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

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