Abstract:
The Rosenbrock system matrix is a structured matrix polynomial that commonly arises in linear system theory and in solving nonlinear eigenvalue problems. It specific block structure reflects underlying physical and theoretical properties of the system being modeled and holds practical significance. In this talk, we address the problem of computing the eigenvalue backward error of the Rosenbrock system under various types of block perturbations. We show that these backward errors can be unified under a class of minimization problems involving the Sum of Two generalized Rayleigh Quotients (SRQ2). For computational purposes and analysis, we reformulate these optimization problems as the minimization of a rational function over the joint numerical range of three Hermitian matrices. This reformulation helps to avoid certain local minima in the original problem and facilitates visualization of the optimal solution. By exploiting the convexity within the joint numerical range, we also derive a characterization of the optimal solution using a Nonlinear Eigenvalue Problem with Eigenvector Dependency (NEPv). The NEPv characterization enables a more efficient solution of the SRQ2 minimization compared to traditional optimization techniques.
This work is a collaboration with Anshul Prajapati and Punit Sharma from IIT Delhi, and Shreemayee Bora from IIT Guwahati.