On CSCS-based iteration methods for Toeplitz system of weakly nonlinear equations

Zhu, Mu-Zheng

doi:10.1016/j.cam.2011.04.038

Cited by 14 publications

(14 citation statements)

References 26 publications

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“…Based on the matrix multi-splitting technique, block and asynchronous two-stage methods are introduced by Bai et al [11]. The Picard circulant and skew-circulant splitting (Picard-CSCS) algorithm and the nonlinear CSCS-like iterative algorithm are presented by Zhu and Zhang [12], when the coefficient matrix A is a Teoplitz matrix. A class of lopsided Hermitian/skew-Hermitian splitting (LHSS) algorithms and a class of nonlinear LHSS-like algorithms are used by Zhu [6] to solve the large and sparse of weakly nonlinear systems.…”

Section: Introductionmentioning

confidence: 99%

An Efficient Iterative Method Based on Two-Stage Splitting Methods to Solve Weakly Nonlinear Systems

et al. 2019

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In this paper, an iterative method for solving large, sparse systems of weakly nonlinear equations is presented. This method is based on Hermitian/skew-Hermitian splitting (HSS) scheme. Under suitable assumptions, we establish the convergence theorem for this method. In addition, it is shown that any faster and less time-consuming two-stage splitting method that satisfies the convergence theorem can be replaced instead of the HSS inner iterations. Numerical results, such as CPU time, show the robustness of our new method. This method is easy, fast and convenient with an accurate solution.

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Section: Introductionmentioning

confidence: 99%

An Efficient Iterative Method Based on Two-Stage Splitting Methods to Solve Weakly Nonlinear Systems

et al. 2019

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“…The system of nonlinear equation (1.1) is said to be weakly nonlinear if the linear term Ax is strongly dominant over the nonlinear term Φ(x) in certain norm. For more details, see [5,10,12,25].…”

Section: Introductionmentioning

confidence: 99%

A Class of Preconditioned TGHSS-Based Iteration Methods for Weakly Nonlinear Systems

Zeng

2016

East Asian J. Appl. Math.

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In this paper, we first construct a preconditioned two-parameter generalized Hermitian and skew-Hermitian splitting (PTGHSS) iteration method based on the two-parameter generalized Hermitian and skew-Hermitian splitting (TGHSS) iteration method for non-Hermitian positive definite linear systems. Then a class of PTGHSS-based iteration methods are proposed for solving weakly nonlinear systems based on separable property of the linear and nonlinear terms. The conditions for guaranteeing the local convergence are studied and the quasi-optimal iterative parameters are derived. Numerical experiments are implemented to show that the new methods are feasible and effective for large scale systems of weakly nonlinear systems.

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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%

“…Hence, computational workloads and computer memory may be saved. For the Toeplitz system of weakly nonlinear (1.1), based on the fact that the coefficient matrix A is Toeplitz, Zhu and Zhang [6] presented the Picard-CSCS and the nonlinear CSCS-like iteration methods. In this paper, we study the HSS iteration method and Toeplitz matrix in depth and establish a new class of splitting iteration methods, called Picard-cSSS and nonlinear cSSS-like iteration methods, respectively, for solving the large scale Toeplitz system of weakly nonlinear equations (1.1).…”

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 CSCS-based iteration methods for Toeplitz system of weakly nonlinear equations

Cited by 14 publications

References 26 publications

An Efficient Iterative Method Based on Two-Stage Splitting Methods to Solve Weakly Nonlinear Systems

An Efficient Iterative Method Based on Two-Stage Splitting Methods to Solve Weakly Nonlinear Systems

A Class of Preconditioned TGHSS-Based Iteration Methods for Weakly Nonlinear Systems

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

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