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A Robust Randomized Indicator Method for Accurate Symmetric Eigenvalue Detection

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Abstract

We propose a robust randomized indicator method for the reliable detection of eigenvalue existence within an interval for symmetric matrices A. An indicator tells the eigenvalue existence based on some statistical norm estimators for a spectral projector. Previous work on eigenvalue indicators relies on a threshold which is empirically chosen, thus often resulting in under or over detection. In this paper, we use rigorous statistical analysis to guide the design of a robust indicator. Multiple randomized estimators for a contour integral operator in terms of A are analyzed. In particular, when A has eigenvalues inside a given interval, we show that the failure probability (for the estimators to return very small estimates) is extremely low. This enables to design a robust rejection indicator based on the control of the failure probability. We also give a prototype framework to illustrate how the indicator method may be applied numerically for eigenvalue detection and may potentially serve as a new way to design randomized symmetric eigenvalue solvers. Unlike previous indicator methods that only detect eigenvalue existence, the framework also provides a way to find eigenvectors with little extra cost by reusing computations from indicator evaluations. Extensive numerical tests show the reliability of the eigenvalue detection in multiple aspects.

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Data Availability

The data for the numerical tests is available upon reasonable request.

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Funding

The research of J. Sun is funded in part by Simons Foundation Grant 711922 and NSF Grant 2109949.

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Authors and Affiliations

  1. Division of Biostatistics, Medical College of Wisconsin, Milwaukee, WI, 53226, USA

    Zhongyuan Chen

  2. Department of Mathematical Sciences, Michigan Technological University, Houghton, MI, 49931, USA

    Jiguang Sun

  3. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Jianlin Xia

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  1. Zhongyuan Chen
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  2. Jiguang Sun
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Correspondence to Jianlin Xia.

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Chen, Z., Sun, J. & Xia, J. A Robust Randomized Indicator Method for Accurate Symmetric Eigenvalue Detection. J Sci Comput 100, 48 (2024). https://doi.org/10.1007/s10915-024-02599-x

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