1988/01/01 by Nicholas J. Higham · 2 citations
Mathematics · Computer Science · Engineering · #Numerical methods for differential equations #Matrix Theory and Algorithms #Advanced Numerical Methods in Computational Mathematics #Vandermonde matrix #Mathematics #Orthogonal polynomials #Algebra over a field #Combinatorics #Applied mathematics #Pure mathematics
paper · doi:10.1093/imanum/8.4.473
openalex publication_date 1988/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
Consider the (n + 1) × (n + 1) Vandermonde-like matrix P=[pi-1(αj-1)], 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 PTa = f respectively, in O(n2) arithmetic operations and O(n) elements of storage. These algorithms generalize those of Björck & Pereyra which apply to the monomial case pi(x). When the pi(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.