Fast Solution of Yandermonde-Like Systems Involving Orthogonal Polynomials

Consider the (n +1) x (n + 1) Vandermonde-like matrix P = [p,--i(o/-i)L where the polynomials po(x),... ,pn(x) satisfy a three-term recurrence relation. We develop algorithms for solving the primal and dual systems, Px = b and Pa =f respectively, in O(n) arithmetic operations and O(n) elements of storage. These algorithms generalize those of Bjorck & Pereyra which apply to the monomial case Pi(x) = x'. When the p,(x) are the Chebyshev polynomials, the algorithms are shown to be numerically unstable. However, it is found empirically that the addition of just one step of iterative refinement is, in single precision, enough to make the algorithms numerically stable.