Path: utzoo!attcan!utgpu!jarvis.csri.toronto.edu!mailrus!wuarchive!gem.mps.ohio-state.edu!rpi!wrf From: wrf@mab.ecse.rpi.edu (Wm Randolph Franklin) Newsgroups: comp.dsp Subject: Re: Taylor vs. Chebyshev polynomials Keywords: Taylor vs. Chebyshev polynomials Message-ID: <1989Oct3.184811.27778@rpi.edu> Date: 3 Oct 89 18:48:11 GMT References: <2421@radio.oakhill.UUCP> <1989Sep28.161516.10353@rpi.edu> <3811@deimos.cis.ksu.edu> Organization: Rensselaer Polytechnic Institute, Troy NY Lines: 93 You're right that my calling Taylor series wimps is excessive. This is especially so for sin(x) which is one of the functions that Taylor series are best suited for since sin is smooth and defined over the whole complex plane with no singularities, and the Taylor coefficients shrink fast. However, even here Chebyshev polynomials are better, and the relative difference increases as the order increases. I scaled sin to that [0,Pi] was mapped to [-1,1] and expanded about the new 0 (for Taylor) or over the interval (for Chebyshev). This is probably the interval you would want to approximate. N abs(Coef of x^n) in Taylor abs(Coef of T(N,x)) in Cheb 0 .7 .6 1 .55 .5 2 .22 .1 3 .06 .01 4 .01 .001 5 .002 .0001 6 .0002 7e-6 7 .00002 3e-7 8 3e-6 2e-8 9 2e-7 9e-10 10 2e-8 3e-11 11 1e-9 1e-12 12 8e-11 4e-14 13 5e-12 1e-15 14 3e-13 3e-17 15 1e-14 9e-19 Note: 1. Truncating the Chebyshev series when it is still expressed in T(n,x) is equivalent (almost) to truncating the Taylor series expressed in x^n. In either case, the error from stopping at any term is usually less than the first unused coefficient. (Picky note to Chebyshev hackers: this is why it is more useful to look at the coefficients of T rather than convert to a power basis.) 2. Although I only retyped 1 or 2 digits for this note, the calcs were done to 20 significant digits to avoid roundoff. 3. The Taylor to the 15th degree is between the 12th and 13th degree Chebyshev -- which is more than one term. Well, maybe you think that the interval favored Chebyshev. So try another interval [-Pi,Pi], scaled to [-1,1]. Note the function is odd. N Taylor Cheb 1 3 1 3 5 1.6 5 3 .2 7 .6 .01 9 .08 .0005 11 .007 .00001 13 .0004 2e-7 15 .00002 2e-9 The Taylor to x^15 worse than the Chebyshev to x^11. ---------------- In all the above I'm using about the best possible function for the Taylor. In fact Chebyshev can do a wide class of functions Taylor can't: - functions with a complex singularity nearby, such as 1/(x^2+.1). - functions which are only differentiable a small number of times. Taylor requires N for N-th degree, Chebyshev requires only 1st order. Admittedly Chebyshev will converge much more slowly in these cases. But it WILL converge, and Taylor WON'T. ---------------- (Now I'm really getting on my hobby-horse). There's something often much better than Chebyshev -- a Pade-Chebyshev expansion, which is a Chebyshev converted to a rational fraction. Its relative betterness also increases with N. However it requires one divide to evaluate. The problem is that Pade expressions are not in the textbooks yet, and even if they were are more complicated to calculate. Also I can't get Maple to do it right now. -- Wm. Randolph Franklin Internet: wrf@ecse.rpi.edu (or @cs.rpi.edu) Bitnet: Wrfrankl@Rpitsmts Telephone: (518) 276-6077; Telex: 6716050 RPI TROU; Fax: (518) 276-6261 Paper: ECSE Dept., 6026 JEC, Rensselaer Polytechnic Inst, Troy NY, 12180