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An Arnoldi-based approach to polynomial and rational least squares problems

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

TL;DR

The paper tackles ill-conditioning in high-degree polynomial and rational least squares by recasting the problems in Krylov subspaces generated by Arnoldi recursions. It builds a unified framework that ties orthogonal polynomials (and Sobolev variants) and orthogonal rational functions to Krylov bases, including Jordan-like blocks for Sobolev cases, and derives corresponding Hessenberg pencils. The main contributions are the formalization of Krylov-induced orthogonal polynomials and Sobolev polynomials, the development of rational and Sobolev rational LS methods with explicit recurrences, and a thorough displacement-structure analysis that underpins efficient implementations. The numerical experiments demonstrate strong accuracy and stability across polynomial and rational LS problems, with re-orthogonalization improving precision, validating the approach for high-degree fitting and derivative-informed problems. An accompanying open-source code base enhances practical impact for researchers and engineers working on robust LS in scientific computing.

Abstract

In this research, we solve polynomial, Sobolev polynomial, rational, and Sobolev rational least squares problems. Although the increase in the approximation degree allows us to fit the data better in attacking least squares problems, the ill-conditioning of the coefficient matrix fuels the dramatic decrease in the accuracy of the approximation at higher degrees. To overcome this drawback, we first show that the column space of the coefficient matrix is equivalent to a Krylov subspace. Then the connection between orthogonal polynomials or rational functions and orthogonal bases for Krylov subspaces in order to exploit Krylov subspace methods like Arnoldi orthogonalization is established. Furthermore, some examples are provided to illustrate the theory and the performance of the proposed approach.

An Arnoldi-based approach to polynomial and rational least squares problems

TL;DR

The paper tackles ill-conditioning in high-degree polynomial and rational least squares by recasting the problems in Krylov subspaces generated by Arnoldi recursions. It builds a unified framework that ties orthogonal polynomials (and Sobolev variants) and orthogonal rational functions to Krylov bases, including Jordan-like blocks for Sobolev cases, and derives corresponding Hessenberg pencils. The main contributions are the formalization of Krylov-induced orthogonal polynomials and Sobolev polynomials, the development of rational and Sobolev rational LS methods with explicit recurrences, and a thorough displacement-structure analysis that underpins efficient implementations. The numerical experiments demonstrate strong accuracy and stability across polynomial and rational LS problems, with re-orthogonalization improving precision, validating the approach for high-degree fitting and derivative-informed problems. An accompanying open-source code base enhances practical impact for researchers and engineers working on robust LS in scientific computing.

Abstract

In this research, we solve polynomial, Sobolev polynomial, rational, and Sobolev rational least squares problems. Although the increase in the approximation degree allows us to fit the data better in attacking least squares problems, the ill-conditioning of the coefficient matrix fuels the dramatic decrease in the accuracy of the approximation at higher degrees. To overcome this drawback, we first show that the column space of the coefficient matrix is equivalent to a Krylov subspace. Then the connection between orthogonal polynomials or rational functions and orthogonal bases for Krylov subspaces in order to exploit Krylov subspace methods like Arnoldi orthogonalization is established. Furthermore, some examples are provided to illustrate the theory and the performance of the proposed approach.
Paper Structure (13 sections, 5 theorems, 67 equations, 4 figures, 5 tables, 8 algorithms)

This paper contains 13 sections, 5 theorems, 67 equations, 4 figures, 5 tables, 8 algorithms.

Table of Contents

  1. Introduction
  2. Weighted polynomial least squares problem
  3. Arnoldi and orthogonal polynomials
  4. Displacement rank and polynomial Krylov subspaces
  5. Weighted Sobolev polynomial least squares problem
  6. Arnoldi and Sobolev orthogonal polynomials
  7. Weighted rational least squares problem
  8. Rational Arnoldi and orthogonal rational functions
  9. Displacement rank and rational Krylov subspaces
  10. Weighted Sobolev rational least squares problem
  11. Rational Arnoldi and Sobolev orthogonal rational functions
  12. Numerical examples
  13. Conclusion

Key Result

Theorem 2.1

Consider the diagonal matrix $Z = diag(\{z_{j}\}_{j=1}^{m})\in \mathbb{C}^{m\times m}$ and the vector $\bold{v}=^\top\in\mathbb{C}^m$. Let $ = Q_{n+1} \in\mathbb{C}^{m\times (n+1)}$ form a nested orthonormal basis for $\mathcal{K}_{n+1}(Z,\bold{v})$. Suppose that polynomials $p_k\in\mathcal{P}_k$ sa

Figures (4)

  • Figure 1: Semi-log representation of the errors for Example \ref{['ex2']} using Arnoldi with re-orthogonalization on first kind Chebyshev nodes.
  • Figure 2: Semi-log representation of the errors for Example \ref{['ex3']}.
  • Figure 3: Semi-log representation of the errors for Example \ref{['ex1']}.
  • Figure 4: Semi-log representation of the errors for Example \ref{['ex5']}.

Theorems & Definitions (14)

  • Theorem 2.1: Krylov induced orthogonal polynomials
  • proof
  • Theorem 3.1: Krylov induced Sobolev orthogonal polynomials
  • proof
  • Definition 4.1: Rational Krylov subspace 2716
  • Theorem 4.1: Rational Arnoldi iteration 27
  • Theorem 4.2: Krylov induced orthogonal rational functions
  • proof
  • Theorem 5.1: Krylov induced Sobolev orthogonal rational functions
  • proof
  • ...and 4 more