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Hongguo Xu

Publications and source records attributed to Hongguo Xu.

7 recordsLinked to original sources
arXiv2025

Invariant subspace perturbations related to defective eigenvalues of $\Delta$-Hermitian and Hamiltonian matrices

Structured perturbation results for invariant subspaces of $\Delta$-Hermitian and Hamiltonian matrices are provided. The invariant subspaces under consideration are associated with the eigenvalues perturbed from a single defective eigenvalue. The results show how the original eigenvectors and generalized eigenvectors are involved in composing such perturbed invariant subspaces and eigenvectors.

math.NA↗
arXiv2023

Invariant subspace perturbation of a matrix with Jordan blocks

We investigate how invariant subspaces corresponding to a single eigenvalue will change when a matrix is perturbed. We focus on the invariant subspaces corresponding to an eigenvalue associated with the Jordan blocks that have the same size. An invariant subspace can be expressed as the range of a full column matrix. We characterize the perturbed invariant subspaces with such matrices expressed in a sum form that exhibits the fractional orders. We also provide the formulas for the coefficient matrices associated with the zero and first fractional orders. The results extend the standard invariant subspace perturbation theory.

math.NA↗
arXiv2022

Port-Hamiltonian Realizations of Nonminimal Linear Time Invariant Systems

Numerical methods for developing port-Hamiltonian representations of general linear time-invariant systems are studied. The approach extends previous port-Hamiltonian characterizations to include the general non-minimal case and the case where the feedthrough term fails to have an invertible symmetric part. The resulting construction is able to identify infeasibility when the system fails to be port-Hamiltonian, and allows for the incorporation of perturbations in order to arrive at a nearby port-Hamiltonian system. Results are illustrated via numerical examples.

math.OC↗
arXiv2017

Port-Hamiltonian descriptor systems

The modeling framework of port-Hamiltonian systems is systematically extended to constrained dynamical systems (descriptor systems, differential-algebraic equations). A new algebraically and geometrically defined system structure is derived. It is shown that this structure is invariant under equivalence transformations, and that it is adequate also for the modeling of high-index descriptor systems. The regularization procedure for descriptor systems to make them suitable for simulation and control is modified to deal with the port-Hamiltonian structure. The relevance of the new structure is demonstrated with several examples.

math.OC↗
arXiv2012

Conditioning of Finite Element Equations with Arbitrary Anisotropic Meshes

Bounds are developed for the condition number of the linear finite element equations of an anisotropic diffusion problem with arbitrary meshes. They depend on three factors. The first, factor proportional to a power of the number of mesh elements, represents the condition number of the linear finite element equations for the Laplacian operator on a uniform mesh. The other two factors arise from the mesh nonuniformity viewed in the Euclidean metric and in the metric defined by the diffusion matrix. The new bounds reveal that the conditioning of the finite element equations with adaptive anisotropic meshes is much better than what is commonly feared. Diagonal scaling for the linear system and its effects on the conditioning are also studied. It is shown that the Jacobi preconditioning, which is an optimal diagonal scaling for a symmetric positive definite sparse matrix, can eliminate the effects of mesh nonuniformity viewed in the Euclidean metric and reduce those effects of the mesh viewed in the metric defined by the diffusion matrix. Tight bounds on the extreme eigenvalues of the stiffness and mass matrices are obtained. Numerical examples are given.

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