Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems

By Zhong-Zhi Bai, Gene H. Golub, Jianyu Pan. Numerische Mathematik, 2004

555 citations recorded by the source corpus.

Subfields: Atomic and Molecular Physics, and Optics, Computational Mechanics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/938f8b17500bc2f60d8f71cadc1fe6763c914683

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In Atomic and Molecular Physics, and Optics

In Computational Mechanics

In Computational Theory and Mathematics

Citations in the corpus

Modulus‐based matrix splitting iteration methods for linear complementarity problems (2010, background): “and F= tridiag(0,1,−1)∈Rm×m being tridiagonal matrices, and ⊗ the Kronecker product symbol; see [19].”
Hermitian and Skew-Hermitian Splitting Methods for Non-Hermitian Positive Definite Linear Systems (2002, methodology): “Recently, the preconditioned HSS iterations and the extension of the HSS method to positive definite and positive semidefinite linear systems have been studied in [6,10,4,11,13,8,3].”
A Generalized Preconditioned MHSS Method for a Class of Complex Symmetric Linear Systems (2013, methodology): “In [7, 10], one can see the preconditioned HSS (PHSS) method and it is systematic analysis.”
A Relaxed HSS Preconditioner for Saddle Point Problems from Meshfree Discretization (2013, background): “A variety of preconditioners have been proposed and studied in many papers, such as block diagonal and block triangular preconditioners [7,19,20,24], constraint preconditioners [8,25], HSS-based preconditioners [3, 6, 12, 13, 30], dimensional split preconditioners [15, 16], H−matrix preconditioners ”
A Preconditioner for Generalized Saddle Point Problems (2004, background): “Some results in this direction can be found in [2, 3, 6].”
On preconditioned iteration methods for complex linear systems (2015, methodology): “The block two-by-two linear system (2) may be efficiently solved in real arithmetics by Krylov subspace iteration methods such as the preconditioned GMRES [18] and QMR [19–21], by alternating splitting iterationmethods such as PHSS [22] and PMHSS [13], or by C-to-R iterationmethods [17], for which h”
On generalized successive overrelaxation methods for augmented linear systems (2005, background): “See [ 7 ,8] and references therein for other efficient alternatives about the augmented linear system (1.1).”
Block Triangular and Skew-Hermitian Splitting Methods for Positive-Definite Linear Systems (2005, methodology): “We remark that unlike the HSS and the PHSS iteration methods in [3, 4], the BTSS iteration methods are divergent for any α > 0 when they are applied to solve the saddle-point problem”
Regularized HSS iteration methods for saddle-point linear systems (2017, methodology): “Here we should point out again that the optimal iteration parameterα for the PHSS iterationmethod is computed by the analytic formula derived in [5,8], which is, however, not computable when m is larger than or equal to 128 due to insufficient computer memory in computing the extremal singular value”
ON NEWTON-HSS METHODS FOR SYSTEMS OF NONLINEAR EQUATIONS WITH POSITIVE-DEFINITE JACOBIAN MATRICES (2010, methodology): “For notational convenience, we also denote by S(F ′(x)) := 1 2 (F ′(x)− F ′(x)∗) the skew-Hermitian part of F ′(x); see [7, 12, 15, 18].”