Stability analysis of algorithms for solving confluent Vandermonde-like systems
SIAM Journal on Matrix Analysis and Applications
Polynomial and matrix computations (vol. 1): fundamental algorithms
Polynomial and matrix computations (vol. 1): fundamental algorithms
Fast inversion of Chebyshev-Vandermonde matrices
Numerische Mathematik
A Divide-and-Conquer Algorithm for the Symmetric TridiagonalEigenproblem
SIAM Journal on Matrix Analysis and Applications
Computing Matrix Eigenvalues and Polynomial Zeros Where the Output is Real
SIAM Journal on Computing
A displacement approach to efficient decoding of algebraic-geometric codes
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Fast reliable algorithms for matrices with structure
Fast reliable algorithms for matrices with structure
Structured matrices and polynomials: unified superfast algorithms
Structured matrices and polynomials: unified superfast algorithms
A Course in Digital Signal Processing
A Course in Digital Signal Processing
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
SIAM Journal on Matrix Analysis and Applications
An Orthogonal Similarity Reduction of a Matrix into Semiseparable Form
SIAM Journal on Matrix Analysis and Applications
Pivoting for structured matrices and rational tangential interpolation
Contemporary mathematics
A Fast Björck-Pereyra-Type Algorithm for Solving Hessenberg-Quasiseparable-Vandermonde Systems
SIAM Journal on Matrix Analysis and Applications
Hi-index | 5.23 |
In this paper, we survey several recent results that highlight an interplay between a relatively new class of quasiseparable matrices and univariate polynomials. Quasiseparable matrices generalize two classical matrix classes, Jacobi (tridiagonal) matrices and unitary Hessenberg matrices that are known to correspond to real orthogonal polynomials and Szego polynomials, respectively. The latter two polynomial families arise in a wide variety of applications, and their short recurrence relations are the basis for a number of efficient algorithms. For historical reasons, algorithm development is more advanced for real orthogonal polynomials. Recent variations of these algorithms tend to be valid only for the Szego polynomials; they are analogues and not generalizations of the original algorithms. Herein, we survey several recent results for the ''superclass'' of quasiseparable matrices, which includes both Jacobi and unitary Hessenberg matrices as special cases. The interplay between quasiseparable matrices and their associated polynomial sequences (which contain both real orthogonal and Szego polynomials) allows one to obtain true generalizations of several algorithms. Specifically, we discuss the Bjorck-Pereyra algorithm, the Traub algorithm, certain new digital filter structures, as well as QR and divide and conquer eigenvalue algorithms.