Hermitian and Skew-Hermitian Splitting Methods for Non-Hermitian Positive Definite Linear Systems

By Zhong-Zhi Bai, Gene H. Golub, Michael Kwok-Po Ng. SIAM Journal on Matrix Analysis and Applications, 2002

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We study efficient iterative methods for the large sparse non-Hermitian positive definite system of linear equations based on the Hermitian and skew-Hermitian splitting of the coefficient matrix. These methods include a Hermitian/skew-Hermitian splitting (HSS) iteration and its inexact variant, the inexact Hermitian/skew-Hermitian splitting (IHSS) iteration, which employs some Krylov subspace methods as its inner iteration processes at each step of the outer HSS iteration. Theoretical analyses show that the HSS method converges unconditionally to the unique solution of the system of linear equations. Moreover, we derive an upper bound of the contraction factor of the HSS iteration which is dependent solely on the spectrum of the Hermitian part and is independent of the eigenvectors of the matrices involved. Numerical examples are presented to illustrate the effectiveness of both HSS and IHSS iterations. In addition, a model problem of a three-dimensional convection-diffusion equation is used to illustrate the advantages of our methods.

Subfields: Atomic and Molecular Physics, and Optics, Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/a5ec59f8bfd7a301c733b90646c99acebabe4e2b

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Citations in the corpus

A Generalized Preconditioned MHSS Method for a Class of Complex Symmetric Linear Systems (2013, methodology): “To overcome this disadvantage and improve the efficiency of the GPMHSS iteration method, similar to the introduced approach in [9] for inexact implementation of HSS iteration method, we can solve the two inner linear systems iteratively.”
A Preconditioner for Generalized Saddle Point Problems (2004, methodology): “In this paper we propose a new approach for preconditioning generalized saddle point problems based on an alternating symmetric/skew-symmetric splitting [2] applied to (1.”
Modulus‐based matrix splitting iteration methods for linear complementarity problems (2010, background): “Then, for all x ∈Rn , ‖x‖P,#=‖Px‖# defines a vector norm on Rn; see [15, 17].”
Regularized HSS iteration methods for saddle-point linear systems (2017, methodology): “We emphasize that the fact that IRHSS iteration method is convergent is not obvious, as the convergence theory presented in this paper only covers the exact iteration method, RHSS; however, the theory developed in [7] for IHSS can be adapted to account for the observed convergence of”
Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems (2004, methodology): “This results in an inexact preconditioned Hermitian/skew-Hermitian splitting (IPHSS) iteration method for the non-Hermitian positive semidefinite system of linear equations (2), and has been previously discussed in [ 2 ].”
On semi-convergence of parameterized Uzawa methods for singular saddle point problems☆ (2009, background): “More related works can be found in [4,6,9,3].”
Modified HSS iteration methods for a class of complex symmetric linear systems (2010, neutral): “We refer to [3,4] for actual implementations and detailed analyses of inexact variants of the HSS iteration method.”
A dimensional split preconditioner for Stokes and linearized Navier-Stokes equations (2011, neutral): “E-mail addresses: benzi@mathcs.emory.edu (M. Benzi), xpguo@math.ecnu.edu.cn (X.-P. Guo).”
Block preconditioning of real-valued iterative algorithms for complex linear systems (2007, background): “HSS M) < 1 for all α > 0, where (•) denotes the spectral radius (Bai et al., 2003).”
ON HERMITIAN AND SKEW-HERMITIAN SPLITTING ITERATION METHODS FOR CONTINUOUS SYLVESTER EQUATIONS * (2010, methodology): “This Hermitian and skew-Hermitian splitting (HSS) iteration method is a matrix variant of the HSS iteration method firstly proposed in [6] for solving systems of linear equations, which are in spirit analogous to the ADI iteration methods [11, 24, 33, 40].”
A Relaxed HSS Preconditioner for Saddle Point Problems from Meshfree Discretization (2013, methodology): “The Relaxed HSS (RHSS) Preconditioner In this section, we first review briefly the HSS iterative method and introduce the HSS preconditioner, for details; see [4,13,30].”
Neurobiology of Aging: Contents (2005, positive): “It was shown in [29] that when H is positive definite, the choice”