Orthogonal polynomials and analytic functions associated to positive definite matrices

Fast algorithms for positive definite matrices structured by orthogonal polynomials

Positive definite matrices structured by orthogonal polynomial systems allow a Cholesky type decomposition of their inverse matrices in O(n2) steps. The algorithm presented in this paper uses the three-term recursion coefficients and the mixed moments of the involved polynomials.

Orthogonal polynomials and analytic functions

For a positive definite infinite matrix A, we study the relationship between its associated sequence of orthonormal polynomials and the asymptotic behaviour of the smallest eigenvalue of its truncation An of size n×n. For the particular case of A being a Hankel or a Hankel block matrix, our results lead to a characterization of positive measures with finite index of determinacy and of completel...

Multiple orthogonal polynomials associated with Macdonald functions

We consider multiple orthogonal polynomials corresponding to two Macdonald functions (modified Bessel functions of the second kind), with emphasis on the polynomials on the diagonal of the Hermite-Padé table. We give some properties of these polynomials: differential properties, a Rodrigues type formula and explicit formulas for the third order linear recurrence relation. 1 Macdonald functions ...

Oscillation Properties of Generalized Characteristic Polynomials for Totally Positive and Positive Definite Matrices

Computing Zeros of Analytic Functions via Modiied Moments Based on Formal Orthogonal Polynomials Computing Zeros of Analytic Functions via Modiied Moments Based on Formal Orthogonal Polynomials Computing Zeros of Analytic Functions via Modified Moments Based on Formal Orthogonal Polynomials

We consider the problem of computing all the zeros of an analytic function that lie in the interior of a Jordan curve, together with their respective multiplicities. Our approach uses modiied moments based on formal orthogonal polynomials. Numerical experiments indicate that it is far superior to classical approaches, which consider the usually ill-conditioned map from the Newton sums to the un...