Preconditioned progressive iterative approximation for tensor product Bézier patches
作者：
Liu, Chengzhi;Liu, Zhongyun;Han, Xuli
期刊：
Mathematics and Computers in Simulation ,2021年185:372-383 ISSN：0378-4754
通讯作者：
Chengzhi Liu
作者机构：
[Liu, Chengzhi] Hunan Univ Humanities Sci & Technol, Sch Math & Finance, Loudi, Peoples R China.;[Liu, Zhongyun] Changsha Univ Sci & Technol, Sch Math & Stat, Changsha, Peoples R China.;[Han, Xuli] Southern Univ Sci & Technol, Dept Math, Shenzhen, Peoples R China.
通讯机构：
[Chengzhi Liu] S;School of Mathematics and Finance, Hunan University of Humanities, Science and Technology, Loudi, China
关键词：
Numerical methods;Tensors;Convergence rates;Iterative approximations;Preconditioners;Tensor products;Iterative methods
摘要：
Based on the diagonally compensated reduction, the preconditioned progressive iterative approximation (PPIA) for tensor product Bézier patches is presented. Due to the effectiveness of the preconditioner, the convergence rate of progressive iterative approximation (PIA) is accelerated significantly. To improve the robustness and reduce the computational complexity of PPIA, the inexact PPIA format for tensor product Bézier patches is presented. Several numerical examples are presented to illustrate the effectiveness of the proposed methods. © 2021 International Association for Mathematics and Computers in Simulation (IMACS)
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英文
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Stationary splitting iterative methods for the matrix equationAXB=C
作者：
Liu, Zhongyun;Li, Zhen;Ferreira, Carla;Zhang, Yulin*
期刊：
Applied Mathematics and Computation ,2020年378:125195 ISSN：0096-3003
通讯作者：
Zhang, Yulin
作者机构：
[Li, Zhen; Liu, Zhongyun] Changsha Univ Sci & Technol, Sch Math & Stat, Changsha 410076, Peoples R China.;[Ferreira, Carla; Zhang, Yulin] Univ Minho, Ctr Matemat, P-4710057 Braga, Portugal.
通讯机构：
[Zhang, Yulin] U;Univ Minho, Ctr Matemat, P-4710057 Braga, Portugal.
关键词：
Hermitian positive definite;H-matrices;Stationary splitting iteration;Induced splitting;Curves fitting
摘要：
Stationary splitting iterative methods for solving AXB = C are considered in this paper. The main tool to derive our new method is the induced splitting of a given nonsingular matrix A = M - N by a matrix H such that (I - H)(-1) exists. Convergence properties of the proposed method are discussed and numerical experiments are presented to illustrate its computational efficiency and the effectiveness of some preconditioned variants. In particular, for certain surface fitting applications our method is much more efficient than the progressive iterative approximation (PIA), a conventional iterative method often used in computer aided geometric design (CAGD). (C) 2020 Elsevier Inc. All rights reserved.
语种：
英文
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Progressive iterative approximation with preconditioners
作者：
Liu, Chengzhi;Liu, Zhongyun*
期刊：
Mathematics ,2020年8(9):1503 ISSN：2227-7390
通讯作者：
Liu, Zhongyun
作者机构：
[Liu, Chengzhi] Cent South Univ, Sch Math & Stat, Changsha 410083, Peoples R China.;[Liu, Chengzhi] Hunan Univ Humanities Sci & Technol, Sch Math & Finance, Loudi 417000, Peoples R China.;[Liu, Zhongyun] Changsha Univ Sci & Technol, Sch Math & Stat, Changsha 410076, Peoples R China.
通讯机构：
[Liu, Zhongyun] C;Changsha Univ Sci & Technol, Sch Math & Stat, Changsha 410076, Peoples R China.
