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Fast and accurate approximations to fractional powers of operators

Lidia Aceto, Paolo Novati

TL;DR

Some rational approximations to the fractional powers of self-adjoint positive operators, arising from the Gauss-Laguerre rules are considered.

Abstract

In this paper we consider some rational approximations to the fractional powers of self-adjoint positive operators, arising from the Gauss-Laguerre rules. We derive practical error estimates that can be used to select a priori the number of Laguerre points necessary to achieve a given accuracy. We also present some numerical experiments to show the effectiveness of our approaches and the reliability of the estimates.

Fast and accurate approximations to fractional powers of operators

TL;DR

Some rational approximations to the fractional powers of self-adjoint positive operators, arising from the Gauss-Laguerre rules are considered.

Abstract

In this paper we consider some rational approximations to the fractional powers of self-adjoint positive operators, arising from the Gauss-Laguerre rules. We derive practical error estimates that can be used to select a priori the number of Laguerre points necessary to achieve a given accuracy. We also present some numerical experiments to show the effectiveness of our approaches and the reliability of the estimates.

Paper Structure

This paper contains 11 sections, 5 theorems, 124 equations, 8 figures.

Table of Contents

  1. Introduction
  2. The Gauss-Laguerre approach
  3. Error analysis for a general function
  4. Error analysis for $\lambda ^{-\alpha }$
  5. First integral $I^{(1)}(\lambda )$
  6. Second integral $I^{(2)}(\lambda )$
  7. Error analysis for $\mathcal{L}^{-\alpha }$
  8. Truncated approaches
  9. A balanced approach
  10. An equalized approach
  11. Conclusions

Key Result

Proposition 1

Let $\gamma ^{\pm }\left( \lambda \right)$ be defined in (gamma) and $\bar{n}=4n+2.$ Denoting by the $\lambda$-dependent factors of $\varepsilon _{n}^{(1)}\left( \lambda \right)$ and $\varepsilon _{n}^{(2)}\left( \lambda \right)$, respectively, then we have

Figures (8)

  • Figure 1: Contour chosen for a function $f$ analytic on or within the parabola $\Gamma _{R}$ with the exception of two simple and conjugated poles located inside $C_{1}$ and $C_{2},$ respectively.
  • Figure 2: Absolute error and its estimate given by (\ref{['esti']}) for $\lambda =10.$
  • Figure 3: Comparison between $\ln \lambda _{n}$ (solid lines) and $\ln {\tilde{\lambda}_n }$ (dahed lines) for $n=10,11, \dots,120$.
  • Figure 4: Error and its estimate given by (\ref{['e5']}) for the operator defined in (\ref{['optest']}).
  • Figure 5: Behavior of the functions $g_{n}^{(1)}(\lambda),g_{n}^{(2)}(\lambda),g_{n}^{(1)}(\lambda)+g_{n}^{(2)}(\lambda)$ for $n=30.$
  • ...and 3 more figures

Theorems & Definitions (10)

  • Proposition 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Remark 1
  • Proposition 5
  • proof