Positive definite Toeplitz matrices, the Arnoldi process for isometric operators, and Gaussian quadrature on the unit circle

Abstract

We show that the well-known Levinson algorithm for computing the inverse Cholesky factorization of positive definite Toeplitz matrices can be viewed as a special case of a more general process. The latter process provides a very efficient implementation of the Arnoldi process when the underlying operator is isometric. This is analogous with the case of Hermitian operators where the Hessenberg matrix becomes tridiagonal and results in the Hermitian Lanczos process. We investigate the structure of the Hessenberg matrices in the isometric case and show that simple modifications of them move all their eigenvalues to the unit circle. These eigenvalues are then interpreted as abscissas for analogs of Gaussian quadrature, now on the unit circle instead of the real line. The trapezoidal rule appears as the analog of the Gauss-Legendre formula.