Structured backward errors for eigenvalues of linear port-Hamiltonian descriptor systems

When computing the eigenstructure of matrix pencils associated with the passivity analysis of perturbed port-Hamiltonian descriptor system using a structured generalized eigenvalue method, one should make sure that the computed spectrum satisfies the symmetries that corresponds to this structure and the underlying physical system. We perform a backward error analysis and show that for matrix pencils associated with port-Hamiltonian descriptor systems and a given computed eigenstructure with the correct symmetry structure there always exists a nearby port-Hamiltonian descriptor system with exactly that eigenstructure. We also derive bounds for how near this system is and show that the stability radius of the system plays a role in that bound

Polynomial eigenvalue solver based on tropically scaled Lagrange linearization

We propose an algorithm to solve polynomial eigenvalue problems via linearization combining several ingredients: a specific choice of linearization, which is constructed using input from tropical algebra and the notion of well-separated tropical roots, an appropriate scaling applied to the linearization and a modified stopping criterion for the $QZ$ iterations that takes advantage of the properties of our scaled linearization. Numerical experiments suggest that our polynomial eigensolver computes all the finite and well-conditioned eigenvalues to high relative accuracy even when they are very different in magnitude.status: publishe

The limit empirical spectral distribution of Gaussian monic complex matrix polynomials

We define the empirical spectral distribution (ESD) of a random matrix polynomial with invertible leading coefficient, and we study it for complex $n \times n$ Gaussian monic matrix polynomials of degree $k$. We obtain exact formulae for the almost sure limit of the ESD in two distinct scenarios: (1) $n \rightarrow \infty$ with $k$ constant and (2) $k \rightarrow \infty$ with $n$ constant. The main tool for our approach is the replacement principle by Tao, Vu and Krishnapur. Along the way, we also develop some auxiliary results of potential independent interest: we slightly extend a result by B\"{u}rgisser and Cucker on the tail bound for the norm of the pseudoinverse of a non-zero mean matrix, and we obtain several estimates on the singular values of certain structured random matrices.Comment: 25 pages, 4 figure

A framework for structured linearizations of matrix polynomials in various bases

We present a framework for the construction of linearizations for scalar and matrix polynomials based on dual bases which, in the case of orthogonal polynomials, can be described by the associated recurrence relations. The framework provides an extension of the classical linearization theory for polynomials expressed in non-monomial bases and allows to represent polynomials expressed in product families, that is as a linear combination of elements of the form $\phi_i(\lambda) \psi_j(\lambda)$, where $\{ \phi_i(\lambda) \}$ and $\{ \psi_j(\lambda) \}$ can either be polynomial bases or polynomial families which satisfy some mild assumptions. We show that this general construction can be used for many different purposes. Among them, we show how to linearize sums of polynomials and rational functions expressed in different bases. As an example, this allows to look for intersections of functions interpolated on different nodes without converting them to the same basis. We then provide some constructions for structured linearizations for $\star$-even and $\star$-palindromic matrix polynomials. The extensions of these constructions to $\star$-odd and $\star$-antipalindromic of odd degree is discussed and follows immediately from the previous results