The Shifted Sine Preconditioner for Real Symmetric Positive

Mathematical Theory and Applications ›› 2017, Vol. 37 ›› Issue (1): 1-6.

The Shifted Sine Preconditioner for Real Symmetric Positive

Liu Zhongyun¹ ，Wu Nianci¹ ，Qin Xiaorong¹，Zhang Yulin²

1.School of Mathematics and Statistics,Changsha University of Science and Technology,Changsha 410004,China; 2.Centro de Matema′tica,Universidade do Minho,4710-057Braga,Portugal

Online:2017-03-30 Published:2020-09-24

Abstract

Abstract:

This paper studies the solution of real symmetric positive definite Toeplitz matrices by the preconditioned conjugate gradient method.Based on the fact that the real symmetric Toeplitz matrix admits a triangular transform splitting(TTS),we propose the shifted Sine preconditioner TS,the spectral properties of the preconditioned matrix are analyzed,and the arithmetic complexity is discussed.The numerical experiments show that our preconditioner is more effective than T.Chan＇s preconditioner[2].

Key words: Toeplitz matrix, Triangular transform splitting, Shifted Sine preconditioner, PCG

Liu Zhongyun , Wu Nianci , Qin Xiaorong, Zhang Yulin. The Shifted Sine Preconditioner for Real Symmetric Positive[J]. Mathematical Theory and Applications, 2017, 37(1): 1-6.

[1]	Liu Zhongyun, Zhang Fang. Extrapolated CSCS Iterations of Large Continuous Sylvester Equations [J]. Mathematical Theory and Applications, 2022, 42(2): 120-126.
[2]	Liu Zhongyun, Xu Weijin, Chen Siheng, Zhang Yulin. A New Preconditioning Technique for Hermitian Toeplitz Systems [J]. Mathematical Theory and Applications, 2018, 38(3-4): 50-58.
[3]	Xu Wenhua. Parallel Algorithm to the Inverse of Cyclic Toeplitz Matrices [J]. Mathematical Theory and Applications, 2016, 36(4): 125-128.

The Shifted Sine Preconditioner for Real Symmetric Positive

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