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