The condition of Vandermonde-like matrices involving orthogonal polynomials☆

The condition number (relative to the Frobenius norm) of the n X n matrix p” = [Pi-ltxj)lY,j-l is investigated, where p,( .) = p,( .; dX) are orthogonal polynomials with respect to some weight distribution dX, and xi are pairwise distinct real numbers. If the nodes x j are the zeros of p,, , the condition number is either expressed, or estimated from below and above, in terms of the Christoffel numbers for dh, depending on whether the p, are normalized or not. For arbitrary real xi and normalized p, a lower bound of the condition number is obtained in terms of the Christoffel function evaluated at the nodes. Numerical results are given for minimizing the condition number as a function of the nodes for selected classical distributions dX.