关键词：
PIA;WPIA;diagonally compensated reduction;preconditioner;inexact solver;convergence
摘要：
The progressive iterative approximation (PIA) plays an important role in curve and surface fitting. By using the diagonally compensated reduction of the collocation matrix, we propose the preconditioned progressive iterative approximation (PPIA) to improve the convergence rate of PIA. For most of the normalized totally positive bases, we show that the presented PPIA can accelerate the convergence rate significantly in comparison with the weighted progressive iteration approximation (WPIA) and the progressive iterative approximation with different weights (DWPIA). Furthermore, we propose an inexact variant of the PPIA (IPPIA) to reduce the computational complexity of the PPIA. We introduce the inexact solver of the preconditioning system by employing some state-of-the-art iterative methods. Numerical results show that both the PPIA and the IPPIA converge faster than the WPIA and DWPIA, while the elapsed CPU times of the PPIA and IPPIA are less than those of the WPIA and DWPIA. © 2020 by the authors.
语种：
英文
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Fast solvers for tridiagonal Toeplitz linear systems
作者：
Liu, Zhongyun;Li, Shan;Yin, Yi;Zhang, Yulin*
期刊：
Computational and Applied Mathematics ,2020年39(4):1-10 ISSN：2238-3603
通讯作者：
Zhang, Yulin
作者机构：
[Liu, Zhongyun; Li, Shan; Zhang, Yulin] Changsha Univ Sci & Technol, Sch Math & Stat, Changsha 410076, Peoples R China.;[Yin, Yi] Hunan Coll Informat, Dept Basic Courses, Changsha 410200, Peoples R China.;[Zhang, Yulin] Univ Minho, Ctr Matemat, P-4710057 Braga, Portugal.
通讯机构：
[Zhang, Yulin] U;Univ Minho, Ctr Matemat, P-4710057 Braga, Portugal.
关键词：
Tridiagonal Toeplitz matrices;Diagonally dominant;Schur complement;Block LU factorization;Pivoting;15A23;15B05;65F05;65F10
摘要：
Let A be a tridiagonal Toeplitz matrix denoted by A= Tritoep (β, α, γ). The matrix A is said to be: strictly diagonally dominant if | α| > | β| + | γ| , weakly diagonally dominant if | α| ≥ | β| + | γ| , subdiagonally dominant if | β| ≥ | α| + | γ| , and superdiagonally dominant if | γ| ≥ | α| + | β|. In this paper, we consider the solution of a tridiagonal Toeplitz system Ax= b, where A is subdiagonally dominant, superdiagonally dominant, or weakly diagonally dominant, respectively. We first consider the case of A being subdiagonally dominant. We transform A into a block 2 × 2 matrix by an elementary transformation and then solve such a linear system using the block LU factorization. Compared with the LU factorization method with pivoting, our algorithm takes less flops, and needs less memory storage and data transmission. In particular, our algorithm outperforms the LU factorization method with pivoting in terms of computing efficiency. Then, we deal with superdiagonally dominant and weakly diagonally dominant cases, respectively. Numerical experiments are finally given to illustrate the effectiveness of our algorithms. © 2020, SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional.
语种：
英文
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A multi-step Smith-inner-outer iteration algorithm for solving coupled continuous Markovian jump Lyapunov matrix equations
作者：
Tian, Zhaolu* ;Wang, Junxin;Dong, Yinghui;Liu, Zhongyun
期刊：
Journal of the Franklin Institute ,2020年357(6):3656-3679 ISSN：0016-0032
通讯作者：
Tian, Zhaolu
作者机构：
[Tian, Zhaolu; Wang, Junxin] Shanxi Univ Finance & Econ, Coll Appl Math, Taiyuan 030006, Peoples R China.;[Dong, Yinghui] Suzhou Univ Sci & Technol, Dept Math & Phys, Suzhou, Peoples R China.;[Liu, Zhongyun] Changsha Univ Sci & Technol, Sch Math & Stat, Changsha 410114, Peoples R China.
通讯机构：
[Tian, Zhaolu] S;Shanxi Univ Finance & Econ, Coll Appl Math, Taiyuan 030006, Peoples R China.
