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A Golub-Welsch version for simultaneous Gaussian quadrature

Walter Van Assche

TL;DR

This paper extends the Golub–Welsch framework to simultaneous Gaussian quadrature for two measures by using a banded Hessenberg matrix whose eigenvalues give the quadrature nodes. The quadrature weights are recovered from left and right eigenvectors via a linking constant matrix, leveraging the Christoffel–Darboux kernel for multiple orthogonal polynomials that ties type II and type I polynomials together. A self-contained proof is provided for the case of two measures (an Angelesco-type setting), and numerical illustrations are given for weights related to modified Bessel functions. The approach enables stable, efficient computation of nodes and weights for simultaneous quadrature and points to natural extensions to more than two measures.

Abstract

The zeros of type II multiple orthogonal polynomials can be used for quadrature formulas that approximate $r$ integrals of the same function $f$ with respect to $r$ measures $μ_1,\ldots,μ_r$ in the spirit of Gaussian quadrature. This was first suggested by Borges in 1994, even though he does not mention multiple orthogonality. We give a method to compute the quadrature nodes and the quadrature weights which extends the Golub-Welsch approach using the eigenvalues and left and right eigenvectors of a banded Hessenberg matrix. This method was already described by Coussement and Van Assche in 2005 but it seems to have gone unnoticed. We describe the result in detail for $r=2$ and give some examples.

A Golub-Welsch version for simultaneous Gaussian quadrature

TL;DR

This paper extends the Golub–Welsch framework to simultaneous Gaussian quadrature for two measures by using a banded Hessenberg matrix whose eigenvalues give the quadrature nodes. The quadrature weights are recovered from left and right eigenvectors via a linking constant matrix, leveraging the Christoffel–Darboux kernel for multiple orthogonal polynomials that ties type II and type I polynomials together. A self-contained proof is provided for the case of two measures (an Angelesco-type setting), and numerical illustrations are given for weights related to modified Bessel functions. The approach enables stable, efficient computation of nodes and weights for simultaneous quadrature and points to natural extensions to more than two measures.

Abstract

The zeros of type II multiple orthogonal polynomials can be used for quadrature formulas that approximate integrals of the same function with respect to measures in the spirit of Gaussian quadrature. This was first suggested by Borges in 1994, even though he does not mention multiple orthogonality. We give a method to compute the quadrature nodes and the quadrature weights which extends the Golub-Welsch approach using the eigenvalues and left and right eigenvectors of a banded Hessenberg matrix. This method was already described by Coussement and Van Assche in 2005 but it seems to have gone unnoticed. We describe the result in detail for and give some examples.
Paper Structure (19 sections, 2 theorems, 98 equations, 4 figures)

This paper contains 19 sections, 2 theorems, 98 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Multiple orthogonal polynomials
  3. Examples extending classical orthogonal polynomials
  4. Modified Bessel functions $(K_\nu,K_{\nu+1})$
  5. Modified Bessel functions $(I_{\nu},I_{\nu+1})$
  6. Simultaneous Gaussian quadrature
  7. Recurrence relations
  8. Golub-Welsch type result
  9. Examples
  10. Bessel functions $K_\nu$
  11. Bessel functions $I_\nu$
  12. Concluding remarks
  13. Declarations
  14. Ethical approval
  15. Availability of supporting data
  16. ...and 4 more sections

Key Result

Theorem 1

If we take for $\{x_{k,2n}, 1 \leq k \leq 2n \}$ the zeros of the type II multiple orthogonal $P_{n,n}$ for the two measures $(\mu_1,\mu_2)$ and use interpolatory quadrature, then the quadrature is exact for polynomials $f$ of degree $\leq 3n-1$.

Figures (4)

  • Figure 6.1: Quadrature nodes and quadrature weights for $(w_1,w_2)=x(\rho_0,\rho_1)$.
  • Figure 6.2: Simultaneous quadrature to $I_1$ and $I_2$
  • Figure 6.3: Quadrature nodes and quadrature weights for $(w_1,w_2)=(\omega_{0,1},\omega_{1,1})$.
  • Figure 6.4: Simultaneous quadrature to $J_1$ and $J_2$

Theorems & Definitions (6)

  • Definition 1: type II MOP
  • Definition 2: type I MOP
  • Theorem 1: Borges Borges
  • proof
  • Theorem 2
  • proof