Matrix-Less Eigensolver for Large Structured Matrices

Giovanni Barbarino, Melker Claesson, Sven-Erik Ekström, Carlo Garoni, and David Meadon

August 2021

Abstract:

Sequences of structured matrices of increasing size arise in many scientific applications and especially in the numerical discretization of linear differential problems. We assume as a working hypothesis that the eigenvalues of a matrix X_n belonging to a sequence of this kind are given by a regular expansion. Based on the working hypothesis, which is proved to be plausible through numerical experiments, we propose an eigensolver for the computation of the eigenvalues of X_n for large n. The performance of the eigensolver-which is called matrix-less because it does not operate on the matrix X_n-is illustrated on several numerical examples, with a special focus on matrices arising from the discretization of differential problems, and turns out to be quite satisfactory in all cases. In a sense, this is an a posteriori proof of the reasonableness of the working hypothesis as well as a testimony of the fact that the spectra of large structured matrices are much more "regular" than one might expect.

Note: Updated 2021-09-03 and 2021-09-08.

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