Book ChapterDOI
Improving the Accuracy of the Generalized Schur Algorithm
S. Chandrasekaran,Ali H. Sayed +1 more
- pp 223-232
TLDR
The modified Schur algorithm proposed in this work essentially achieves the bound for a large class of structured matrices that satisfy R-FRF T = GJG T, where J is a signature matrix, F is a stable lower-triangular matrix, and G is a generator matrix.Abstract:
We show how to stabilize the generalized Schur algorithm for the Cholesky factorization of positive-definite structured matrices R that satisfy R-FRF T = GJG T , where J is a signature matrix, F is a stable lower-triangular matrix, and G is a generator matrix. We use a perturbation analysis to indicate the best accuracy that can be expected from any finite precision algorithm that uses the generator matrix as the input data. We then show that the modified Schur algorithm proposed in this work essentially achieves this bound for a large class of structured matrices.read more
References
More filters
Journal ArticleDOI
A state-space approach to adaptive RLS filtering
Ali H. Sayed,Thomas Kailath +1 more
TL;DR: This article is to show how several different variants of the recursive least-squares algorithm can be directly related to the widely studied Kalman filtering problem of estimation and control.
Journal ArticleDOI
Displacement structure: theory and applications
Thomas Kailath,Ali H. Sayed +1 more
TL;DR: This survey paper describes how strands of work that are important in two different fields, matrix theory and complex function theory, have come together in some work on fast computational algorithms for matrices with what the authors call displacement structure, and develops a fast triangularization procedure.
Journal ArticleDOI
Fast Gaussian elimination with partial pivoting for matrices with displacement structure
TL;DR: The algorithms proposed in this paper suggest an attractive alternative to look-ahead approaches, where one has to jump over ill-conditioned leading submatrices, which in the worst case requires O(n 3 ) operations.
Journal ArticleDOI
The Numerical Stability of the Levinson-Durbin Algorithm for Toeplitz Systems of Equations
TL;DR: The numerical stability of the Levinson-Durbin algorithm for solving the Yule-Walker equations with a positive-definite symmetric Toeplitz matrix is studied and arguments based on the analytic results of an error analysis for fixed-point and floating-point arithmetics show that the algorithm is stable and comparable to the Cholesky algorithm.