A Matrix-less Method for Approximating the Eigenvectors of Toeplitz-like Matrices
Abstract: Matrix-less methods (MLM) have successfully been used to efficiently approximate the eigenvalues of certain classes of structured matrices. Specifically, the method has been used to approximate the eigenvalues of Toeplitz and Toeplitz-like matrices. The method exploits the inherent structure of the eigenvalues, which is maintained when the matrix size changes, and thus can use the eigenvalues of a set of smaller matrices to approximate the eigenvalues for much larger matrices. In this thesis, we investigate whether there exists a similar structure for the eigenvectors for some of these matrices and if we can apply an MLM such that we can efficiently approximate the eigenvectors of Toeplitz(-like) matrices. We here study symmetric Toeplitz(-like) matrices generated by monotone symbols. Specifically, we investigate the use of this method on four different types matrix sequences: the (1) Laplacian matrix in one dimension, the closely related (2) bi-Laplacian matrix, (3) Isogeometric analysis discretisation matrices and also (4) a `full' matrix related to the discretisation of fractional diffusion. For the first three types of investigated matrix sequences, it is found that the matrix-less method is able to well-approximate the eigenvectors, where as for the fourth case (fractional derivative related matrix sequence) where it does not work, we discuss some potential adjustments that may allow for MLM to work in that case as well. We also discuss why the method cannot be immediately used for Toeplitz matrices with non-monotone symbols. We conclude the thesis with suggestions for future avenues of research including how to possibly deal with matrix sequences generated by non-monotone symbols.
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