Structure preserving piecewise polynomial interpolation for definite matrices
type: Item
instance of: scholarly article
title: Structure preserving piecewise polynomial interpolation for definite matrices
zbMATH Open document ID: 0807.15019
DOI: 10.1016/0024-3795(94)90182-1
author: Luca Dieci
published in: Linear Algebra and its Applications
publication date: 1994-08-15T00:00:00Z
review text: The problem considered is as follows. We know a smooth Hermitian positive (semi)definite matrix function \(A(t)\) \((t \in \mathbb{R}\), \(A \in \mathbb{C}^{n \times n})\) only at a discrete set of points: \(\{A_ i\}^{N + 1}_{i = 1}\), \(A_ i = A(t_ i)\) where \(t_ i < t_{i + 1}\). We are interested in finding an interpolatory polynomial \(P(t)\) for this sequence (i.e. \(P(t_ i) = A_ i)\) such that: (i) \(P(t)\) is a globally smooth \((C^ k\) with \(k \geq 1)\) function for \(t \in I = [t_ 1, t_{N + 1}]\), (ii) \(P(t)\) is positive (semi)definite for all \(t \in I\), and (iii) if the matrices \(A_ i\) are locally ordered with respect to the natural ordering of (semi)definite matrices, then the interpolant must preserve such an ordering (more precisely, if, for example, \(A_ i \geq A_{i + 1}\) then also \(P(t') \geq P(t'')\) for all \(t'\) and \(t''\) such that \(t_ i \leq t' \leq t'' \leq t_{i + 1})\). The piecewise linear interpolant satisfies (ii) and (iii) but not (i). The author considers piecewise cubic (and higher degree) interpolants which satisfy (i)--(iii). Included in the discussion is the order of approximation of such interpolants as the step size of the subintervals goes to zero. Motivation comes from numerical integration of Hermitian Lyapunov and Riccati differential equations, where the exact solution is known to be positive (semi)definite.
reviewed by: Joseph A. Ball
Mathematics Subject Classification ID: 15A54
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 614108
zbMATH Keywords: polynomial interpolation
zbMATH Keywords: smooth Hermitian positive (semi)definite matrix function
zbMATH Keywords: Hermitian Lyapunov and Riccati differential equations
describes a project that uses: pchip
MaRDI profile type: Publication
full work available at URL: 0024-3795(94)90182-1
OpenAlex ID: W2023652622
cites work: Optimal state estimation in high noise
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Wikidata QID: Q126594617
description: scientific article; zbMATH DE number 614108