关键词：
Matrix algebra;Comparison result;Convergence conditions;Inner-outer iterations;Iteration algorithms;Lyapunov matrix equations;Markovian jumps;Numerical algorithms;Sylvester matrix equations;Iterative methods
摘要：
In this paper, combining the Smith method [30] with inner-outer (IO) iteration [4], a multi-step Smith-IO (MSIO) iteration algorithm for solving the Sylvester matrix equation and coupled Lyapunov matrix equations (CLMEs) is proposed. Moreover, some convergence conditions and comparison results are also given for the MSIO iteration algorithm. Finally, several numerical examples are used to verify the effectiveness of the proposed numerical algorithms. © 2020 The Franklin Institute
语种：
英文
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The eigen-structures of real (skew) circulant matrices with some applications
作者：
Liu, Zhongyun;Chen, Siheng;Xu, Weijin;Zhang, Yulin*
期刊：
Computational and Applied Mathematics ,2019年38(4):1-13 ISSN：2238-3603
通讯作者：
Zhang, Yulin
作者机构：
[Chen, Siheng; Xu, Weijin; Liu, Zhongyun] Changsha Univ Sci & Technol, Sch Math & Stat, Changsha 410076, Hunan, Peoples R China.;[Zhang, Yulin] Univ Minho, Ctr Matemat, P-4710057 Braga, Portugal.
通讯机构：
[Zhang, Yulin] U;Univ Minho, Ctr Matemat, P-4710057 Braga, Portugal.
关键词：
Real Schur form;Real circulant matrices;Real skew-circulant matrices;Real Toeplitz matrices;CSCS iteration
摘要：
The circulant matrices and skew-circulant matrices are two special classes of Toeplitz matrices and play vital roles in the computation of Toeplitz matrices. In this paper, we focus on real circulant and skew-circulant matrices. We first investigate their real Schur forms, which are closely related to the family of discrete cosine transform (DCT) and discrete sine transform (DST). Using those real Schur forms, we then develop some fast algorithms for computing real circulant, skew-circulant, and Toeplitz matrix-real vector multiplications. Also, we develop a DCT-DST version of circulant and skew-circulant splitting (CSCS) iteration for real positive definite Toeplitz systems. Compared with the fast Fourier transform (FFT) version of CSCS iteration, the DCT-DST version is more efficient and saves a half storage. Numerical experiments are presented to illustrate the effectiveness of our method. © 2019, SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional.
语种：
英文
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Some remarks on Jacobi and Gauss–Seidel-type iteration methods for the matrix equation AXB=C
作者：
Liu, Zhongyun* ;Zhou, Yang;Zhang, Yuelan;Lin, Lu;Xie, Dongxiu
期刊：
Applied Mathematics and Computation ,2019年354:305-307 ISSN：0096-3003
通讯作者：
Liu, Zhongyun
作者机构：
[Liu, Zhongyun; Zhou, Yang; Zhang, Yuelan] Changsha Univ Sci & Technol, Sch Math & Stat, Changsha 410076, Hunan, Peoples R China.;[Lin, Lu] Xiamen Univ, Sch Math Sci, Xiamen 361005, Peoples R China.;[Xie, Dongxiu] Beijing Informat Sci & Technol Univ, Sch Sci, Beijing 100192, Peoples R China.
通讯机构：
[Liu, Zhongyun] C;Changsha Univ Sci & Technol, Sch Math & Stat, Changsha 410076, Hunan, Peoples R China.
关键词：
Matrix equation;Classical splitting;Convergence;Norm
摘要：
Tian, et al. proposed in [5] several Jacobi and Gauss-Seidel-type iterative methods for solving matrix equation AXB = C. Those methods were demonstrated to be effective by the given numerical experiments. However, we find that there is a technical error in the proof of the main theorem (Theorem 3.3). In this note we first show this erratum by an example. Then we establish a new convergence theorem which contains the Theorem 3.3 in [5] as a special case. (C) 2019 Elsevier Inc. All rights reserved.
