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On SuccessiveOverrelaxation Acceleration Of The Hermitian And SkewHermitian Splitting Iterations
 Department of Computer Science, Stanford University
, 2002
"... We further generalize the technique for constructing the Hermitian/skewHermitian splitting (HSS) iteration method for solving large sparse nonHermitian positive definite system of linear equations to the normal/skewHermitian (NS) splitting obtaining a class of normal/skewHermitian splitting (NSS ..."
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We further generalize the technique for constructing the Hermitian/skewHermitian splitting (HSS) iteration method for solving large sparse nonHermitian positive definite system of linear equations to the normal/skewHermitian (NS) splitting obtaining a class of normal/skewHermitian splitting (NSS) iteration methods. Theoretical analyses show that the NSS method converges unconditionally to the exact solution of the system of linear equations. Moreover, we derive an upper bound of the contraction factor of the NSS iteration which is dependent solely on the spectrum of the normal splitting matrix, and is independent of the eigenvectors of the matrices involved. We present a successive overrelaxation (SOR) acceleration scheme for the NSS iteration. Convergence conditions for this SOR scheme are derived under the assumption that the eigenvalues of the corresponding block Jacobi iteration matrix lie in certain regions in the complex plane. Numerical examples show that the SOR technique can significantly accelerate the convergence rate of the NSS iteration method.
Simultaneous preconditioning and symmetrization of nonsymmetric linear systems
, 2008
"... Motivated by the theory of selfduality which provides a variational formulation and resolution for non selfadjoint partial differential equations [6, 7], we propose new templates for solving large nonsymmetric linear systems. The method consists of combining a new scheme that simultaneously preco ..."
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Motivated by the theory of selfduality which provides a variational formulation and resolution for non selfadjoint partial differential equations [6, 7], we propose new templates for solving large nonsymmetric linear systems. The method consists of combining a new scheme that simultaneously preconditions and symmetrizes the problem, with various well known iterative methods for solving linear and symmetric problems. The approach seems to be efficient when dealing with certain illconditioned, and highly nonsymmetric systems. 1 Introduction and main results Many problems in scientific computing lead to systems of linear equations of the form, Ax = b where A ∈ R n×n is a nonsingular but sparse matrix, and b is a given vector in R n, (1) and various iterative methods have been developed for a fast and efficient resolution of such systems. The Conjugate Gradient Method (CG) which is the oldest and best known of the nonstationary iterative methods,