arXiv subjects
Publications and source records attributed to Anshul Prajapati.
We study linear time-invariant dissipative Hamiltonian differential-algebraic systems. We characterize when the systems are robustly asymptotically stable and derive exact conditions and bounds when this property is lost under structure-preserving perturbations.
We study linear time-invariant Dissipative Hamiltonian (DH) systems arising in energy-based modeling of dynamical systems. An advantage of DH systems is that they are always stable due to the structure of their coefficient matrices, and, under further weak conditions, even asymptotically stable. In this paper, we discuss the computation of the stability radii for a given asymptotically stable DH system; i.e., the smallest structured perturbation that puts a DH system on the boundary of the region of asymptotic stability, so that it has purely imaginary eigenvalues. We obtain explicit computable formulas for various structured stability radii. For this, the problem of computing stability radii is reformulated in terms of minimizing the Rayleigh quotient of a Hermitian matrix or the sum of two generalized Rayleigh quotients of Hermitian semidefinite matrices. This reformulation results in the problem of minimizing the largest eigenvalue of an eigenvector-dependent Hermitian matrix or minimizing the smallest eigenvalue of a Hermitian matrix which depends on the eigenvector. It is also demonstrated (via numerical experiments) that, under structure-preserving perturbations, the asymptotic stability of a DH system is much more robust than under general perturbations, since the distance to instability is typically much larger when structure-preserving perturbations are considered. Finally, similar results are obtained for optimally robust representations of stable systems.
We derive the necessary and sufficient conditions for the simple eigenvalues of rational matrix functions with symmetry structure to have the same normwise condition number with respect to arbitrary and structure-preserving perturbations. We obtain an exact expression for the structured condition number of simple eigenvalues of symmetric, skew-symmetric and even/odd rational matrix functions, and tight bounds are obtained for simple eigenvalues of Hermitian, skew-Hermitian, even/odd, and palindromic rational matrix functions.
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.
In this paper, we compute the structured eigenvalue backward error of a Rosenbrock system matrix $S(z)=\left[\begin{array}{cc} A-zI & B \\ C & P(z) \end{array}\right]$ for a given scalar $\lambda\in \mathbb C$. We have developed simplified formulas for the structured eigenvalue backward error of the Rosenbrock system matrix, considering both full and partial block perturbations. These formulas involve computing structured $\mu$-values of a rectangular matrix under rectangular-block-diagonal perturbations. For the reformulated $\mu$-value problem, we provide an explicit expression using partial isometric matrices and also obtain a computable upper bound, which is equal to the $\mu$-value when the pertrubation matrix has no more than three blocks at the diagonal. The results are illustrated through numerical experiments.
We derive computable formulas for the structured backward errors of a complex number $λ$ when considered as an approximate eigenvalue of rational matrix polynomials that carry a symmetry structure. We consider symmetric, skew-symmetric, T-even, T-odd, Hermitian, skew-Hermitian, $*$-even, $*$-odd, and $*$-palindromic structures. Numerical experiments show that the backward errors with respect to structure-preserving and arbitrary perturbations are significantly different.
For a Hermitian matrix $H \in \mathbb C^{n,n}$ and symmetric matrices $S_0, S_1,\ldots,S_k \in \mathbb C^{n,n}$, we consider the problem of computing the supremum of $\left\{ \frac{v^*Hv}{v^*v}:~v\in \mathbb C^{n}\setminus \{0\},\,v^TS_iv=0~\text{for}~i=0,\ldots,k\right\}$. For this, we derive an estimation in the form of minimizing the second largest eigenvalue of a parameter depending Hermitian matrix, which is exact when the eigenvalue at the optimal is simple. The results are then applied to compute the eigenvalue backward errors of higher degree matrix polynomials with T-palindromic, T-antipalindromic, T-even, T-odd, and skew-symmetric structures. The results are illustrated by numerical experiments.
We study the structured distance to singularity for a given regular matrix pencil $A+sE$, where $(A,E)\in \mathbb S \subseteq (\mathbb C^{n,n})^2$. This includes Hermitian, skew-Hermitian, $*$-even, $*$-odd, $*$-palindromic, T-palindromic, and dissipative Hamiltonian pencils. We present a purely linear algebra-based approach to derive explicit computable formulas for the distance to the nearest structured pencil $(A-Δ_A)+s(E-Δ_E)$ such that $A-Δ_A$ and $E-Δ_E$ have a common null vector. We then obtain a family of computable lower bounds for the unstructured and structured distances to singularity. Numerical experiments suggest that in many cases, there is a significant difference between structured and unstructured distances. This approach extends to structured matrix polynomials with higher degrees.