Tridiagonal approach to the algebraic environment of Toeplitz matrices. I: Basic results. (English) Zbl 0728.65020

The paper investigates families of “symmetric predictor polynomials” which arise in conjunction with nonnegative-definite Hermitian Toeplitz matrices and associated reflection coefficients. The linear prediction problem, which essentially amounts to computing the first column of the inverse of the Toeplitz matrix, is usually solved by means of the Toeplitz algorithm. This is equivalent to a diagonal reduction of the matrix, while the methods proposed here correspond to a tridiagonal reduction or equivalently to a three-term recurrence relation of the newly introduced polynomials. The new methods are claimed to be able to solve the linear prediction problem very efficiently.


65F05 Direct numerical methods for linear systems and matrix inversion
65C99 Probabilistic methods, stochastic differential equations
60G25 Prediction theory (aspects of stochastic processes)
62M20 Inference from stochastic processes and prediction
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