Numerical technique increases the efficiencies of the numerical methods involving Toeplitz matrices by reducing the number of multiplications required by an N-order Toeplitz matrix from N-cubed to N-squared multiplications. Some efficient algorithms are given

Bareiss, E. H.

English

NASA Technical Reports Server

September 1969	 Brief 69-10415 
•
119Y
	
AEC-NASA TECH BRIEF 
FES
AEC-NASA Tech Briefs describe innovations resulting from the research and development program 
of the U.S. AEC or from AEC-NASA interagency efforts. They are issued to encourage commercial 
application. Tech Briefs are published by NASA and may be purchased, at 15 cents each, from 
the Clearinghouse for Federal Scientific and Technical Information, Springfield, Virginia 22151. 
Numerical Inversion of Finite Toeplitz Matrices and

Vector Toeplitz Matrices 
.
A numerical technique is described (1) that sim-
plifies nth-order Toeplitz matrix inversion; it 
increases the efficiencies of the numerical methods, 
involving Toeplitz matrices, by reducing the number 
of multiplications required by an i%-order Toeplitz 
matrix from N-cubed to N-squared multiplications. 
Many problems in mathematical physics, statistics, 
and algebra lead to the need for the inverse of finite 
Toeplitz or Hankel matrices. The problems involving 
convolutions, integral equations with difference 
kernels, and least-squares approximations by poly-
nomials are well known. Although there is abundant 
literature on the mathematical properties of Toeplitz 
matrices, references to the problem of numerical 
inversion seem to be few. The efficiencies of numerical 
methods involving . Toeplitz or Hankel matrices are 
often judged on the assumption that the inversion 
of a Toeplitz matrix of order-n requires roughly 
multiplications. 
The purpose of this report (1) is introduction of a 
new method by which the exact inversion can be 
accomplished simply by use of on the order of n2 
multiplications. Some efficient algorithms are given. 
Extension is made to vector Toeplitz matrices that 
occurred in the author's work. 
Vector Toeplitz matrices are defined as rectangular 
matrices whose elements are vectors and whose 
diagonals consist of like elements, except for the 
vector elements in the last row, which may be ob-
tained by omission of the last components of corre-
sponding full vector elements. 
Reference: 
1. E. H. Bareiss, ANL-7440 (Argonne National 
Laboratory, June 1968) 
Notes: 
1. This method may interest persons concerned with 
mathematical physics. 
2. Inquiries concerning this innovation may be di-
rected to:
	 - 
Office of Industrial Cooperation 
Argonne National Laboratory 
9700 South Cass Avenue 
Argonne, Illinois 60439 
Reference: B69-10415 
Source: E. H. Bareiss

Applied Mathematics Division
(ARG-10445) 
Patent status: 
Inquiries concerning rights for commercial use of 
this innovation may be made to: 
Mr. George H. Lee, Chief 
Chicago Patent Group 
U. S. Atomic Energy Commission 
Chicago Operations Office 
9800 South Cass Avenue 
Argonne, Illinois 60439
Category 02 
This document was prepared under the sponsorship of the Atomic	 liability resulting from the use of the information Contained in this 
Energy Commission and/or the National Aeronautics and Space document, or warrants that the use of any information, apparatus, 
Administration. Neither the United States Government nor any per-
	 method, or process disclosed in this document may not infringe pri-
son acting on behalf of the United States Government assumes any vately owned rights.
https://ntrs.nasa.gov/search.jsp?R=19690000408 2020-03-16T18:16:16+00:00Z


Numerical inversion of finite Toeplitz matrices and vector Toeplitz matrices

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server

Crossref

UNT (University of North Texas) Digital Library