Tridiagonal Approach to the Algebraic Environment of Toeplitz Matrices, Part I: Basic Results

Abstract

This paper contains a thorough investigation of a family of symmetric “predictor polynomials” associated with a nonnegative-definite Toeplitz matrix. These polynomials are constructed from the classical predictors and from the values assumed by some dual predictors in a fixed point of unit modulus; the appropriate duality is induced by changing the sequence of reflection coefficients into its conjugate mirror image, within a unit modulus factor. The central theme of the paper is a well-defined three-term recurrence relation satisfied by these symmetric polynomials; it motivates the “tridiagonal” terminology. The properties of the recurrence are studied in detail; special attention is paid to the important issue of computing the recurrence coefficients from the reflection coefficients. It is shown how this three-term recurrence formula produces an efficient solution method, called the split Levinson algorithm, for the linear prediction problem.