Research Seminar in Mathematics - Local linearizations of rational matrices with application to nonlinear eigenvalue problems

Speaker

Prof. Froilán M. Dopico, Universidad Carlos III de Madrid.

Abstract

The numerical solution of nonlinear eigenvalue problems (NLEP) has attracted considerable attention

since 2004, mainly as a consequence of the influential reference "Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods" (GAMM Mitt. Ges. Angew. Math. Mech., 2004) by V. Mehrmann and H. Voss. A variety of methods have been developed for these problems and the most successful ones can be found in the recent survey "The nonlinear eigenvalue problem" (Acta Numer., 2017) by S. G\"{u}ttel and F. Tissuer. Both for dense-medium-size and for large-scale problems the preferred methods consist of three steps: 

(1) to approximate the NLEP by a rational eigenvalue problem (REP) in a certain region; 

(2) to construct a linear eigenvalue problem (LEP) that has the same eigenvalues of the REP in the region of interest; 

(3) to compute via the QZ method for dense problems or via structured rational Krylov methods for large-scale problems the eigenvalues of the LEP. 

The purpose of this talk is to develop a mathematical local theory of linearizations of REPs that allows us, among other things, to establish rigorously the properties of the LEPs that have been used for solving NLEPs in a number of recent references.