SOAR: A Second-order Arnoldi Method for the Solution of the Quadratic Eigenvalue Problem

Abstract

Abstract. We first introduce a second-order Krylov subspace Gn(A, B; u) based on a pair of square matrices A and B and a vector u. The subspace is spanned by a sequence of vectors defined via a second-order linear homogeneous recurrence relation with coefficient matrices A and B and an initial vector u. It generalizes the well-known Krylov subspace Kn(A; v), which is spanned by a sequence of vectors defined via a first-order linear homogeneous recurrence relation with a single coefficient matrix A and an initial vector v. Then we present a second-order Arnoldi (SOAR) procedure for generating an orthonormal basis of Gn(A, B; u). By applying the standard RayleighRitz orthogonal projection technique, we derive an SOAR method for solving a large-scale quadratic eigenvalue problem (QEP). This method is applied to the QEP directly. Hence it preserves essential structures and properties of the QEP. Numerical examples demonstrate that the SOAR method outperforms convergence behaviors of the Krylov subspace-based Arnoldi method applied to the linearized QEP.

An Automatic Model Order Reduction of a Uwb Antenna System

“…Table 2 lists the final results. Depended on these 4 poles s i and residues R i , reduced models of Q 1 and Q 2 can be established according to (18). The approximate results shown in Fig.…”

“…Compared to the mature level of the techniques for the linear system [17], the development of second-order techniques is a relatively new topic, but has raised growing interest, including Second Order Arnoldi [18], Second order dominant pole identification [19] and Quadratic Arnoldi algorithm [11]. In this paper, Quadratic Arnoldi algorithm (Q-arnoldi) are adopted as the numerical method, in terms of its remarkable feature of favoring the convergence to the particular eigenvalues using the strategies of implicit restarting or purging [17].…”

Structure-Preserving Model Reduction

“…The work of Su and Craig [17] has spawned several recent research papers on model reduction of second-order systems and quadratic eigenvalue problems, includ- ing [3,4,5,18]. But the attempt to preserve meaningful substructures as in (2.3) -(2.6) for any general linear systems, not necessarily from linearizing a second-order system, appears to be conceived first by [10].…”

“…Item 4 of Theorem 3 was implicitly stated in [3,4,17]. It gives a relation between span( X 1 ) and span( X 2 ); so does Item 1.…”

Model Reduction for RF MEMS Simulation

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