# Higher order spectral symbols and eigenvalues approximation for Toeplitz matrices – Giovanni Barbarino (Scuola Normale Superiore)

#### Abstract

Room: Sala Seminari Est, Dipartimento di Informatica Abstract. When modeling natural phenomena with linear partial differential equations, the discretized system of equations is generally represented by a sequence of matrices, often coupled with a function called spectral symbol that enclosures asymptotic spectral information of the sequence. In fact, when sampling the symbol, we obtain an approximation of the spectrum of the matrices, with an error having magnitude inversely proportional with respect to the matrix size. Under particular hypothesis, the sequence presents an higher-order asymptotic formula for the eigenvalues, that let us precisely compute the eigenvalues of large matrices with a technique of interpolation-extrapolation at a low computational cost and that is also parallelizable. This algorithm seems also to work on different kind of matrices, but with irregularities on the extreme eigenvalues, for which a more accurate analysis is needed.

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