Levinson parallel algorithm: A Finite-Dimensional Approach with an Infinite-Dimensional Perspective

Abstract

A normalization of the generators of the defect spaces of an isometry is obtained, a version of the Levinson algorithm for Toeplitz block matrices in the infinite-dimensional case is built. Additionally, a factorization of the inverse of the Toeplitz matrix by blocks is obtained. Under this methodology, the obtained recurrences in the infinite dimensional case coincide with the case of the finite dimension, and an autoregressive linear filter to estimate stationary second-order stochastic processes is obtained, usually, the area extension in statistics, applications to spectral estimation, analysis of functional data and prediction problems among other applications is required. The parallelized algorithm for computing multiplications and inverses of block matrices is developed using the Pthreads POSIX library. Two real examples of the literature is illustrated, the parameters of a VAR$(1)$ model and an autoregressive process of order $5$ (AR $(5)$) are estimated. The predicted values in each case are obtained. The estimated quality of the parallelized algorithm is validated, the $T^{RC}$ test as a measure of goodness of fit is used, negligible estimation errors are shown. The performance of the parallel algorithm by the acceleration and efficiency factors is measured, an increase of $8\%$ in speed with respect to the sequential version and the most efficient for $P = 2$ threads are shown.