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In this paper, we propose the palindromic doubling algorithm (PDA) for the palindromic generalized eigenvalue problem (PGEP) A x= Ax. We establish a complete convergence theory of the PDA for PGEPs without unimodular eigenvalues, or with unimodular eigenvalues of partial multiplicities two (one or two for eigenvalue 1). Some important applications from the vibration analysis and the optimal control for singular descriptor linear systems will be presented to illustrate the feasibility and efficiency of the PDA.

In this paper, we present an efficient QR algorithm for solving the linear response eigenvalue problem H x= x, where H is -symmetric with respect to 0= diag (I n, I n). Based on newly introduced -orthogonal transformations, the QR algorithm preserves the -symmetric structure of H throughout the whole process, and thus guarantees the computed eigenvalues to appear pairwise (, ) as they should. With the help of a newly established implicit -orthogonality theorem, we incorporate the implicit multi-shift technique to accelerate the convergence of the QR algorithm. Numerical experiments are given to show the effectiveness of the algorithm.

We propose an efficient eigensolver for computing densely distributed spectra of the two-dimensional transmission eigenvalue problem (TEP), which is derived from Maxwell's equations with Tellegen media and the transverse magnetic mode. The governing equations, when discretized by the standard piecewise linear finite element method, give rise to a large-scale quadratic eigenvalue problem (QEP). Our numerical simulation shows that half of the positive eigenvalues of the QEP are densely distributed in some interval near the origin. The quadratic JacobiDavidson method with a so-called non-equivalence deflation technique is proposed to compute the dense spectrum of the QEP. Extensive numerical simulations show that our proposed method processes the convergence efficiently, even when it needs to compute more than 5000 desired eigenpairs. Numerical results also illustrate that the computed

We consider the solution of large-scale discrete-time algebraic Riccati equations with numerically low-ranked solutions. The structure-preserving doubling algorithm will be adapted, with the iterates for A not explicitly computed but in the recursive form Ak= A2 k 1 D (1) k

For the steady-state solution of an integraldifferential equation from a two-dimensional model in transport theory, we shall derive and study a nonsymmetric algebraic Riccati equation B--XF--F+ X+ XB+ X= 0, where F I-s PD, B-(b I+ s P) D-and B+ b I+ s PD+ with a nonnegative matrix P, positive diagonal matrices D, and nonnegative parameters f, b b/(1-f) and s s/(1-f). We prove the existence of the minimal nonnegative solution X under the physically reasonable assumption f+ b+ s P (D++ D-)< 1, and study its numerical computation by fixed-point iteration, Newtons method and doubling. We shall also study several special cases; eg when b = 0 and P is low-ranked, then X= s 2 UV is low-ranked and can be computed using more efficient iterative processes in U and V. Numerical examples will be given to illustrate our theoretical results.

For the steady-state solution of a differential equation from a one-dimensional multistate model in transport theory, we shall derive and study a nonsymmetric algebraic Riccati equation <i>B</i> <i>XF</i> <i>F</i>⁺<i>X</i> + <i>XB</i>⁺<i>X</i> = 0, where <i>F</i> (<i>I</i> <i>F</i>)<i>D</i> and <i>B</i> <i>BD</i> with positive diagonal matrices <i>D</i> and possibly low-ranked matrices <i>F</i> and <i>B</i>. We prove the existence of the minimal positive solution <i>X</i>^* under a set of physically reasonable assumptions and study its numerical computation by fixed-point iteration, Newtons method and the doubling algorithm. We shall also study several special cases. For example when <i>B</i> and <i>F</i> are low ranked then X*=(i=14UiViT) with low-ranked <i>U_i</i> and <i>V_i</i> that can be computed using more efficient iterative processes. Numerical examples will be given to illustrate our theoretical results.

In Chu (Syst Control Lett 56:303314, 2007), the pole assignment problem was considered for the control system \dot{x} = Ax + Bu with linear state-feedback \dot{x} = Ax + Bu An algorithm using the Schur form has been proposed, producing good suboptimal solutions which can be refined further using optimization. In this paper, the algorithm is improved, incorporating the minimization of the feedback gain \dot{x} = Ax + Bu It is also extended for the pole assignment of the descriptor system \dot{x} = Ax + Bu with linear state- and derivative-feedback \dot{x} = Ax + Bu Newton refinement for the solutions is discussed and several illustrative numerical examples are presented.

Calculation of band structure of three dimensional photonic crystals amounts to solving large-scale Maxwell eigenvalue problems, which are notoriously challenging due to high multiplicity of zero eigenvalue. In this paper, we try to address this problem in such a broad context that band structure of three dimensional isotropic photonic crystals with all 14 Bravais lattices can be efficiently computed in a unified framework. We uncover the delicate machinery behind several key results of our work and on the basis of this new understanding we drastically simplify the derivations, proofs and arguments in our framework. In this work particular effort is made on reformulating the Bloch boundary condition for all 14 Bravais lattices in the redefined orthogonal coordinate system, and establishing eigen-decomposition of discrete partial derivative operators by systematic use of commutativity among them, which has been overlooked previously, and reducing eigen-decomposition of double-curl operator to the canonical form of a 3x3 complex skew-symmetric matrix under unitary congruence. With the validity of the novel nullspace free method in the broad context, we perform some calculations on one benchmark system to demonstrate the accuracy and efficiency of our algorithm.

We study the generalized eigenvalue problems (GEPs) derived from modeling the surface acoustic wave in piezoelectric materials with periodic inhomogeneity. The eigenvalues appear in the reciprocal pairs due to periodic boundary conditions in the modeling. By transforming the GEP into a T-palindromic quadratic eigenvalue problem (TPQEP), the reciprocal relationship of the eigenvalues can be maintained. In this paper, we outline four recently developed structure-preserving algorithms, SA, SDA, TSHIRA and GTSHIRA, for solving the TPQEP. Numerical comparisons on the accuracy and the computational costs of these algorithm are presented. The eigenvalues close to unit circle on the complex plane are of interest in the area of filter and sensor designs. Our numerical results show that the Arnoldi-type structure-preserving algorithms TSHIRA and GTSHIRA with re-symplectic and re-bi-isotropic processes

We propose a structure-preserving doubling algorithm for a quadratic eigenvalue problem arising from the stability analysis of time-delay systems. We are particularly interested in the eigenvalues on the unit circle, which are difficult to estimate. The convergence and backward error of the algorithm are analyzed and three numerical examples are presented. Our experience shows that our algorithm is efficient in comparison to the few existing approaches for small to medium size problems.