Stability and McMillan degree for rational matrix interpolants
type: Item
instance of: scholarly article
title: Stability and McMillan degree for rational matrix interpolants
zbMATH Open document ID: 0791.30035
DOI: 10.1016/0024-3795(94)90325-5
author: Joseph A. Ball
author: Jeongook Kim
published in: Linear Algebra and its Applications
publication date: 1994-07-07T00:00:00Z
review text: Let \(\mathbb{C}_ +\) denote the right hand plane. Fix \(n\) points \(z_ 1,\dots,z_ n\) in \(\mathbb{C}_ +\) and a positive constant \(c\). Let \(w_ 1,\dots,w_ n\) be an \(n\)-tuple of complex numbers. The classical Nevanlinna-Pick interpolation problem asks to find conditions under which there is an analytic function \(f:\mathbb{C}_ +\to D(c)\), satisfying \(f(z_ j)=w_ j\), \(1 \leq j \leq n\), where \(D(c)\) is the disk centered at zero, of radius \(c\). The necessary and sufficient condition, which goes back to the original papers of Pick and respectively Nevanlinna is that the matrix \[ \left( {c^ 2-w_ iw_ j \over z_ i+\overline z_ j} \right)_{1 \leq i, j \leq n} \] to be positive semidefinite. In that case Nevanlinna has constructed in an algorithmic way a canonical rational solution to the problem. Motivated by linear systems' theory, the authors study the above problem with symmetric data (that is \(n\) is odd and \((n-1)/2\) pairs of the interpolation data satisfy \(z_ i=-\overline z_ j\), \(c^ 2=w_ i \overline w_ j)\) and with minimial McMillan degree (maximum polynomial degree of the numerator and denumerator when the solution \(f\) is rational and expressed as an irreducible quotient of polynomials). The main result asserts that, if a solution of this problem exists and the minimal degree solution is unique, then it is stable and it coincides with the constructive Nevanlinna solution. In fact the matrix valued case (when \(w_ i\) are \(m \times n\) matrices) is analyzed and settled by well known methods at this moment. See for references the monograph: \textit{J. A. Ball}, \textit{I. Gohberg} and \textit{L. Rodman}, Interpolation of rational matrix functions (1990; Zbl 0708.15011).
reviewed by: Mihai Putinar
Mathematics Subject Classification ID: 30E05
Mathematics Subject Classification ID: 93B20
zbMATH DE Number: 519711
MaRDI profile type: Publication
full work available at URL: 0024-3795(94)90325-5
OpenAlex ID: W2061320774
cites work: Rational interpolation and the Euclidean algorithm
cites work: On the Scalar Rational Interpolation Problem
cites work: On the problem of stable rational interpolation
cites work: Interpolation problems, extensions of symmetric operators and reproducing kernel spaces. II
cites work: On the solution of the minimal rational interpolation problem
cites work: Q5666975
cites work: Errata to: Two-sided tangential interpolation of real rational matrix functions
cites work: Column reduced rational matrix functions with given null-pole data in the complex plane
cites work: On Majorization, Factorization, and Range Inclusion of Operators on Hilbert Space
cites work: Q3032797
cites work: A Topological Approach to Nevanlinna–Pick Interpolation
cites work: Directional interpolation approach to H&lt;sup&gt;∞&lt;/sup&gt;-Optimization and robust stabilization
cites work: Conjugation, interpolation and model-matching in H∞
cites work: An indefinite metric in the Nevanlinna-Pick problem
cites work: An interpolation theory approach to \(H^{\infty}\) controller degree bounds
cites work: Fast projection methods for minimal design problems in linear system theory
description: scientific article; zbMATH DE number 519711