Büyük boyutlu matrislerin yaklaşık spectral yarıçapı

The epsilon-pseudospectral radius of a square matrix A is the maximal modulus attained among the points that belong to its epsilon-pseudospectrum, the set consisting of the eigenvalues of all matrices within the epsilon neighborhood of A with respect to the matrix 2-norm. We present a numerical algorithm to estimate this quantity for a large scale matrix, and analyze it. We tackle the problem by viewing it as a nonsmooth eigenvalue optimization problem, and constructing piecewise quadratic overestimators for the eigenvalue functions. Overall algorithm is an extension of a recent work by Meerbergen, Mengi, Michiels and Van Beeumen, which concerns the epsilon-pseudospectral abscissa in the more general nonlinear eigenvalue problem setting. The approach presented benefits from a subspace and a restart idea introducedby Kressner and Vandereycken, and Meerbergen et al. in other contexts. We observe convergence to a point with largest modulus locally at a superlinear rate.