Path: utzoo!attcan!utgpu!jarvis.csri.toronto.edu!mailrus!cs.utexas.edu!tut.cis.ohio-state.edu!ucsd!nosc!trout.nosc.mil!broman
From: broman@schroeder.nosc.mil (Vincent Broman)
Newsgroups: comp.dsp
Subject: Re: Taylor vs. Chebyshev polynomials
Message-ID: 
Date: 3 Oct 89 15:43:21 GMT
References: <2421@radio.oakhill.UUCP> <1989Sep28.161516.10353@rpi.edu> <3811@deimos.cis.ksu.edu>
Sender: nobody@nosc.NOSC.MIL
Reply-To: broman@nosc.mil
Followup-To: comp.dsp
Organization: Naval Ocean Systems Center, San Diego
Lines: 30
In-reply-to: mac@harris.cis.ksu.edu's message of 3 Oct 89 01:03:47 GMT

[mac@ksuvax1.cis.ksu.edu posts a misunderstanding about telescoping
power series with Chebychev polynomials.]

Actually, if one could always reduce the degree of a polynomial
approximation by one, retaining the same max error, then repeating the
process would give us a zeroth degree approximation with the same max
error as the original Nth degree polynomial provided.
Doesn't seem reasonable.

The method looks more like this:
Generate a Taylor series giving significantly better accuracy ofapproximation
than is needed over the interval of interest, say \epsilon / 2.
Express this (N-th degree) polynomial as a weighted sum
of Chebychev polynomials:
\sum_{k=0}^N \alpha_k T_k ( x).
This step is just a change of basis in the vector space of N-th degree
polynomials.
Now choose M such that \sum_{k=M+1}^N \| \alpha_k \| <= \epsilon / 2,
i.e. the tail of the series is small enough.
Then we use the M-th degree polynomial \sum_{k=0}^M \alpha_k T_k ( x)
as our approximating polynomial, and the error will be less
than or equal to \epsilon / 2 + \epsilon / 2 = \epsilon.

Generally M will be much smaller than N; one is trading off
accurate approximation of the function derivatives near one point
for accurate approximation of function values over the whole interval.


Vincent Broman,  code 632, Naval Ocean Systems Center, San Diego, CA 92152, USA
Phone: +1 619 553 1641    Internet: broman@nosc.mil   Uucp: sdcsvax!nosc!broman