Abstract: This article is devoted to the numerical solution of
large-scale quadratic eigenvalue problems. Such problems arise in
a wide variety of applications, such as the dynamic analysis of
structural mechanical systems, acoustic systems, fluid mechanics,
and signal processing. We first introduce a generalized second-order
Krylov subspace based on a pair of square matrices and two initial
vectors and present a generalized second-order Arnoldi process for
constructing an orthonormal basis of the generalized second-order
Krylov subspace. Then, by using the projection technique and the
refined projection technique, we propose a restarted generalized
second-order Arnoldi method and a restarted refined generalized
second-order Arnoldi method for computing some eigenpairs of largescale
quadratic eigenvalue problems. Some theoretical results are also
presented. Some numerical examples are presented to illustrate the
effectiveness of the proposed methods.
Abstract: The IDR(s) method based on an extended IDR theorem was proposed by Sonneveld and van Gijzen. The original IDR(s) method has excellent property compared with the conventional iterative methods in terms of efficiency and small amount of memory. IDR(s) method, however, has unexpected property that relative residual 2-norm stagnates at the level of less than 10-12. In this paper, an effective strategy for stagnation detection, stagnation avoidance using adaptively information of parameter s and improvement of convergence rate itself of IDR(s) method are proposed in order to gain high accuracy of the approximated solution of IDR(s) method. Through numerical experiments, effectiveness of adaptive tuning IDR(s) method is verified and demonstrated.
Abstract: In the present paper, we propose numerical methods for solving the Stein equation AXC - X - D = 0 where the matrix A is large and sparse. Such problems appear in discrete-time control problems, filtering and image restoration. We consider the case where the matrix D is of full rank and the case where D is factored as a product of two matrices. The proposed methods are Krylov subspace methods based on the block Arnoldi algorithm. We give theoretical results and we report some numerical experiments.