Chebyshev-Vandermonde systems

Abstract

A Chebyshev-Vandermonde matrix \[ V = [ p j ( z k ) ] j , k = 0 n ∈ C ( n + 1 ) × ( n + 1 ) V = [{p_j}({z_k})]_{j,k = 0}^n \in {\mathbb {C}^{(n + 1) \times (n + 1)}} \] is obtained by replacing the monomial entries of a Vandermonde matrix by Chebyshev polynomials p j {p_j} for an ellipse. The ellipse is also allowed to be a disk or an interval. We present a progressive scheme for allocating distinct nodes z k {z_k} on the boundary of the ellipse such that the Chebyshev-Vandermonde matrices obtained are reasonably well-conditioned. Fast progressive algorithms for the solution of the Chebyshev-Vandermonde systems are described. These algorithms are closely related to methods recently presented by Higham. We show that the node allocation is such that the solution computed by the progressive algorithms is fairly insensitive to perturbations in the right-hand side vector. Computed examples illustrate the numerical behavior of the schemes. Our analysis can also be used to bound the condition number of the polynomial interpolation operator defined by Newton’s interpolation formula. This extends earlier results of Fischer and the first author.