Technical Report 2021-005

Matrix-Less Eigensolver for Large Structured Matrices

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

August 2021

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 Xn 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 Xn for large n. The performance of the eigensolver-which is called matrix-less because it does not operate on the matrix Xn-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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