An algebraic method to solve the minimal partial realization problem for scalar sequences
type: Item
instance of: scholarly article
title: An algebraic method to solve the minimal partial realization problem for scalar sequences
zbMATH Open document ID: 0651.15011
DOI: 10.1016/0024-3795(88)90312-6
published in: Linear Algebra and its Applications
publication date: 1988-01-01T00:00:00Z
review text: Let \(h_ 1,...,h_ N\) be a finite sequence of complex numbers. The minimal partial realization problem is to find the least n and complex numbers \(f_ 0,...,f_ n\) with \(f_ n=1\) such that \(f_ 0h_ r+f_ 1h_{r+1}+...+f_ nh_{n+r}=0\), for \(r=1,2,...,N-n\). This is related in an obvious way to finding rational approximations to \(h_ 1z^{- 1}+h_ 2z^{-2}+...+h_ Nz^{-N}\). After showing that an infinite Hankel matrix of rank n can be written as a product of a matrix with n columns and a matrix with n rows, the authors describe a straightforward method of solving the partial realization problem using linear algebra. Reviewer's remark: Although they mention Padé approximations in passing, the authors do not compare their method with other methods of solving the same problem. A well-known algorithm of E. R. Berlekamp and J. L. Massey uses an adaptation of the extended Euclidean algorithm for rational series to solve the partial realization problem; see, for example, Section 8.6 of \textit{R. Lidl} and \textit{H. Niederreiter} [Finite Fields (1984; Zbl 0554.12010)]. The reviewer has found it to be very effective over infinite as well as finite fields.
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 41A21
zbMATH DE Number: 4061415
zbMATH Keywords: minimal partial realization problem
zbMATH Keywords: rational approximations
zbMATH Keywords: infinite Hankel matrix
zbMATH Keywords: Padé approximations
author: Marc van Barel
author: Adhemar Bultheel
MaRDI profile type: Publication
full work available at URL: 0024-3795(88)90312-6
OpenAlex ID: W2115391265
cites work: Padé techniques for model reduction in linear system theory: a survey
cites work: Q3254327
cites work: On the partial realization problem
cites work: Q3859649
cites work: Hankel matrices and the infinite companion
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description: scientific article; zbMATH DE number 4061415