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英文
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全正矩阵的多水平预处理子
作者：
刘仲云;周孜
期刊：
邵阳学院学报(自然科学版) ,2019年16(2):1-5 ISSN：1672-7010
作者机构：
长沙理工大学 数学与统计学院,湖南 长沙,410114;[周孜; 刘仲云] 长沙理工大学
关键词：
全正矩阵;Schur补;多水平预处理子;法方程
摘要：
研究CG型方法解全正线性方程组。基于全正矩阵的Schur补也是全正矩阵这一性质,文中通过近似Schur补的方法构造多水平预处理子。数值实验表明预处理矩阵有较好的谱聚集性质,共轭梯度法求解预处理线性方程组有很好的收敛性。
语种：
中文
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两步渐进迭代逼近法和收敛性
作者：
刘仲云;李杰
期刊：
邵阳学院学报(自然科学版) ,2019年16(06):1-6 ISSN：1672-7010
作者机构：
长沙理工大学数学与统计学院,湖南长沙,410114;[李杰; 刘仲云] 长沙理工大学
关键词：
曲线拟合;控制多边形;加权渐进迭代逼近法;两步渐进迭代逼近法;收敛性
摘要：
文中提出了一种更新拟合曲线控制多边形的新方法。该方法通过使用当前控制点和上一步控制多边形的控制顶点以及拟合误差来获得新的控制多边形,证明了对于任意标准化的全正基,两步渐进迭代逼近法比加权渐进迭代逼近法具有更快的收敛速度。数值实验表明了新方法的有效性。
语种：
中文
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The general inner-outer iteration method based on regular splittings for the PageRank problem
作者：
Tian, Zhaolu* ;Liu, Yong;Zhang, Yan;Liu, Zhongyun;Tian, Maoyi*
期刊：
Applied Mathematics and Computation ,2019年356:479-501 ISSN：0096-3003
通讯作者：
Tian, Zhaolu;Tian, Maoyi
作者机构：
[Tian, Zhaolu] Shanxi Univ Finance & Econ, Coll Appl Math, Taiyuan 030006, Shanxi, Peoples R China.;[Liu, Yong] Shanxi Univ, Inst Loess Plateau, Taiyuan 030006, Shanxi, Peoples R China.;[Tian, Maoyi; Zhang, Yan] Shandong Univ Sci & Technol, Geomat Coll, Qingdao 266590, Shandong, Peoples R China.;[Liu, Zhongyun] Changsha Univ Sci & Technol, Sch Math & Stat, Changsha 410114, Hunan, Peoples R China.
通讯机构：
[Tian, Zhaolu; Tian, Maoyi] S;Shanxi Univ Finance & Econ, Coll Appl Math, Taiyuan 030006, Shanxi, Peoples R China.;Shandong Univ Sci & Technol, Geomat Coll, Qingdao 266590, Shandong, Peoples R China.
关键词：
PageRank;Inner-outer iteration;Regular splitting;Preconditioner;Convergence
摘要：
In this paper, combined the regular splittings of the coefficient matrix I - alpha P with the inner-outer iteration framework [9], a general inner-outer (GIO) iteration method is presented for solving the PageRank problem. Firstly, the AOR and modified AOR (MAOR) methods for solving the PageRank problem are constructed, and several comparison results are also given. Next, the GIO iteration scheme is developed, and its overall convergence is analyzed in detail. Furthermore, the preconditioner derived from the GIO iteration can be used to accelerate the Krylov subspace methods, such as GMRES method. Finally, some numerical experiments on several PageRank problems are provided to illustrate the efficiency of the proposed algorithm. (C) 2019 Elsevier Inc. All rights reserved.
