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Polynomial interpolation and Gaussian quadrature for matrix valued functions

Walter Van Assche, Ann Sinap

TL;DR

This work extends Gaussian quadrature from scalars to matrix-valued functions by developing a rigorous framework of matrix polynomials, orthogonality, and interpolation. The core idea is to exploit zeros and rootvectors of orthonormal matrix polynomials to build quadrature formulas with maximal degree of precision $2n-1$, accompanied by explicit quadrature coefficients constructed from Jordan pairs and reproducing kernels. The approach yields both general interpolation theory for matrix polynomials and a practical high-precision quadrature rule for matrix integrals, with proven convergence for continuous matrix-valued integrands. This provides a principled method for accurate evaluation of matrix-valued integrals in applications requiring noncommutative or multi-dimensional data handling.

Abstract

The techniques for polynomial interpolation and Gaussian quadrature are generalized to matrix-valued functions. It is shown how the zeros and rootvectors of matrix orthonormal polynomials can be used to get a quadrature formula with the highest degree of precision.

Polynomial interpolation and Gaussian quadrature for matrix valued functions

TL;DR

This work extends Gaussian quadrature from scalars to matrix-valued functions by developing a rigorous framework of matrix polynomials, orthogonality, and interpolation. The core idea is to exploit zeros and rootvectors of orthonormal matrix polynomials to build quadrature formulas with maximal degree of precision , accompanied by explicit quadrature coefficients constructed from Jordan pairs and reproducing kernels. The approach yields both general interpolation theory for matrix polynomials and a practical high-precision quadrature rule for matrix integrals, with proven convergence for continuous matrix-valued integrands. This provides a principled method for accurate evaluation of matrix-valued integrals in applications requiring noncommutative or multi-dimensional data handling.

Abstract

The techniques for polynomial interpolation and Gaussian quadrature are generalized to matrix-valued functions. It is shown how the zeros and rootvectors of matrix orthonormal polynomials can be used to get a quadrature formula with the highest degree of precision.

Paper Structure

This paper contains 9 sections, 15 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Matrix polynomials
  3. Orthogonal matrix polynomials on the real line
  4. Polynomial interpolation
  5. Polynomial interpolation in general
  6. The interpolation problem of Lagrange
  7. Gaussian quadrature
  8. Quadrature formulas
  9. Convergence

Key Result

Proposition 2.1

The vectors $v_{0},v_{1},\ldots,v_{k}$ form a right Jordan chain of the monic matrix polynomial $\hat{P}(x) = Ix^{n} + A_{n-1}x^{n-1} + \ldots + A_{1}x + A_{0}$ corresponding to $x_{0}$ if and only if $v_{0} \not = 0$ and where $X_{0} = \left( \right)$ is a $p \times (k+1)$ matrix and $J_{0}$ is a Jordan block of size $(k+1) \times (k+1)$ with $x_{0}$ on the main diagonal.

Theorems & Definitions (15)

  • Proposition 2.1: go:1982, p. 27
  • Proposition 2.2: go:1982, p. 45
  • Theorem 2.3: go:1982, p. 58
  • Proposition 2.4
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 4.3
  • ...and 5 more