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Structured generalized eigenvalue condition numbers for parameterized quasiseparable matrices

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Abstract

In this paper, when A and B are {1;1}-quasiseparable matrices, we consider the structured generalized relative eigenvalue condition numbers of the pair \((A, \, B)\) with respect to relative perturbations of the parameters defining A and B in the quasiseparable and the Givens-vector representations of these matrices. A general expression is derived for the condition number of the generalized eigenvalue problems of the pair \((A,\, B)\), where A and B are any differentiable function of a vector of parameters with respect to perturbations of such parameters. Moreover, the explicit expressions of the corresponding structured condition numbers with respect to the quasiseparable and Givens-vector representation via tangents for \(\{1; 1\}\)-quasiseparable matrices are derived. Our proposed condition numbers can be computed efficiently by utilizing the recursive structure of quasiseparable matrices. We investigate relationships between various condition numbers of structured generalized eigenvalue problem when A and B are {1;1}-quasiseparable matrices. Numerical results show that there are situations in which the unstructured condition number can be much larger than the structured ones.

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Acknowledgements

The authors thank Prof. Dopico and Dr. Pomés for sending Matlab codes of [12]. We would like to thank two referees for their constructive comments, which led to improvements of our manuscript. Especially, suggestions on how to compute the structured generalized eigenvalue condition numbers efficiently were proposed by two referees, which initiated us into the study of the corresponding works in this manuscript. Also we are in debt to the reviewer for correcting many grammatical typos that have contributed to improve the presentation of the original manuscript.

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  1. Qing-Le Meng

    Present address: School of Mathematical Sciences, Xiamen University, Xiamen, 361005, People’s Republic of China

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  1. School of Mathematics and Statistics, Northeast Normal University, No. 5268 Renmin Street, Chang Chun, 130024, People’s Republic of China

    Huai-An Diao & Qing-Le Meng

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  1. Huai-An Diao
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Correspondence to Huai-An Diao.

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Communicated by Daniel Kressner.

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This work was supported by the Fundamental Research Funds for the Central Universities under Grant 2412017FZ007.

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Diao, HA., Meng, QL. Structured generalized eigenvalue condition numbers for parameterized quasiseparable matrices. Bit Numer Math 59, 695–720 (2019). https://doi.org/10.1007/s10543-019-00748-5

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