Date: Thu, 6 Apr 95 03:03:20 CDT From: rusin (Dave Rusin) To:
Subject: Re: Chebychev companion matrix >Thanks for your answer.
What you describe works well with the >coefficients of the
polynomial p(x)=a_0+a_1x+...+a_nx^n, but the point >is that my
polynomial is given as p(x)=c_0+c_1T_1(x)+...+c_nT_n(x), >T_i(x)
Chebychev polynomials, and I don't want to compute the a's from
>the c's. Obviously, (it seems to me) doing what you prescribes
with >the c-coefficients, does not give the roots of p(x). (What
it gives, I >cannot immediately see.) Right. I don't rememebr
just what the T_i are, but perhaps this will help. If p is any
polynomial, then the quotient ring V = C[x]/(p(x)) is a finite
dimensional vector space; indeed, if p has degree n, then V is
spanned by 1, x, x^2, ..., x^(n-1). The way the companion matrix
comes into play is that it's the matrix that goes with the
linear transformation which multiplies everything by x. (If p =
Prod(x-r^i), then another basis is the set of polynomials
bi(x)=p(x)/(x-ri), w