical stability.

dave
==============================================================================

From: hrubin@odds.stat.purdue.edu (Herman Rubin)
Subject: Re: Smoothing function
Date: 26 Jan 2000 21:30:50 -0500
Newsgroups: sci.math.num-analysis

In article <ylaya9czwd2.fsf@shekel.mcl.cs.columbia.edu>,
Art Werschulz  <agw@shekel.mcl.cs.columbia.edu> wrote:
>Hi.

>Laurent Guerby <guerby@acm.org> writes:

>> I need to to find a function [0,1]->[0,1] that has the following
>> properties:

>> * f(0)=0, f(1)=1
>> * all derivatives (or at least up to some order n) are 0 in 0 and 1
>> * as fast as possible to compute (64 bits IEEE FP)

>The function f(x) = exp(-1/(x^2(1-x)^2)) does the trick.  See Gelbaum
>and Olmstead, "Counterexamples in Analysis", pg 40 (item 11).

This function is 0 at both 0 and 1.  One which does what is asked
for is 

	g(x) = exp(-1/x^2)/[exp(-1/x^2) + exp(-1/(1-x)^2)]
-- 
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hrubin@stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558