A weakly stable algorithm for general Toeplitz systems

We show that a fast algorithm for the QR factorization of a Toeplitz or Hankel matrix A is weakly stable in the sense that R^T.R is close to A^T.A. Thus, when the algorithm is used to solve the semi-normal equations R^T.Rx = A^Tb, we obtain a weakly stable method for the solution of a nonsingular Toeplitz or Hankel linear system Ax = b. The algorithm also applies to the solution of the full-rank Toeplitz or Hankel least squares problem.Comment: 17 pages. An old Technical Report with postscript added. For further details, see http://wwwmaths.anu.edu.au/~brent/pub/pub143.htm

Formally biorthogonal polynomials and a look-ahead Levinson algorithm for general Toeplitz systems

Systems of linear equations with Toeplitz coefficient matrices arise in many important applications. The classical Levinson algorithm computes solutions of Toeplitz systems with only O(n(sub 2)) arithmetic operations, as compared to O(n(sub 3)) operations that are needed for solving general linear systems. However, the Levinson algorithm in its original form requires that all leading principal submatrices are nonsingular. An extension of the Levinson algorithm to general Toeplitz systems is presented. The algorithm uses look-ahead to skip over exactly singular, as well as ill-conditioned leading submatrices, and, at the same time, it still fully exploits the Toeplitz structure. In our derivation of this algorithm, we make use of the intimate connection of Toeplitz matrices with formally biorthogonal polynomials

Block pivoting implementation of a symmetric Toeplitz solver

Toeplitz matrices are characterized by a special structure that can be exploited in order to obtain fast linear system solvers. These solvers are difficult to parallelize due to their low computational cost and their closely coupled data operations. We propose to transform the Toeplitz system matrix into a Cauchy-like matrix since the latter can be divided into two independent matrices of half the size of the system matrix and each one of these smaller arising matrices can be factorized efficiently in multicore computers. We use OpenMP and store data in memory by blocks in consecutive positions yielding a simple and efficient algorithm. In addition, by exploiting the fact that diagonal pivoting does not destroy the special structure of Cauchy-like matrices, we introduce a local diagonal pivoting technique which improves the accuracy of the solution and the stability of the algorithm.This work was partially supported by the Spanish Ministerio de Ciencia e Innovacion (Project TIN2008-06570-C04-02 and TEC2009-13741), Vicerrectorado de Investigacion de la Universidad Politecnica de Valencia through PAID-05-10 (ref. 2705), and Generalitat Valenciana through project PROMETEO/2009/2013.Alonso-Jordá, P.; Dolz Zaragozá, MF.; Vidal Maciá, AM. (2014). Block pivoting implementation of a symmetric Toeplitz solver. Journal of Parallel and Distributed Computing. 74(5):2392-2399. https://doi.org/10.1016/j.jpdc.2014.02.003S2392239974

Stability of Levinson algorithm for Toeplitz-like systems

Numerical stability of the Levinson algorithm generalized for Toeplitzlike systems, is studied. Arguments based on the analytic results of an error analysis for floating point arithmetic produce an exponential upper bound on the norm of the residual vector. The base of such exponential function can be small for a class of matrices containing point row diagonally dominant matrices. Numerical experiments show that, for this class, Gaussian elimination by row and Levinson algorithm have residuals of the same order of magnitude. As expected, the empirical results point out that the theoretical bound is too pessimistic

Extracción paralela de valores propios en matrices Toeplitz simétricas usando hardware gráfico

Adaptación e implementación del algoritmo de extracción de valores propios shift-and-invert 2-way Lanczos aplicado a matrices Toeplitz simétricas usando entorno de programación paralela en GPUs CUDA.Gracia Gil, L. (2008). Extracción paralela de valores propios en matrices Toeplitz simétricas usando hardware gráfico. http://hdl.handle.net/10251/13620Archivo delegad

Digital signal processing algorithms and structures for adaptive line enhancing

Imperial Users onl

Algorithms in time series

The use of finitely parametrized linear models such as ARMA models in analysing time series data has been extensively studied and in recent years there has been an increasing emphasis on the development of fast regression—based algorithms for the problem of model identification. In this thesis we investigate the statistical properties of pseudo—linear regression algorithms in the context of off-line and online (real-time) identification. A review of these procedures is presented in Part I in relation to the problem of identifying an appropriate ARMA model from observed time series data. Thus, criteria introduced by Akaike and Rissanen are important here to ensure a model of sufficient complexity is selected, based on the data. In chapter 1 we survey published results pertaining to the statistical properties of identification procedures in the off-line context and show there are important differences as concerns the asymptotic performance of certain parameter estimation algorithms. However, to effect the identification process in real-time recursive estimation algorithms are required. Furthermore, these procedures need to be adaptive to be applicable in practice. This is discussed in chapter 2. Technical results and limit theorems required for the theoretical analysis conducted in Part II are collated in chapter 3. Chapters 4 and 5 of Part II are therefore devoted to the detailed investigation of particular algorithms discussed in Part I. Chapter 4 deals with off-line parameter estimation algorithms and in chapter 5, the important idea of a Description Length Principle introduced by Rissanen, is examined in the context of the recursive estimation of autoregressions. Empirical evidence from simulation experiments are also reported in each chapter and in chapter 5, aspects of speech analysis are incorporated in the simulation study. The simulation results bear out the theory and the proofs of asymptotic results are given at the end of the chapter