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From: saswss@hotellng.unx.sas.com (Warren Sarle)
Subject: Re: Fast test for "quadraticness"
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In article <4i6626$f87@ixnews2.ix.netcom.com>, jdadson@ix.netcom.com(Jive Dadson ) writes:
|> ...
|> Given a twice differentiable function from Rn -> R+, is there any quick
|> way to determine the function's "quadraticness" at a point? I'm not
|> sure what I mean by "quadraticness", but a quadratic function would get
|> a perfect score, regardless of where the function evaluation-point was
|> chosen. (It is "perfectly quadratic everywhere".) "Locally
|> approximately quadriatic" functions would get good scores. Hey, I said
|> it was vague.

You need to look at the 3rd derivatives.  See Bates, D.M. and Watts,
D.G. (1988) Nonlinear Regression Analysis & Its Applications, NY: Wiley.
 
BTW, the various trust-region algorithms and Levenberg-Marquardt
algorithms automatically adapt to the degree of local quadraticness.

-- 

Warren S. Sarle       SAS Institute Inc.   The opinions expressed here
saswss@unx.sas.com    SAS Campus Drive     are mine and not necessarily
(919) 677-8000        Cary, NC 27513, USA  those of SAS Institute.
