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Constructing Sobolev orthonormal rational functions via an updating procedure

Amin Faghih, Marc Van Barel, Niel Van Buggenhout, Raf Vandebril

TL;DR

The paper reframes the construction of Sobolev orthonormal rational functions with prescribed poles as a Hessenberg pencil inverse eigenvalue problem derived from a discretized Sobolev inner product via rational Gauss quadrature. It introduces an updating procedure that sequentially incorporates nodes, using unitary plane rotations to maintain the Hessenberg structure and pole placement, and contrasts it with an alternative approach that starts from a discretized full Hessenberg problem. Numerical experiments show the updating method achieves high recurrence and orthogonality accuracy, closely matching Krylov-based methods while enabling memory-efficient updates and flexible pole management. This work provides a rigorous matrix-analytic pathway for generating SORFs with controlled poles, with implications for Sobolev-aware approximations and rational Krylov subspace methods.

Abstract

In this paper, we generate the recursion coefficients for rational functions with prescribed poles that are orthonormal with respect to a continuous Sobolev inner product. Using a rational Gauss quadrature rule, the inner product can be discretized, thus allowing a linear algebraic approach. The presented approach involves reformulating the problem as an inverse eigenvalue problem involving a Hessenberg pencil, where the pencil will contain the recursion coefficients that generate the sequence of Sobolev orthogonal rational functions. This reformulation is based on the connection between Sobolev orthonormal rational functions and the orthonormal bases for rational Krylov subspaces generated by a Jordan-like matrix. An updating procedure, introducing the nodes of the inner product one after the other, is proposed and the performance is examined through some numerical examples.

Constructing Sobolev orthonormal rational functions via an updating procedure

TL;DR

The paper reframes the construction of Sobolev orthonormal rational functions with prescribed poles as a Hessenberg pencil inverse eigenvalue problem derived from a discretized Sobolev inner product via rational Gauss quadrature. It introduces an updating procedure that sequentially incorporates nodes, using unitary plane rotations to maintain the Hessenberg structure and pole placement, and contrasts it with an alternative approach that starts from a discretized full Hessenberg problem. Numerical experiments show the updating method achieves high recurrence and orthogonality accuracy, closely matching Krylov-based methods while enabling memory-efficient updates and flexible pole management. This work provides a rigorous matrix-analytic pathway for generating SORFs with controlled poles, with implications for Sobolev-aware approximations and rational Krylov subspace methods.

Abstract

In this paper, we generate the recursion coefficients for rational functions with prescribed poles that are orthonormal with respect to a continuous Sobolev inner product. Using a rational Gauss quadrature rule, the inner product can be discretized, thus allowing a linear algebraic approach. The presented approach involves reformulating the problem as an inverse eigenvalue problem involving a Hessenberg pencil, where the pencil will contain the recursion coefficients that generate the sequence of Sobolev orthogonal rational functions. This reformulation is based on the connection between Sobolev orthonormal rational functions and the orthonormal bases for rational Krylov subspaces generated by a Jordan-like matrix. An updating procedure, introducing the nodes of the inner product one after the other, is proposed and the performance is examined through some numerical examples.

Paper Structure

This paper contains 12 sections, 2 theorems, 47 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Inverse eigenvalue problem formulation
  3. Updating procedure
  4. Embedding of partial solutions
  5. Adjusting the basis matrix
  6. Restore Hessenberg structure
  7. Adding and swapping poles
  8. Alternative strategy
  9. Numerical experiments
  10. Degree of exactness
  11. Comparison of three algorithms
  12. Conclusion

Key Result

Theorem 2.1

FaVBVBVa24 Consider the Jordan-like matrix and the vector Let $Q_{n}= \in\mathbb{C}^{m\times n}$ form a nested orthonormal basis for $\mathcal{K}_{n}(J,w;\Xi)$, $n<m$. Assume that the rational functions $r_k\in\mathcal{R}^\Xi_k$ are defined on the spectrum of $J$ satisfying $q_k = r_{k-1}(J)w$. Then $\{r_k\}_{k=0}^{n-1}$ is the set of Sobolev orthonormal

Figures (2)

  • Figure 1: Orthogonality error for each entry of the moment matrix (minus the identity matrix of the same size) generated by the continuous (left) and discretized (right) Sobolev-Gegenbauer inner product. The first 3 SORFs with respect to the continuous inner product are computed, which require the computation of a $10\times 10$ Hessenberg pencil. As predicted by the degree of exactness of the discretization, the orthogonality error for the continuous inner product increases rapidly beyond the $3\times 3$ leading principal subpencil, whereas the orthogonality error for the discretized inner product is small everywhere.
  • Figure 2: Error metrics for increasing the size of the sequence of SORFs requested. SORFs for the Gegenbauer-Sobolev inner product with $\lambda=1$ and $\mu=2$. The poles are $\{-\omega,\omega,-2\omega,2\omega,\dots\}$ with $\omega = 1.1$. The error metrics for the recurrence relation $E_m(r)$ and for the poles $E_m(p)$ indicate high accuracy. Krylov subspace method ($\circ$), Updating procedure ($\ast$), and the procedure in Section \ref{['sec:SORFviaSOP']} ( +).

Theorems & Definitions (9)

  • Example 2.1
  • Example 2.2
  • Definition 2.1: Rational Krylov subspace Ru84
  • Theorem 2.1
  • Example 2.3
  • Proposition 2.1
  • Example 3.1
  • Example 3.2
  • Example 3.3