Matrix biorthogonal polynomials on the unit circle and non-Abelian Ablowitz-Ladik hierarchy

Adler and van Moerbeke \cite{AVM} described a reduction of 2D-Toda hierarchy called Toeplitz lattice. This hierarchy turns out to be equivalent to the one originally described by Ablowitz and Ladik \cite{AL} using semidiscrete zero-curvature equations. In this paper we obtain the original semidiscrete zero-curvature equations starting directly from the Toeplitz lattice and we generalize these computations to the matrix case. This generalization lead us to the semidiscrete zero-curvature equations for the non-abelian (or multicomponent) version of Ablowitz-Ladik equations \cite{GI}. In this way we extend the link between biorthogonal polynomials on the unit circle and Ablowitz-Ladik hierarchy to the matrix case.Comment: 23 pages, accepted on publication on J. Phys. A., electronic link: http://stacks.iop.org/1751-8121/42/36521

Quadrature Strategies for Constructing Polynomial Approximations

Finding suitable points for multivariate polynomial interpolation and approximation is a challenging task. Yet, despite this challenge, there has been tremendous research dedicated to this singular cause. In this paper, we begin by reviewing classical methods for finding suitable quadrature points for polynomial approximation in both the univariate and multivariate setting. Then, we categorize recent advances into those that propose a new sampling approach and those centered on an optimization strategy. The sampling approaches yield a favorable discretization of the domain, while the optimization methods pick a subset of the discretized samples that minimize certain objectives. While not all strategies follow this two-stage approach, most do. Sampling techniques covered include subsampling quadratures, Christoffel, induced and Monte Carlo methods. Optimization methods discussed range from linear programming ideas and Newton's method to greedy procedures from numerical linear algebra. Our exposition is aided by examples that implement some of the aforementioned strategies

ISSN 1068-9613. STRONG RANK REVEALING CHOLESKY FACTORIZATION

Abstract. For any symmetric positive definite ¢¤£¥ ¢ matrix ¦ we introduce a definition of strong rank revealing Cholesky (RRCh) factorization similar to the notion of strong rank revealing QR factorization developed in the joint work of Gu and Eisenstat. There are certain key properties attached to strong RRCh factorization, the importance of which is discussed by Higham in the context of backward stability in his work on Cholesky decomposition of semidefinite matrices. We prove the existence of a pivoting strategy which, if applied in addition to standard Cholesky decomposition, leads to a strong RRCh factorization, and present two algorithms which use pivoting strategies based on the idea of local maximum volumes to compute a strong RRCh decomposition. Key words. Cholesky decomposition, LU decomposition, QR decomposition, rank revealing, numerical rank, singular values, strong rank revealing QR factorization. AMS subject classifications. 65F30. 1. Introduction. Cholesk