The T-palindromic quadratic eigenvalue problem $(\lambda^2 B + \lambda C + A)x = 0$, with $A,B,C \in \mathbb{C}^{n \times n}$, $C^T = C$ and $B^T = A$, governs the vibration behaviour of trains. Other palindromic eigenvalue problems, quadratic or higher order, arise from applications in surface acoustic wave filters, optimal control of discrete-time systems and crack modelling. Numerical solution of palindromic eigenvalue problems is challenging, with unacceptably low accuracy from the basic linearization approach. In this survey paper, we shall talk about the history of palindromic eigenvalue problems, in terms of their history, applications, numerical solution and generalization. We shall also speculate on some future directions of research.

We investigate the perturbation of the palindromic eigenvalue problem for the matrix quadratic $P(\lambda) \equiv \lambda^2 A_1^T + \lambda A_0 + A_1$, with $A_0,\, A_1 \in \mathbb{C}^{n \times n}$ and $A_0^T = A_0$. The perturbation of eigenvalues and eigenvectors, in terms of palindromic matrix polynomials and palindromic linearizations, are discussed using Sun's implicit function approach.

A rational Krylov algorithm for eigenvalue computation is described. It is usable on a real matrix pencil with complex eigenvalues and builds up a real basis. The main purpose is to get real reduced models of a real linear dynamic system. Two variants are described, one where two real vectors are added to the Krylov space in each step and another where just one real vector is added in each step.

Results are reported from one small example that has been used earlier and where the solution is known, and one more realistic example, a linear descriptor system from a computational fluid dynamics application.

The SHIRA method of Mehrmann and Watkins belongs among the structure preserving Krylov subspace methods for solving skew-Hamiltonian eigenvalue problems. It can also be applied to Hamiltonian eigenproblems by considering a suitable transformation. Structure-induced shift-and-invert techniques are employed to steer the algorithm towards the interesting region of the spectrum. However, the shift cannot be altered in the middle of the computation without discarding the information that has been accumulated so far. This paper shows how SHIRA can be combined with ideas from Ruhe’s Rational Krylov algorithm to yield a method that permits an adjustment of shift after every step of the computation, adding greatly to the flexibility of the algorithm. We call this new method Rational SHIRA. A numerical example is presented to demonstrate its efficiency.

The Rayleigh-Ritz-type approach of the contour integral (CIRR) eigensolver is extended to be generally applicable to non-Hermitian systems. The CIRR method can extract only the eigenvalues in a given domain, which was previously formulated for non-degenerated Hermitian systems. In this method, the Ritz space for the domain is constructed by numerical evaluation of a contour integral. The effect of the numerical approximation is analyzed from the viewpoint of a filter operator, which supports the use of moderate approximations. The numerical accuracy of the original moment-based approach is also assured. A block version of the CIRR method is proposed with a detailed algorithm, which allows us to resolve degenerated systems.

This paper is concerned with solving large-scale eigenvalue problems by algebraic sub-structuring and contour integral. We combine Algebraic Sub-structuring (AS) method and the Contour Integral Rayleigh-Ritz (CIRR) method. The AS method calculates approximate eigenpairs fast and has been shown to be efficient for vibration and acoustic analysis. However, the application areas of this method have been limited because its accuracy is usually lower than other methods. On the other hand, if the appropriate domains are chosen, the CIRR method produces accurate solutions. However, it is difficult to choose these domains without the information of eigenvalue distribution. We propose a combination of AS and CIRR such as the AS method is used as a method for profiling a distribution of eigenvalues, and the accurate solutions are produced by the CIRR method using the information of eigenvalue distribution provided by AS. We show our method is effective from the result of applying this method to the molecular orbital calculations.

In the present paper, we consider a parallel method for computing interior eigenvalues and corresponding eigenvectors of generalized eigenvalue problems arisen from molecular orbital computation in biochemistry applications. Matrices in such applications are sparse but often have a relatively large number of nonzero elements, and we may require some eigenpairs in a specific part of the spectrum. We use contour integration to construct a desired subspace. Properties of the subspace obtained by numerical integration are discussed, and a parallel implementation is then presented. We report the numerical aspects and parallel performance of the proposed method with matrices derived from molecular orbital computation.

A nonlinear Rayleigh-Ritz iterative (NRRIT) method for solving nonlinear eigenvalue problems is studied in this paper. It is an extension of the nonlinear Arnoldi algorithm due to Heinrich Voss. The efficiency of the NRRIT method is demonstrated by comparing with the inverse iteration method to solve a highly nonlinear eigenvalue problem arising from finite element electromagnetic simulation in accelerator modeling.

The total least squares (TLS) method is a successful approach for linear problems if both the system matrix and the right hand side are contaminated by some noise. For ill-posed TLS problems regularization is necessary to stabilize the computed solution. In this paper we summarize two iterative methods which are based on a sequence of eigenproblems. The focus is on efficient implementation with particular emphasis on the reuse of information gained during the convergence history.

This paper concerns the eigenvalue embedding problem (EEP) of updating a symmetric finite-element model so that a few troublesome eigenvalues are replaced by some chosen ones, while the remaining large number of eigenvalues and eigenvectors of the original model do not change. Based on the theory established in [2], by sufficiently utilizing the inherent freedom of the EEP, an expression of the parameterized solution to the EEP is derived. This expression is then used to develop a novel numerical method for solving the EEP, in which the parameters in the solutions are optimized in some sense. This method not only utilizes the freedom of the EEP but also removes the limitation of the method proposed in [6]. The results of our numerical experiments show that the present algorithm is feasible and efficient, and can outperform the iterative method in [3] and the method in [6].

In this paper we propose a structured doubling algorithm for solving discrete-time algebraic Riccati equations without the invertibility of control weighting matrices. In addition, we prove that the convergence of the SDA algorithm is linear with ratio less than or equal $\frac{1}{2}$ when all unimodular eigenvalues of the closed-loop matrix are semi-simple. Numerical examples are shown to illustrate the feasibility and efficiency of the proposed algorithm.

Several beautiful formulae for the solutions of some nonlinear matrix equations were proposed by Yongdo Lim in 2007, in terms of the matrix golden means. Numerically, these formulae will not be applicable when some matrices involved are ill-conditioned. In this note, we propose to partially fill in this gap of applicability with the structured doubling algorithm, under some favourable conditions. We also discuss how some pre-processing or scaling procedures can be applied to the matrix equations, to improve their condition. More generally, we also explore the possibility of computing the matrix golden mean using structured doubling algorithms. Some numerical examples will be presented for illustrative purposes.

Numerical algorithms for the computation of an upper bound of the largest structured singular value arising from the $\mu$-synthesis control problem are developed. Since the computation for the largest structured singular value has been shown to be an NP-hard problem in literatures, we concentrate the study on the computation for an upper bound of the largest structured singular value. A Newton's type method is proposed. Some theoretical results related to the method are investigated. Numerical implementation shows the efficiency of the method.

Nano-scale quantum dot simulations result in large-scale polynomial eigenvalue problems. It remains unclear how these problems can be solved efficiently. We fill this gap in capability partially by proposing a polynomial Jacobi-Davidson method framework, including several varied schemes for solving the associated correction equations. We investigate the performance of the proposed Jacobi-Davidson methods for solving the polynomial eigenvalue problems and several Krylov subspace methods for solving the linear eigenvalue problems with the use of various linear solvers and preconditioning schemes. This study finds the most efficient scheme combinations for different types of target problems.