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Gaussian quadrature rules for composite highly oscillatory integrals

Menghan Wu, Haiyong Wang

TL;DR

The paper tackles the computation of composite highly oscillatory integrals of the form $I[f]=\int_a^b f(x)(g\circ\phi_\omega)(x)\,dx$ with large $\omega$. It develops two Gaussian quadrature rules: the first uses orthogonal polynomials with respect to the weight $(g\circ\phi_\omega)$, yielding spectral or algebraic convergence depending on the smoothness of $f$; the second targets $\omega$-dependent accuracy by subtracting the leading asymptotic term and using polynomials orthogonal to $(g\circ\phi_\omega)(x)-\rho_0/2$, with an error of $O(\omega^{-n-1})$ under suitable hypotheses. The work also analyzes node trajectories, endpoint convergence, and provides asymptotic error estimates validated by numerical experiments, offering a robust framework for efficiently evaluating composite oscillatory integrals. These methods extend the Gaussian quadrature toolkit beyond standard Fourier-type oscillations and have potential impact for simulations in electronics and wave propagation where composite oscillations arise.

Abstract

Highly oscillatory integrals of composite type arise in electronic engineering and their calculations is a challenging problem. In this paper, we propose two Gaussian quadrature rules for computing such integrals. The first one is constructed based on the classical theory of orthogonal polynomials and its nodes and weights can be computed efficiently by using tools of numerical linear algebra. We show that the rate of convergence of this rule depends solely on the regularity of the non-oscillatory part of the integrand. The second one is constructed with respect to a sign-changing function and the classical theory of Gaussian quadrature can not be used anymore. We explore theoretical properties of this Gaussian quadrature, including the trajectories of the quadrature nodes and the convergence rate of these nodes to the endpoints of the integration interval, and prove its asymptotic error estimate under suitable hypotheses. Numerical experiments are presented to demonstrate the performance of the proposed methods.

Gaussian quadrature rules for composite highly oscillatory integrals

TL;DR

The paper tackles the computation of composite highly oscillatory integrals of the form with large . It develops two Gaussian quadrature rules: the first uses orthogonal polynomials with respect to the weight , yielding spectral or algebraic convergence depending on the smoothness of ; the second targets -dependent accuracy by subtracting the leading asymptotic term and using polynomials orthogonal to , with an error of under suitable hypotheses. The work also analyzes node trajectories, endpoint convergence, and provides asymptotic error estimates validated by numerical experiments, offering a robust framework for efficiently evaluating composite oscillatory integrals. These methods extend the Gaussian quadrature toolkit beyond standard Fourier-type oscillations and have potential impact for simulations in electronics and wave propagation where composite oscillations arise.

Abstract

Highly oscillatory integrals of composite type arise in electronic engineering and their calculations is a challenging problem. In this paper, we propose two Gaussian quadrature rules for computing such integrals. The first one is constructed based on the classical theory of orthogonal polynomials and its nodes and weights can be computed efficiently by using tools of numerical linear algebra. We show that the rate of convergence of this rule depends solely on the regularity of the non-oscillatory part of the integrand. The second one is constructed with respect to a sign-changing function and the classical theory of Gaussian quadrature can not be used anymore. We explore theoretical properties of this Gaussian quadrature, including the trajectories of the quadrature nodes and the convergence rate of these nodes to the endpoints of the integration interval, and prove its asymptotic error estimate under suitable hypotheses. Numerical experiments are presented to demonstrate the performance of the proposed methods.
Paper Structure (6 sections, 3 theorems, 44 equations, 8 figures)

This paper contains 6 sections, 3 theorems, 44 equations, 8 figures.

Table of Contents

  1. Introduction
  2. The first Gaussian quadrature rule
  3. Polynomials orthogonal with respect to $(g\circ \phi_{\omega})(x)$
  4. Convergence analysis of Gaussian quadrature
  5. The second Gaussian quadrature rule
  6. Concluding remarks

Key Result

Theorem 2.2

Let $\{x_k\}_{k=1}^{n}$ and $\{x_k^{\mathrm{GL}}\}_{k=1}^{n}$ denote the zeros of $p_n^{\omega}(x)$ and $P_n(x)$, respectively, and assume that they are ordered according to the same rule. Then, for each $k\in\{1,\ldots,n\}$, we have

Figures (8)

  • Figure 1: The nodes $\{x_k\}_{k=1}^{n}$ as a function of $\omega$ for $n=5,10,20$.
  • Figure 2: The nodes $\{x_k\}_{k=1}^{n}$ as a function of $n$ for $\omega=50,100,500$.
  • Figure 3: The absolute errors $|x_k-x_k^{\mathrm{GL}}|$ with $k=1,\ldots,n$ scaled by $\omega$ for $n=5,10,20$.
  • Figure 4: Absolute errors of the Gaussian quadrature rule \ref{['eq:FirstGauss']} as a function of $n$ for $\omega=50,100,200,500,1000$. The first and second rows are plotted in a log scale and the last row is plotted in a log-log scale.
  • Figure 5: The trajectories of the zeros of $q_4^{\omega}(x)$ and $q_8^{\omega}(x)$ as a function of $\omega$ (top) and the absolute errors between the zeros and the corresponding endpoint scaled by $\omega$ (bottom). Here $g(x)=\ln(x+4)$, $\phi_{\omega}(x)=\cos(\omega x)$, $[a,b]=[-1,1]$ and $\omega$ ranges from $20$ to $100$.
  • ...and 3 more figures

Theorems & Definitions (9)

  • Remark 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • Theorem 3.1
  • proof