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<p>In this paper we give an entirely matrix based analysis of the mathematical properties of the non-Hermitian Lanczos algorithm. Based on the spectral properties of the given matrix and its relation to the initial vectors, we transparently explain the “mechanics” of the algorithm, the causes of all possible types of breakdowns, and the techniques behind the look-ahead process. Our approach yields new proofs of several well-known results, for example of the original version of Taylor’s Mismatch Theorem [<CitationRef CitationID="CR39">39</CitationRef>], as well as some new results, for example about the attainable “look-ahead patterns”. We also analyze properties and possible simplifications of the algorithm for several special matrix classes, in particular <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(H\)</EquationSource> </InlineEquation>-selfadjoint matrices. Throughout the paper, we illustrate our findings with explicitly computed examples.</p>

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A matrix based analysis of the non-Hermitian Lanczos algorithm with applications to special matrix classes

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年4月14日

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  • Jörg Liesen,
  • Justus Ramme

摘要

In this paper we give an entirely matrix based analysis of the mathematical properties of the non-Hermitian Lanczos algorithm. Based on the spectral properties of the given matrix and its relation to the initial vectors, we transparently explain the “mechanics” of the algorithm, the causes of all possible types of breakdowns, and the techniques behind the look-ahead process. Our approach yields new proofs of several well-known results, for example of the original version of Taylor’s Mismatch Theorem [39], as well as some new results, for example about the attainable “look-ahead patterns”. We also analyze properties and possible simplifications of the algorithm for several special matrix classes, in particular \(H\) -selfadjoint matrices. Throughout the paper, we illustrate our findings with explicitly computed examples.