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A polynomial Jacobi–Davidson solver with support for non-monomial bases and deflation

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Abstract

Large-scale polynomial eigenvalue problems can be solved by Krylov methods operating on an equivalent linear eigenproblem (linearization) of size \(d\cdot n\), where d is the polynomial degree and n is the problem size, or by projection methods that keep the computation in the n-dimensional space. Jacobi–Davidson belongs to the latter class of methods, and, since it is a preconditioned eigensolver, it may be competitive in cases where explicitly computing a matrix factorization is exceedingly expensive. However, a fully fledged implementation of polynomial Jacobi–Davidson has to consider several issues, including deflation to compute more than one eigenpair, use of non-monomial bases for the case of large degree polynomials, and handling of complex eigenvalues when computing in real arithmetic. We discuss these aspects and present computational results of a parallel implementation in the SLEPc library.

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Acknowledgements

We thank Eloy Romero for useful comments on an initial version of the manuscript. The computational experiments of Sect. 6 were carried out on the supercomputer Tirant 3 belonging to Universitat de València. The authors of [17] are acknowledged for kindly providing the code of their polynomial Jacobi–Davidson solver, which served as inspiration for building our own solver, as well as the matrices coming from the quantum dot simulation used in Sect. 6.

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  1. Departament de Sistemes Informàtics i Computació, Universitat Politècnica de València, Camí de Vera s/n, 46022, Valencia, Spain

    Carmen Campos & Jose E. Roman

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  1. Carmen Campos
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  2. Jose E. Roman
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Correspondence to Jose E. Roman.

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This work was supported by Agencia Estatal de Investigación (AEI) under Grant TIN2016-75985-P, which includes European Commission ERDF funds.

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Campos, C., Roman, J.E. A polynomial Jacobi–Davidson solver with support for non-monomial bases and deflation. Bit Numer Math 60, 295–318 (2020). https://doi.org/10.1007/s10543-019-00778-z

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