Updating and downdating of orthonormal polynomial vectors and some applications (Q2784761)

Let \(z_i\in \mathbb C \) and \(w_i >0\), \(i=1,\dots ,n\), be complex points and weights, respectively, and consider the inner product \((f,g)=\sum _{i=1}^n \overline {f(z_i)}g(z_i)w_i \). This paper is concerned with the problem of developing \textit{updating } and \textit{downdating} procedures to compute a sequence of orthonormal polynomial vectors with respect to a discrete inner product when the points \(z_i\) can be anywhere in \(\mathbb C\). The authors propose some algorithms and indicate how the amount of computational work can be reduced when all points \(z_i\) are on the real line or all on the unit circle. The usefulness of the orthonormal polynomial vectors is illustrated using them to solve a data fitting problem, to solve a system of two polynomial equations in two variables and to solve a Bezout equation.NEWLINENEWLINEFor the entire collection see [Zbl 0972.00035].