A fast algorithm for computing the smallest eigenvalue of a symmetric positive‐definite Toeplitz matrix

Abstract

SUMMARYRecent progress in signal processing and estimation has generated considerable interest in the problem of computing the smallest eigenvalue of a symmetric positive-definite (SPD) Toeplitz matrix.An algorithm for computing upper and lower bounds to the smallest eigenvalue of a SPD Toeplitz matrix has been recently derived (Linear Algebra Appl. 2007; DOI: 10.1016 DOI: 10. /j.laa.2007. The algorithm relies on the computation of the R factor of the Q R factorization of the Toeplitz matrix and the inverse of R. The simultaneous computation of R and R −1 is efficiently accomplished by the generalized Schur algorithm. In this paper, exploiting the properties of the latter algorithm, a numerical method to compute the smallest eigenvalue and the corresponding eigenvector of SPD Toeplitz matrices in an accurate way is proposed.

Eigenvalue-Spread-Based Combination Rule for Distributed Blind Equalization in Networked System

“…The resultingR k corresponds to an estimate of R k . Then, we can accomplish the eigenvalue decomposition ofR k by utilizing several algorithms [28], [29]. Finally, the resulting maximum and minimum eigenvalues are substituted in Eq.…”