From: David Kastrup Newsgroups: sci.math Subject: Re: cos((n)arcos(theta)) Date: 24 Oct 1997 12:40:44 +0200 J Giffen writes: > I have a value, say, B, such that > > B = cos((n)arcos(theta)). > > I want the cos and the arcos to cancel into each other. > How, then, do I get the n out of there? Oh, you just write B=T_n(x), the nth Chebychev polynomial of the first kind. Write them as x^n - (n choose 2) x^{n-2} (1-x^2) + (n choose 4) x^{n-4} (1-x^2)^2 - (n choose 6) x^{n-6} (1-x^2)^3 ... Or generate them via the recursion T_{n+1}(x) = 2x T_n(x) - T_{n-1}(x) with T_0(x) = 1 T_1(x) = x ... In short, you'll get a polynomial of degree n, and the general form of it is not quite trivial. The zeros are all simple and real and in the interval (-1,1). Chebychev polynomials are very often used for polynomial approximations because they are the least "bulgy": 2^{-n+1} T_n(x) is the polynomial of degree n and with a_n=1 that shows the least absolute maximal deviation from 0 in (-1,1). Cutting of an expansion in Chebychev polynomials (which can be easily calculated recursively with very little computational impact) leads to less maximum absolute error in function approximation than cutting off, say, a Taylor expansion (or almost any other polynomial expansion) at the same point. More than you probably ever wanted to hear about those buggers. Sorry. -- David Kastrup Phone: +49-234-700-5570 Email: dak@neuroinformatik.ruhr-uni-bochum.de Fax: +49-234-709-4209 Institut f�r Neuroinformatik, Universit�tsstr. 150, 44780 Bochum, Germany