SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2529-2548, December 2024.
Abstract. We reformulate the Lanczos tau method for the discretization of time-delay systems in terms of a pencil of operators, allowing for new insights into this approach. As a first main result, we show that, for the choice of a shifted Legendre basis, this method is equivalent to Padé approximation in the frequency domain. We illustrate that Lanczos tau methods straightforwardly give rise to sparse, self-nesting discretizations. Equivalence is also demonstrated with pseudospectral collocation, where the nonzero collocation points are chosen as the zeros of orthogonal polynomials. The importance of such a choice manifests itself in the approximation of the [math]-norm, where, under mild conditions, supergeometric convergence is observed and, for a special case, superconvergence is proved, both of which are significantly faster than the algebraic convergence reported in previous work.

中文翻译：

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时滞系统的 Lanczos Tau 框架：重新审视 Padé 近似和搭配


SIAM 数值分析杂志，第 62 卷，第 6 期，第 2529-2548 页，2024 年 12 月。

抽象。我们重新制定了 Lanczos tau 方法，用于根据算子铅笔对时滞系统进行离散化，从而对这种方法有新的见解。作为第一个主要结果，我们表明，对于移位勒让德基的选择，该方法等效于频域中的帕德近似。我们说明了 Lanczos tau 方法直接导致了稀疏、自嵌套的离散化。伪谱搭配也证明了等效性，其中非零搭配点被选为正交多项式的零。这种选择的重要性体现在对 [数学] 范数的近似中，其中，在温和的条件下，观察到超几何收敛，并且在特殊情况下，证明了超收敛，这两者都比以前工作中报道的代数收敛要快得多。