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From: "John D'Errico" <woodchips@rochester.rr.com>
Newsgroups: comp.soft-sys.matlab
Subject: Re: interpolating/smoothing w/ monotonically increasing
Date: Thu, 13 Aug 2009 18:43:20 +0000 (UTC)
Organization: John D'Errico (1-3LEW5R)
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"Bruno Luong" <b.luong@fogale.findmycountry> wrote in message <h61jq9$5hq$1@fred.mathworks.com>...
> A remark that might be relevant: once the points are adjusted monotonically, a Hermit cublic interpolation (PCHIP) can be used for higher order interpolation properly. The monotonic will be prevailed.
> 
> Bruno

To some extent this is true, at least if you will set the
knots of the spline as the same as the data points.

However it is arguably better to use a monotone cubic
model in the first place. Even if the knots are still
coincident with all of the data points, this will still
result in a C2 cubic. Thus you will get a monotone
approximant that both fits the data as well as
possible and is C2. Since pchip is only C1, this may
be an issue. And application of pchip to a general
set of monotone points will not result in a C2 curve.
It will still be only C1.

I once did a minor survey, and I found that people
could indeed see the difference between a C1 and a
C2 cubic fit. Sometimes that difference is subtle, but
it is very much visible.

And it is not adequate to make the data points
monotone and then to run a standard interpolating
spline through them, since there is no way to
assure that the spline will still be truly monotone.

Finally, use of a tool like SLM allows you to build
many other constraints into the curve fit. Must it
pass through a specific point? Or perhaps it must
have a zero slope at one end of the curve? You
can also control the smoothness of the curve by a
judicious choice of the knots with SLM. 

John