Decay properties of matrix functions: the influence of polynomial approximation and eigenvalue distribution

• Date: 23. 5. 2024
• Time: 14:00 - 15:30
• Address:
  Sokolovská 83, Praha
• Room: K1
• Speaker: Michele Rinelli

Matrix functions of the form f(A) are often encountered in numerical linear algebra and applied sciences. In many applications, the matrix argument A is sparse, i.e., among all the entries, only a few are nonzeros. Usually, even if A is sparse, f(A) is actually a dense matrix. However, it is often observed a decay when moving away from the main diagonal, or with respect to the sparsity pattern of A. In such case, we are able to approximate the matrix function with a sparse matrix, up to an error that depends on the decay rate, potentially leading to huge computational savings. In this seminar, we delve into the relation between the decay properties of f(A) and the best uniform polynomial approximations of f over a suitable set containing the spectrum of A. We consider in detail some specific matrix functions, such as the inverse and the sign. We explore the influence of the eigenvalue distribution of A on the decay, showing that the contribution of isolated eigenvalues is negligible, and predicting a superexponential decay for certain clustered spectra.