语种：
英文
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特殊块三对角Toeplitz线性方程组的精化迭代法及收敛性
作者：
刘仲云;李莉
期刊：
邵阳学院学报(自然科学版) ,2019年16(3):1-5 ISSN：1672-7010
作者机构：
长沙理工大学 数学与统计学院,湖南 长沙,410114;[李莉; 刘仲云] 长沙理工大学
关键词：
全正矩阵;块三对角Toeplitz矩阵;精化迭代法;收敛性
摘要：
基于迭代精化的基本思想,利用系数矩阵的特殊结构,提出了求解特殊块三对角Toeplitz线性方程组的方法-精化迭代法,它大大提高了解的精确值。该方法的特点是方法简单、稳定性好、解精度高、收敛速度快。最后,将此方法应用于三次均匀B样条曲面拟合,数值实验体现了其高效性。
语种：
中文
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Trigonometric transform splitting methods for real symmetric Toeplitz systems
作者：
Liu, Zhongyun* ;Wu, Nianci;Qin, Xiaorong;Zhang, Yulin
期刊：
Computers & Mathematics with Applications ,2018年75(8):2782-2794 ISSN：0898-1221
通讯作者：
Liu, Zhongyun
作者机构：
[Liu, Zhongyun; Wu, Nianci; Qin, Xiaorong] Changsha Univ Sci & Technol, Sch Math & Stat, Changsha 410076, Hunan, Peoples R China.;[Zhang, Yulin] Univ Minho, Ctr Matemat, P-4710057 Braga, Portugal.
通讯机构：
[Liu, Zhongyun] C;Changsha Univ Sci & Technol, Sch Math & Stat, Changsha 410076, Hunan, Peoples R China.
关键词：
Sine transform;Cosine transform;Matrix splitting;Iterative methods;Real Toeplitz matrices
摘要：
In this paper we study efficient iterative methods for real symmetric Toeplitz systems based on the trigonometric transformation splitting (TTS) of the real symmetric Toeplitz matrix A. Theoretical analyses show that if the generating function f of the n×n Toeplitz matrix A is a real positive even function, then the TTS iterative methods converge to the unique solution of the linear system of equations for sufficient large n. Moreover, we derive an upper bound of the contraction factor of the TTS iteration which is dependent solely on the spectra of the two TTS matrices involved. Different from the CSCS iterative method in Ng (2003) in which all operations counts concern complex operations when the DFTs are employed, even if the Toeplitz matrix A is real and symmetric, our method only involves real arithmetics when the DCTs and DSTs are used. The numerical experiments show that our method works better than CSCS iterative method and much better than the positive definite and skew-symmetric splitting (PSS) iterative method in Bai et al. (2005) and the symmetric Gauss–Seidel (SGS) iterative method. © 2018 Elsevier Ltd
语种：
英文
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Hermitian Toeplitz线性方程组的新预处理方法
作者：
刘仲云;徐伟进;陈思恒;张育林
期刊：
Shuxue Lilun yu Yingyong ,2018年38(03):50-58 ISSN：1006-8074
作者机构：
长沙理工大学数学与统计学院,湖南长沙,410004;Minho大学数学中心葡萄牙,4710-057;[张育林] Minho大学;[陈思恒; 徐伟进; 刘仲云] 长沙理工大学
关键词：
Hermitian Toeplitz矩阵;预处理共轭梯度方法
摘要：
本文研究解Hermitian Toeplitz线性方程组Ax =b的预处理共轭梯度法.基于Hermitian Toeplitz 矩阵可通过酉相似转化为一个实Toeplitz矩阵与一个Hankel矩阵的和(UAU*=T+H)的结论,我们首先将Ax=b转化为实线性方程组(T+H)[x1,x2]=[b1,b2].然后,我们提出一个新预处理子来求解这两个方程组.特剐地,我们采用DCT和DST求解,只涉及到实运算.我们分析预处理矩阵的谱性质,并讨论每步迭代的计算复杂度.数值实验表明该预处理子是有效的.
语种：
中文
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实对称正定Toeplitz矩阵的带位移的Sine预处理子
作者：
刘仲云;吴念慈;秦小蓉;张育林
期刊：
Shuxue Lilun yu Yingyong ,2017年37(1):1-6 ISSN：1006-8074
作者机构：
长沙理工大学数学与统计学院,湖南长沙,410004;Minho大学数学中心,葡萄牙,4710-057;[张育林] Minho大学;[秦小蓉; 吴念慈; 刘仲云] 长沙理工大学
关键词：
Toeplitz矩阵;三角变换分裂;带位移的Sine预处理子;预处理共轭梯度方法
摘要：
本文研究了求解实对称正定Toeplitz线性方程组的预处理共轭梯度法.基于实对称Toeplitz矩阵都有一个三角变换分裂（TTS）的事实,我们提出了带位移的Sine预处理子TS,分析了预处理矩阵的谱性质,并讨论了每步迭代的计算复杂度.数值实验表明该预处理子比T.Chan预处理子[2]更有效.
