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.
Rights
This publication is a work of the U.S. Government as defined in Title 17, United States Code, Section 101. Copyright protection is not available for this work in the United States.Collections
Related items
Showing items related by title, author, creator and subject.
-
Toward joint Medical Logistics 2010 and beyond: process innovation and redesign of Class VIII supply chain at a medical logistics company
Augustitus, Mary A.; Stevenson, Sonja L. (Monterey, California. Naval Postgraduate School, 2000-12);The purpose of this thesis is to evaluate current Class VIII supply chain procedures at a U.S. Medical Logistics Company (Med Log Co), process map the "as is" baseline process and propose possible "to be" process redesign ... -
Expeditionary Mine Countermeasures (ExMCM) C4I Requirements (Continuation)
Das, Arijit (Monterey, California: Naval Postgraduate SchoolMonterey, California. Naval Postgraduate School, 2019-12); NPS-19-N065-AThe ExMCM is a broad program providing an innovative approach to the Mine Warfare mission area, required to operate with both U.S Navy and U.S. Marine Corps forces. The large number of sonar imagery files (from the MK18 ... -
Applying systems engineering methodologies to the creative process
Stone, Adam H. (Monterey, California: Naval Postgraduate School, 2014-09);The application of systems engineering methodologies to the creative process provides opportunities to improve the creative capabilities of individuals and organizations. Through creativity and systems engineering research, ...