  We address the problem of computing the eigenvalue backward error of the
Rosenbrock system matrix under various types of block perturbations. We
establish computable formulas for these backward errors using a class of
minimization problems involving the Sum of Two generalized Rayleigh Quotients
(SRQ2). For computational purposes and analysis, we reformulate such
optimization problems as minimization of a rational function over the joint
numerical range of three Hermitian matrices. This reformulation eliminates
certain local minimizers of the original SRQ2 minimization and allows for
convenient visualization of the solution. Furthermore, by exploiting the
convexity within the joint numerical range, we 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. Our
numerical experiments demonstrate the benefits and effectiveness of the NEPv
approach for SRQ2 minimization in computing eigenvalue backward errors of
Rosenbrock systems.