语种：
中文
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$H{\rm{ - }}$Preconditioner for $H{\rm{ - }}$matrices
作者：
刘仲云;于静;张艳;张育林
期刊：
应用数学 ,2017年30(1):144-150 ISSN：1001-9847
作者机构：
School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, 410004;Centro de Matematica, Universidade do Minho, Portugal, Braga, 4710-057;[刘仲云; 于静; 张艳] School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, 410004;[张育林] Centro de Matematica, Universidade do Minho, Portugal, Braga, 4710-057
关键词：
严格对角占优矩阵;H矩阵;Jacobi分裂;预处理子;预处理共轭梯度方法(PCG)
摘要：
设$A$为一实对称正定的严格对角占优矩阵. 设$A = D - B$ 为$A$ 的Jacobi分裂.为了求解线性方程组${A_x} = b$, 在新提出的预处理子的基础上,我们采用预处理共轭梯度方法(PCG)来求解该问题. 新提出的预处理子${p_\upsilon } = D + v\upsilon {\upsilon ^T}$, 其中$\upsilon = {\left| B \right|_e}$, $e = {\left({1, \cdots 1} \right)^{\rm{T}}}$, $v = {\upsilon ^{\rm{T}}}B\upsilon /||\upsilon ||_2^4$,且$v$ 使$||c\upsilon {\upsilon ^{\rm{T}}} - B|{|_F}$ 达到极小. 我们得到了预处理矩阵$P_\upsilon ^{ - 1}$$A$ 特征值的上下界,它的界比JIN提出的预处理子的界简单紧凑. 数值结果表明我们的预处理子的有效性.
语种：
中文
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The Jacobi and Gauss–Seidel-type iteration methods for the matrix equation AXB=C
作者：
Tian, Zhaolu* ;Tian, Maoyi;Liu, Zhongyun;Xu, Tongyang
期刊：
Applied Mathematics and Computation ,2017年292(C):63-75 ISSN：0096-3003
通讯作者：
Tian, Zhaolu
作者机构：
[Tian, Zhaolu] Taiyuan Univ Technol, Coll Math, Taiyuan 030024, Peoples R China.;[Tian, Maoyi] Shandong Univ Sci & Technol, Geomat Coll, Qingdao 266590, Peoples R China.;[Liu, Zhongyun] Changsha Univ Sci & Technol, Sch Math & Stat, Changsha 410114, Hunan, Peoples R China.;[Xu, Tongyang] Shanxi Univ Finance & Econ, Sch Informat Management, Taiyuan 030006, Peoples R China.
通讯机构：
[Tian, Zhaolu] T;Taiyuan Univ Technol, Coll Math, Taiyuan 030024, Peoples R China.
关键词：
Jacobi-type iteration;Gauss Seidel-type iteration;Preconditioned;Kronecker products
摘要：
In this paper, the Jacobi and Gauss-Seidel-type iteration methods are proposed for solving the matrix equation AXB = C, which are based on the splitting schemes of the matrices A and B. The convergence and computational cost of these iteration methods are discussed. Furthermore, we give the preconditioned Jacobi and Gauss-Seidel-type iteration methods. Numerical examples are given to demonstrate the efficiency of these methods proposed in this paper. (C) 2016 Elsevier Inc. All rights reserved.
语种：
英文
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The Shifted Classical Circulant and Skew Circulant Splitting Iterative Methods for Toeplitz Matrices
作者：
Liu, Zhongyun* ;Qin, Xiaorong;Wu, Nianci;Zhang, Yulin
期刊：
CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES ,2017年60(4):807-815 ISSN：0008-4395
通讯作者：
Liu, Zhongyun
作者机构：
[Liu, Zhongyun] Changsha Univ Sci & Technol, Sch Math & Stat, Changsha 410076, Hunan, Peoples R China.;[Wu, Nianci; Qin, Xiaorong; Zhang, Yulin] Univ Minho, Ctr Matemat, P-4710057 Braga, Portugal.
