Scholarlink

<p>Gegenbauer–Sobolev polynomials are an important class of Sobolev orthogonal polynomials. We prove a novel recurrence relation for this sequence of polynomials that allows a more accurate construction of the sequence. This new recurrence relation allows us to decompose the recurrence matrix of the sequence as the product of three structured matrices. By computing the eigenvalues of the recurrence matrix, we obtain the zeros of the Gegenbauer–Sobolev polynomials. However, the eigenvalue problem formulated with this recurrence matrix is ill-conditioned and thus cannot lead to reliable approximations of the actual zeros. We deal with this ill-conditioning by reformulating the eigenvalue problem as a generalized eigenvalue problem, involving the three structured matrices that decompose the recurrence matrix, and applying a balancing technique to this generalized eigenvalue problem.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Recurrence relations and zeros of Gegenbauer–Sobolev orthogonal polynomials

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2026年8月3日

查看全文
  • Niel Van Buggenhout,
  • Teresa Laudadio,
  • Nicola Mastronardi,
  • Francisco Marcellán,
  • Paul Van Dooren

摘要

Gegenbauer–Sobolev polynomials are an important class of Sobolev orthogonal polynomials. We prove a novel recurrence relation for this sequence of polynomials that allows a more accurate construction of the sequence. This new recurrence relation allows us to decompose the recurrence matrix of the sequence as the product of three structured matrices. By computing the eigenvalues of the recurrence matrix, we obtain the zeros of the Gegenbauer–Sobolev polynomials. However, the eigenvalue problem formulated with this recurrence matrix is ill-conditioned and thus cannot lead to reliable approximations of the actual zeros. We deal with this ill-conditioning by reformulating the eigenvalue problem as a generalized eigenvalue problem, involving the three structured matrices that decompose the recurrence matrix, and applying a balancing technique to this generalized eigenvalue problem.