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Eigenvalue backward errors of Rosenbrock systems and optimization of sums of Rayleigh quotient

Ding Lu, Anshul Prajapati, Punit Sharma, Shreemayee Bora

TL;DR

This work develops a comprehensive framework for computing eigenvalue backward errors of the Rosenbrock system matrix under block perturbations by reducing the problem to a Sum of Two generalized Rayleigh Quotients (SRQ2) minimization. The SRQ2 problem is reformulated as a minimization over the joint numerical range of three Hermitian matrices, exploiting convexity to derive a nonlinear eigenvalue problem with eigenvector dependency (NEPv) that provides an efficient route to the solution. The authors derive explicit formulas for backward errors under full and partial perturbations (one, two, or three blocks), and demonstrate how these expressions translate into SRQ2 optimizations. Numerical experiments validate the NEPv approach, showing speedups over traditional optimization methods and offering geometric visualization via the joint numerical range. Overall, the paper delivers computable backward-error formulas, a unifying JNR–NEPv framework, and practical tools for backward error analysis in rational eigenvalue problems arising from Rosenbrock systems.

Abstract

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.

Eigenvalue backward errors of Rosenbrock systems and optimization of sums of Rayleigh quotient

TL;DR

This work develops a comprehensive framework for computing eigenvalue backward errors of the Rosenbrock system matrix under block perturbations by reducing the problem to a Sum of Two generalized Rayleigh Quotients (SRQ2) minimization. The SRQ2 problem is reformulated as a minimization over the joint numerical range of three Hermitian matrices, exploiting convexity to derive a nonlinear eigenvalue problem with eigenvector dependency (NEPv) that provides an efficient route to the solution. The authors derive explicit formulas for backward errors under full and partial perturbations (one, two, or three blocks), and demonstrate how these expressions translate into SRQ2 optimizations. Numerical experiments validate the NEPv approach, showing speedups over traditional optimization methods and offering geometric visualization via the joint numerical range. Overall, the paper delivers computable backward-error formulas, a unifying JNR–NEPv framework, and practical tools for backward error analysis in rational eigenvalue problems arising from Rosenbrock systems.

Abstract

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.
Paper Structure (15 sections, 11 theorems, 93 equations, 2 figures, 3 tables)

This paper contains 15 sections, 11 theorems, 93 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Contribution and outline of the paper
  3. Notations
  4. Backward Error Analysis of Rosenbrock Systems
  5. Backward Error with Full Perturbation
  6. Backward Error with Partial Perturbations
  7. Perturbing a Single Block
  8. Perturbing Any Two Blocks
  9. Perturbing Any Three Blocks
  10. Minimization of Sum of Two Rayleigh Quotients
  11. Minimization Over Joint Numerical Range
  12. Local Optimality Condition and NEPv
  13. Non-differentiable Minimizers
  14. Numerical Experiments
  15. Concluding Remarks

Key Result

Lemma 1

\newlabellem:mapping0 Let $x\in {\mathbb C}^n$ , $b\in {\mathbb C}^m$ , $\Omega\!=\!\{D\in {\mathbb C}^{m,n} \colon D x\!=\!b\}$, and ${ \widehat{D}\!=\!\mathop{\mathrm{argmin}}\limits_{D\in \Omega}\|D\|_F^2}$.

Figures (2)

  • Figure 1: Joint numerical range $W(\mathcal{M})$ (bounded by gray surface, based on $N=800$ sample boundary points) and level surface $\{y\colon g(y) = g(y_{\star})\}$ (colored) at solution $y_{\star}$ (marked 'o') of optimization problem \ref{['eq:eg1b']}. The left plot is with $y_{\star}=y_{\star}^{(1)}$ and the right with $y_{\star}=y_{\star}^{(2)}$, both from \ref{['eq:ystar']}.
  • Figure 2: Rosenbrock system \ref{['eq:rosenbrocklin']} with $n=100$ and $r=10$ to compute the backward error $\eta(\lambda)$ by \ref{['eq:allbloexp']}. Left: Convergence history of residual norms $\| H(x_k) x_k - s_k x_k\|_2$ with $s_k = x_k^*H(x_k)x_k$ for iterative $x_k$ by level-shifted SCF \ref{['eq:lsscf']} and Riemannian Trust-Region method. Reported are $20$ repeated runs with randomly generated starting vectors $x_0$. Right: Illustration of the corresponding JNR minimization \ref{['eq:optw']}, with approximate joint numerical range $W(\mathcal{M})$ (gray surface) and level surface of $g(y):=y_1 + y_2/y_3$ (color surface) at the solution $y_{\star}=\rho_{\!\!_{\mathcal{M}}}(x_{\star})$ (marked 'o').

Theorems & Definitions (32)

  • Lemma 1
  • Proof 1
  • Definition 2
  • Remark 1
  • Theorem 3
  • Proof 2
  • Definition 4
  • Remark 2
  • Lemma 5
  • Proof 3
  • ...and 22 more