On HSS-based iteration methods for weakly nonlinear systems

Bai, Zhong‐Zhi; Yang, Xi

doi:10.1016/j.apnum.2009.06.005

Cited by 85 publications

(65 citation statements)

References 17 publications

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“…This method is based on the modified Newton method as outer iteration and the double PMHSS method as inner iteration. If the Jacobian matrices are real, there are many efficient methods, such as [1,13,14] and some deformed methods. In this paper, we have also established the convergence of the Newton-DPMHSS iteration.…”

Section: Discussionmentioning

confidence: 99%

MN-DPMHSS iteration method for systems of nonlinear equations with block two-by-two complex Jacobian matrices

2017

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Preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) method is an unconditionally convergent iteration method for solving large sparse complex symmetric systems of linear equation. Motivated by the PMHSS method, we develop a new method of solving a class of linear equations with block two-by-two complex coefficient matrix by introducing two coefficients, noted as DPMHSS. By making use of the DPMHSS iteration as the inner solver to approximately solve the Newton equations, (MN-DPMHSS) method for solving large systems of nonlinear equations. We analyze the local convergence properties under the Hölder continuous conditions, which is weaker than Lipschitz assumptions. Numerical results are given to confirm the effectiveness of our method.

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Section: Discussionmentioning

confidence: 99%

MN-DPMHSS iteration method for systems of nonlinear equations with block two-by-two complex Jacobian matrices

2017

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“…The HSS method is very efficient and robust for solving non-Hermitian positive definite systems of linear equations; see [20][21][22][23][24][25][26]. Hence, Yang and Wu [27] recently proposed a Uzawa-HSS method to solve nonsingular saddle-point problems by employing one-step of HSS iteration to approximate the solution of non-Hermitian positive definite sub-system involved in the Uzawa method; see Ref.…”

Section: Introductionmentioning

confidence: 99%

On semi-convergence of the Uzawa–HSS method for singular saddle-point problems

Yang

2015

Applied Mathematics and Computation

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“…Here, the system of nonlinear equations (1.1) is said to be Toeplitz weakly nonlinear if the linear term Ax is strongly dominant over the nonlinear term φ(x ) in certain norm and A is a Toeplitz matrix; see [1][2][3][4][5][6].…”

Section: Introductionmentioning

confidence: 99%

A class of iteration methods based on the HSS for Toeplitz systems of weakly nonlinear equations

Zhu

2015

Journal of Computational and Applied Mathematics

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For Toeplitz systems of weakly nonlinear equations, combining the separability and strong dominance between the linear and the nonlinear terms with the Hermitian and skew-Hermitian splitting (HSS) iteration technique, we establish two nonlinear composite iteration schemes, called Picard-cSSS and nonlinear cSSS-like iteration methods, which are based on a special case of the HSS, where the symmetric part H = 1 2 (A + A T ) is a centrosymmetric matrix and the skew-symmetric part H = 1 2 (A − A T ) is a skew-centrosymmetric matrix. The advantages of these methods are that they can transfer the linear sub-systems involved in inner iteration to two linear systems of half an order, besides, fast methods are available for computing the two half-steps involved in the inner iteration. Numerical results are provided, to further show that both Picard-cSSS and nonlinear cSSS-like iteration methods are feasible and effective.

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On HSS-based iteration methods for weakly nonlinear systems

Cited by 85 publications

References 17 publications

MN-DPMHSS iteration method for systems of nonlinear equations with block two-by-two complex Jacobian matrices

MN-DPMHSS iteration method for systems of nonlinear equations with block two-by-two complex Jacobian matrices

On semi-convergence of the Uzawa–HSS method for singular saddle-point problems

A class of iteration methods based on the HSS for Toeplitz systems of weakly nonlinear equations

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