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A FEAST SVDsolver Based on Chebyshev–Jackson Series for Computing Partial Singular Triplets of Large Matrices

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Abstract

The FEAST eigensolver is extended to the computation of the singular triplets of a large matrix A with the singular values in a given interval. The resulting FEAST SVDsolver is subspace iteration applied to an approximate spectral projector of \(A^TA\) corresponding to the desired singular values in a given interval, and constructs approximate left and right singular subspaces corresponding to the desired singular values, onto which A is projected to obtain Ritz approximations. Differently from a commonly used contour integral-based FEAST solver, we propose a robust alternative that constructs approximate spectral projectors by using the Chebyshev–Jackson polynomial series, which are shown to be symmetric positive semi-definite with the eigenvalues in [0, 1]. We prove the pointwise convergence of this series and give compact estimates for pointwise errors of it and the step function that corresponds to the exact spectral projector of interest. We present error bounds for the approximate spectral projector and reliable estimates for the number of desired singular triplets, prove the convergence of the resulting FEAST SVDsolver, and propose practical selection strategies for determining the series degree and the subspace dimension. The solver and results on it are directly applicable or adaptable to the real symmetric and complex Hermitian eigenvalue problem. Numerical experiments illustrate that the FEAST SVDsolver is substantially more efficient than the contour integral-based FEAST SVDsolver, and it is also more robust and stable than the latter.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

    Zhongxiao Jia & Kailiang Zhang

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  1. Zhongxiao Jia
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  2. Kailiang Zhang
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Correspondence to Zhongxiao Jia.

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The two authors declare that they have no financial interests, and they read and approved the final manuscript. The algorithmic Matlab code is available upon reasonable request from the corresponding author.

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Jia, Z., Zhang, K. A FEAST SVDsolver Based on Chebyshev–Jackson Series for Computing Partial Singular Triplets of Large Matrices. J Sci Comput 97, 21 (2023). https://doi.org/10.1007/s10915-023-02342-y

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  • DOI: https://doi.org/10.1007/s10915-023-02342-y

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