Showing papers by "Zhong-Zhi Bai published in 2015"

On preconditioned iteration methods for complex linear systems

Abstract: A complex system of linear equations arises in many important applications. We further explore algebraic and convergence properties and present analytical and numerical comparisons among several available iteration methods such as C-to-R and PMHSS for solving such a class of linear systems. Theoretical analyses and computational results show that reformulating a complex linear system into an equivalent real form is a feasible and effective approach, for which we can construct, analyze, and implement accurate, efficient, and robust preconditioned iteration methods.

On the Numerical Behavior of Matrix Splitting Iteration Methods for Solving Linear Systems

Abstract: We study the numerical behavior of stationary one-step or two-step matrix splitting iteration methods for solving large sparse systems of linear equations. We show that inexact solutions of inner linear systems associated with the matrix splittings may considerably influence the accuracy of the approximate solutions computed in finite precision arithmetic. For a general stationary matrix splitting iteration method, we analyze two mathematically equivalent implementations and discuss the conditions when they are componentwise or normwise forward or backward stable. We distinguish two different forms of matrix splitting iteration methods and prove that one of them is significantly more accurate than the other when employing inexact inner solves. The theoretical results are illustrated by numerical experiments with an inexact one-step and an inexact two-step splitting iteration method.

On the Solvability Condition and Numerical Algorithm for the Parameterized Generalized Inverse Eigenvalue Problem

Abstract: We discuss the parameterized generalized inverse eigenvalue problem (PGIEP): For given matrices $A_i$, $B_i \in {\bf C}^{n \times n}$ ($i=0, 1, \ldots, n$), find complex numbers $c_i \in {\bf C}$ ($i=1, 2, \ldots, n$) such that the generalized eigenvalue problem $(A_0+\sum_{i=1}^nc_iA_i)x =\lambda (B_0+\sum_{i=1}^nc_iB_i)x$ has the prescribed eigenvalues $\lambda_1, \lambda_2,\ldots, \lambda_n$. We show that this problem is equivalent to a multiparameter eigenvalue problem if the given eigenvalues $\lambda_1, \lambda_2,\ldots, \lambda_n$ are distinct. Applying the theory about the multiparameter eigenvalue problem, we obtain sufficient conditions that guarantee the existence of a solution of the PGIEP. In addition, we propose a smooth LU decomposition for a matrix depending on several parameters and discuss its algebraic property. Based on these theoretical results, we present a numerical algorithm for solving the PGIEP and prove its locally quadratic convergence. Numerical implementations show that the n...