Friday, July 14

IC6
Efficient Algorithms for Quasiseparable Matrices and Quasiseparable Polynomials

9:15 AM - 10:00 AM
Room: Georgian - ML
Chair: Peter Clarkson, University of Kent, United Kingdom

An interplay between orthogonal polynomials and structured matrices (such as Toeplitz, Hankel, Vandermonde, tridiagonal, unitary Hessenberg) is classical. In particular, tridiagonal matrices capture the recurrence relations for real orthogonal polynomials, while unitary Hessenberg matrices capture the recurrence relations for the Szego polynomials, i.e, polynomials orthogonal on the unit circle.

In this talk we describe a relatively new class of quasiseparable matrices that includes both tridiagonal and unitary Hessenberg matrices as special cases. Correspondingly, the associated ``quasiseparable'' polynomials generalize both real orthogonal polynomials and the Szego polynomials.

We describe generalizations of several eigenvalue algorithms for quasiseparable matrices, i.e, of a QR algorithm for quasiseparable and for shifted quasiseparable matrices, bisection method, and a divide-and-conquer algorithm. Since every symmetric matrix can be reduced to a quasiseparable one, the above methods solve the general eigenvalue problem as well.

Time permitting we will describe quasiseparable generalizations of several more well-known algorithms, e.g., of Bjorck-Pereyra and of Traub.

The papers and preprints are available at http://www.math.uconn.edu/~olshevsky/new.php

Joint works with Tom Bella, Yuli Eidelman and Israel Gohberg.

Vadim Olshevsky
University of Connecticut

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