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Volume 34, Issue 3
Positive Definite and Semi-Definite Splitting Methods for Non-Hermitian Positive Definite Linear Systems

Na Huang & Changfeng Ma

DOI: 10.4208/jcm.1511-m2015-0299

J. Comp. Math., 34 (2016), pp. 300-316.

Published online: 2016-06

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  • Abstract

In this paper, we further generalize the technique for constructing the normal (or positive definite) and skew-Hermitian splitting iteration method for solving large sparse non-Hermitian positive definite system of linear equations. By introducing a new splitting, we establish a class of efficient iteration methods, called positive definite and semi-definite splitting (PPS) methods, and prove that the sequence produced by the PPS method converges unconditionally to the unique solution of the system. Moreover, we propose two kinds of typical practical choices of the PPS method and study the upper bound of the spectral radius of the iteration matrix. In addition, we show the optimal parameters such that the spectral radius achieves the minimum under certain conditions. Finally, some numerical examples are given to demonstrate the effectiveness of the considered methods.

  • Keywords

Linear systems, Splitting method, Non-Hermitian matrix, Positive definite matrix, Positive semi-definite matrix, Convergence analysis.

  • AMS Subject Headings

65F10, 65F30, 65F50.

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

hna@cau.edu.cn (Na Huang)

macf@fjnu.edu.cn (Changfeng Ma)

  • BibTex
  • RIS
  • TXT
@Article{JCM-34-300, author = {Huang , Na and Ma , Changfeng}, title = {Positive Definite and Semi-Definite Splitting Methods for Non-Hermitian Positive Definite Linear Systems}, journal = {Journal of Computational Mathematics}, year = {2016}, volume = {34}, number = {3}, pages = {300--316}, abstract = {

In this paper, we further generalize the technique for constructing the normal (or positive definite) and skew-Hermitian splitting iteration method for solving large sparse non-Hermitian positive definite system of linear equations. By introducing a new splitting, we establish a class of efficient iteration methods, called positive definite and semi-definite splitting (PPS) methods, and prove that the sequence produced by the PPS method converges unconditionally to the unique solution of the system. Moreover, we propose two kinds of typical practical choices of the PPS method and study the upper bound of the spectral radius of the iteration matrix. In addition, we show the optimal parameters such that the spectral radius achieves the minimum under certain conditions. Finally, some numerical examples are given to demonstrate the effectiveness of the considered methods.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1511-m2015-0299}, url = {http://global-sci.org/intro/article_detail/jcm/9797.html} }
TY - JOUR T1 - Positive Definite and Semi-Definite Splitting Methods for Non-Hermitian Positive Definite Linear Systems AU - Huang , Na AU - Ma , Changfeng JO - Journal of Computational Mathematics VL - 3 SP - 300 EP - 316 PY - 2016 DA - 2016/06 SN - 34 DO - http://doi.org/10.4208/jcm.1511-m2015-0299 UR - https://global-sci.org/intro/article_detail/jcm/9797.html KW - Linear systems, Splitting method, Non-Hermitian matrix, Positive definite matrix, Positive semi-definite matrix, Convergence analysis. AB -

In this paper, we further generalize the technique for constructing the normal (or positive definite) and skew-Hermitian splitting iteration method for solving large sparse non-Hermitian positive definite system of linear equations. By introducing a new splitting, we establish a class of efficient iteration methods, called positive definite and semi-definite splitting (PPS) methods, and prove that the sequence produced by the PPS method converges unconditionally to the unique solution of the system. Moreover, we propose two kinds of typical practical choices of the PPS method and study the upper bound of the spectral radius of the iteration matrix. In addition, we show the optimal parameters such that the spectral radius achieves the minimum under certain conditions. Finally, some numerical examples are given to demonstrate the effectiveness of the considered methods.

Huang , Na and Ma , Changfeng. (2016). Positive Definite and Semi-Definite Splitting Methods for Non-Hermitian Positive Definite Linear Systems. Journal of Computational Mathematics. 34 (3). 300-316. doi:10.4208/jcm.1511-m2015-0299
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