Solving parametric eigenvalue problems using a Homotopy-Chebyshev method

Yunyun Wu and Yayun Li

Mathematics and Computers in Simulation (MATCOM), 2026, vol. 248, issue C, 260-269

Abstract: This paper presents a robust numerical method for solving parametric eigenvalue problems, where the system matrix depends smoothly on a parameter, and its eigenvalues and eigenvectors correspondingly vary with this parameter. While existing approaches based on Taylor or Chebyshev expansions provide accurate local approximations, they often struggle with large parameter variations, eigenvalue crossings, and near-degenerate cases. To address these limitations, we propose a homotopy-Chebyshev method that traces eigenpair solutions along a continuous parameter path, thereby enhancing robustness across wide parameter ranges. The method combines the global approximation capability of Chebyshev polynomials with the stability of homotopy continuation, overcoming limitations inherent in purely expansion-based approaches. Numerical experiments demonstrate that our homotopy approach outperforms pure expansion-based methods across extensive parameter intervals while remaining computationally tractable for moderately large systems. In practical applications such as structural dynamics, the method enables accurate prediction of vibration frequencies under varying material properties, significantly improving reliability over traditional perturbation techniques. The algorithm’s robustness makes it particularly suitable for applications in uncertainty quantification, structural dynamics, and other fields where parametric dependence plays a critical role.

Keywords: Parametric eigenvalue problems; Homotopy continuation; Chebyshev approximation; Bordered systems; Eigenvalue tracking; Path-following methods (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:248:y:2026:i:c:p:260-269

DOI: 10.1016/j.matcom.2026.04.019

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