Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Best polynomial approximation for non-autonomous linear ODEs in the $\star$-product framework

Stefano Pozza

TL;DR

This work formulates and analyzes the problem of obtaining the best $n$-degree $\star$-polynomial approximation to the solution of linear non-autonomous ODEs within the $\star$-product framework. By leveraging a matrix spectral decomposition in the $\star$-setting and introducing a $\star$-inner product and norm, the authors reduce the approximation to scalar exponential problems and derive a Bernstein-type error bound: the $L_2$-error decays at least as fast as the best uniform approximation of the exponential on a suitably defined interval $\mathcal{J}$, with explicit constants depending on an ellipse parameter $\chi$. The main result shows exponential (and even super-exponential) convergence of the $\star$-polynomial approximants under mild analyticity assumptions, linking the performance to well-understood polynomial approximation theory. These findings illuminate the numerical behavior of polynomial-based methods in the $\star$-framework and pave the way for rigorous Krylov-subspace analyses of time-ordered exponential problems in non-autonomous settings.

Abstract

We present the first formulation of the optimal polynomial approximation of the solution of linear non-autonomous systems of ODEs in the framework of the so-called $\star$-product. This product is the basis of new approaches for the solution of such ODEs, both in the analytical and the numerical sense. The paper shows how to formally state the problem and derives upper bounds for its error.

Best polynomial approximation for non-autonomous linear ODEs in the $\star$-product framework

TL;DR

This work formulates and analyzes the problem of obtaining the best -degree -polynomial approximation to the solution of linear non-autonomous ODEs within the -product framework. By leveraging a matrix spectral decomposition in the -setting and introducing a -inner product and norm, the authors reduce the approximation to scalar exponential problems and derive a Bernstein-type error bound: the -error decays at least as fast as the best uniform approximation of the exponential on a suitably defined interval , with explicit constants depending on an ellipse parameter . The main result shows exponential (and even super-exponential) convergence of the -polynomial approximants under mild analyticity assumptions, linking the performance to well-understood polynomial approximation theory. These findings illuminate the numerical behavior of polynomial-based methods in the -framework and pave the way for rigorous Krylov-subspace analyses of time-ordered exponential problems in non-autonomous settings.

Abstract

We present the first formulation of the optimal polynomial approximation of the solution of linear non-autonomous systems of ODEs in the framework of the so-called -product. This product is the basis of new approaches for the solution of such ODEs, both in the analytical and the numerical sense. The paper shows how to formally state the problem and derives upper bounds for its error.
Paper Structure (6 sections, 8 theorems, 53 equations, 2 tables)

This paper contains 6 sections, 8 theorems, 53 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Basics and main result
  3. Matrix spectral decomposition and the $\star$-product
  4. Approximation of the $\star$-resolvent
  5. Proof of Theorem \ref{['thm:main']}
  6. Conclusion

Key Result

Theorem 2.1

\newlabelthm:main Consider the initial value problem eq:ode, with $\| \tilde{v} \|_2 = 1$. Let $\tilde{\lambda}_1(t), \dots, \tilde{\lambda}_N(t)$ be the eigenvalues and $\tilde{Q}(t) = [\tilde{q}_1(t), \dots, \tilde{q}_N(t)]$ the related eigenvectors of $\tilde{A}(t) \in \mathbb{C}^{N \times N}$. and denote with $E_n(\mathcal{J})$ the minimal uniform error of the polynomial approximation of the

Theorems & Definitions (16)

  • Theorem 2.1: Main result
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof
  • Theorem 3.3: Properties of the $\star$-norm
  • proof
  • ...and 6 more