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Rational approximations to fractional powers of self-adjoint positive operators

Lidia Aceto, Paolo Novati

TL;DR

This work provides accurate error bounds by exploiting classical results in approximation theory involving Padé approximants and improves some existing results and the numerical experiments proves its accuracy.

Abstract

We investigate the rational approximation of fractional powers of unbounded positive operators attainable with a specific integral representation of the operator function. We provide accurate error bounds by exploiting classical results in approximation theory involving Padé approximants. The analysis improves some existing results and the numerical experiments proves its accuracy.

Rational approximations to fractional powers of self-adjoint positive operators

TL;DR

This work provides accurate error bounds by exploiting classical results in approximation theory involving Padé approximants and improves some existing results and the numerical experiments proves its accuracy.

Abstract

We investigate the rational approximation of fractional powers of unbounded positive operators attainable with a specific integral representation of the operator function. We provide accurate error bounds by exploiting classical results in approximation theory involving Padé approximants. The analysis improves some existing results and the numerical experiments proves its accuracy.

Paper Structure

This paper contains 6 sections, 7 theorems, 62 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background on the Gauss-Jacobi approach
  3. Error analysis
  4. The case of bounded operators
  5. Numerical experiments
  6. Conclusions

Key Result

Theorem 3.1

For each integer $k\geq 1,$ the exact representation of the truncation error defined in (trerr) is given by in which $\Gamma$ denotes the gamma function and $_{2}F_{1}$ the hypergeometric function.

Figures (5)

  • Figure 1: Error and error bound (\ref{['th1']}) with respect to $k$ for Example 1 with $N=100$, $p=4.$
  • Figure 2: Error comparison for Example \ref{['e1']} using $\tilde{\tau}$ as in (\ref{['tauopt']}) and $\tau _{k}$ as in (\ref{['tauk']}), $p=2,3,4$ (lowest to highest curve), $N=100$ and $\alpha =0.5.$
  • Figure 3: Error comparison for Example \ref{['e2']} using $\tilde{\tau}$ as in (\ref{['tauopt']}) and $\tau _{k}$ as in (\ref{['tk2']}), $N=500$ and $\alpha =0.5.$
  • Figure 4: Selected values for $\tau _{k}$ defined by (\ref{['tk2']}) for Example \ref{['e2']} with $N=500$ and $\alpha =0.5$.
  • Figure 5: Error and error estimate (\ref{['th2']}) with respect to $k$ for Example 2 with $N=200.$

Theorems & Definitions (16)

  • Theorem 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Theorem 3.5
  • proof
  • ...and 6 more