Inversion of Vandermonde-like matrices
An \((n+1)\times (n+1)\) matrix \(V_n\) is called a Vandermonde-like matrix if it takes the form \[ V_n = \left[ \begin{matrix} p_0(x_0) & \cdots & p_n(x_0) \\ \vdots & & \vdots \\ p_0(x_n) & \cdots & p_n(x_n) \end{matrix} \right], \] where the polynomials \(p_i\) satisfy a certain three-term recursion. The nodes \(x_0,\dots, x_n\) are assumed to be distinct, which guarantees the invertibility of \(V_n\). This paper is concerned with finding analytic expressions for the entries of \(V^{-1}_n\). Among other results it is shown that \[ V_n^{-1} = D \left[ \alpha_i^{\{1\}} (x_j) \right]_{i,j = 0}^n \text{diag}\left(\frac{1}{K^{\prime}_{n+1}(x_0)},\dots,\frac{1}{K^{\prime}_{n+1}(x_n)}\right), \] where \(K_i(x) = \prod_{l = 0}^{i-1}(x-x_l)\) and \(\alpha_i^{\{1\}}(x)\) are the so called Clenshaw polynomials associated with \(K_i(x)\), while the diagonal matrix \(D\) constains coefficients from the three-term recursion. A numerical section discusses the influence of the node ordering on the finite-precision aspects of this formula. It is demonstrated that Leja orderings may have a positive influence on the accuracy compared to monotone orderings if the nodes tend to be clustered.
scientific article; zbMATH DE number 2153705