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<p>One common way to solve the polynomial eigenvalue problem (PEP) is to recast it by linearization, as a generalized eigenvalue problem (GEP) which can be solved by a backward stable algorithm such as QZ algorithm. QZ algorithm is backward stable for GEP, but may be backward unstable for PEP when the norms of the coefficient matrices for PEP vary widely. To improve the backward stability of PEP solved by a companion linearization, we combine the linearization with tropical scaling, and we also investigate the backward error of approximate eigenvalues of PEP via the tropically scaled linearization. Furthermore, we derive global upper bounds for the backward errors of approximate eigenvalues of PEP via the companion linearization without and with tropical scaling. In numerical experiments, we illustrate that the backward errors of the computed eigenvalues can be successfully improved by tropical scaling and well predicted by the global upper bounds.</p>

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Global bounds of backward errors of polynomial eigenvalue problem solved by a companion linearization

  • 期刊论文

  • 发表日期：

    2025年3月22日

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  • Ziyin Yang,
  • Zekun Wang,
  • Zongqi Cao,
  • Xiang Wang

摘要

One common way to solve the polynomial eigenvalue problem (PEP) is to recast it by linearization, as a generalized eigenvalue problem (GEP) which can be solved by a backward stable algorithm such as QZ algorithm. QZ algorithm is backward stable for GEP, but may be backward unstable for PEP when the norms of the coefficient matrices for PEP vary widely. To improve the backward stability of PEP solved by a companion linearization, we combine the linearization with tropical scaling, and we also investigate the backward error of approximate eigenvalues of PEP via the tropically scaled linearization. Furthermore, we derive global upper bounds for the backward errors of approximate eigenvalues of PEP via the companion linearization without and with tropical scaling. In numerical experiments, we illustrate that the backward errors of the computed eigenvalues can be successfully improved by tropical scaling and well predicted by the global upper bounds.