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.

翻译：本文研究了在各类块扰动下计算Rosenbrock系统矩阵特征值后向误差的问题。通过建立一类涉及双广义Rayleigh商和（SRQ2）的最小化问题，我们推导出这些后向误差的可计算公式。为便于计算与分析，我们将此类优化问题重构为在三个Hermitian矩阵联合数值域上有理函数的最小化问题。该重构消除了原始SRQ2最小化问题的部分局部极小点，并提供了解的直观可视化方法。进一步地，通过利用联合数值域内的凸性结构，我们借助具有特征向量依赖性的非线性特征值问题（NEPv）推导出最优解的刻画方式。相较于传统优化技术，NEPv刻画方法能更高效地求解SRQ2最小化问题。数值实验验证了NEPv方法在计算Rosenbrock系统特征值后向误差时求解SRQ2最小化问题的优势与有效性。