通讯机构：
[Liu, Zhongyun] C;Changsha Univ Sci & Technol, Sch Math & Stat, Changsha 410076, Hunan, Peoples R China.
关键词：
15A23;65F10;65F15;Hermitian positive definite;CSCS splitting;Gauss-Seidel splitting;iterative method;Toeplitz matrix
摘要：
<jats:title>Abstract</jats:title><jats:p>It is known that every Toeplitz matrix <jats:italic>T</jats:italic> enjoys a circulant and skew circulant splitting (denoted CSCS) i.e., <jats:italic>T = C−S</jats:italic> with <jats:italic>C</jats:italic> a circulantmatrix and <jats:italic>S</jats:italic> a skew circulantmatrix. Based on the variant of such a splitting (also referred to as CSCS), we first develop classical CSCS iterative methods and then introduce shifted CSCS iterative methods for solving hermitian positive deûnite Toeplitz systems in this paper. The convergence of each method is analyzed. Numerical experiments show that the classical CSCS iterative methods work slightly better than the Gauss–Seidel (GS) iterative methods if the CSCS is convergent, and that there is always a constant α such that the shifted CSCS iteration converges much faster than the Gauss–Seidel iteration, no matter whether the CSCS itself is convergent or not.</jats:p>
语种：
英文
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Hermitian Toeplitz矩阵向量乘积的快速算法
作者：
刘仲云;陈思恒;徐伟进;张育林
期刊：
数学理论与应用 ,2017年37(3):38-42 ISSN：1006-8074
作者机构：
长沙理工大学数学与统计学院,湖南长沙,410114;Minho大学数学中心,葡萄牙,4710-057;[张育林] Minho大学;[陈思恒; 徐伟进; 刘仲云] 长沙理工大学
关键词：
Hermitian Toeplitz矩阵;矩阵向量乘法;实运算
摘要：
众所周知,大规模Hermitian Toeplitz矩阵向量乘积Ax可由快速Fourier变换(FFT)进行计算.事实上,Hermitian Toeplitz矩阵在酉相似变换下可约化为一个实的Toeplitz矩阵与Hankel矩阵之和.基于此,本文利用DCT和DST,构造了一个更有效的方法,只需O(n)的复运算.
语种：
中文
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关于对称正定的H-矩阵的预处理方法
作者：
张艳;于静;刘仲云
期刊：
Shuxue Lilun yu Yingyong ,2016年36(1):19-24 ISSN：1006-8074
作者机构：
长沙理工大学数学与统计学院,长沙,410004;[于静; 刘仲云; 张艳] 长沙理工大学
关键词：
严格对角占优矩阵;H—矩阵;预处理矩阵;预条件共轭梯度法
摘要：
本文研究了对称正定的H—矩阵的预处理方法.我们根据对系数矩阵的分裂及其非对角元性质,构造一个新的预处理矩阵Pv=I-μννT,并分析了预处理矩阵的谱性质.数值实验表明该预处理矩阵比T.Chan的预处理矩阵[5]更有效.
语种：
中文
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非埃米特正定Toeplitz矩阵的m—步预处理子（英文）
作者：
刘仲云;于静;张艳;张育林
期刊：
数学理论与应用 ,2016年36(1):25-30 ISSN：1006-8074
作者机构：
长沙理工大学数学与统计学院,长沙,410004;Minho大学数学中心,4710-057,葡萄牙;[张育林] Minho大学;[于静; 刘仲云; 张艳] 长沙理工大学
关键词：
循环与反循环分裂;m—步多项式预处理子;共轭梯度方法;Toeplitz矩阵
摘要：
众所周知，如果A是Toeplitz矩阵，那么矩阵A有一循环与反循环分裂（记为CSCS）[7]，可写为A—C＋S，其中c为循环矩阵，s为反循环矩阵．本文针对某类Toeplitz矩阵，提出了一个m步的预处理子Pm，这个预处理子P。是基于CSCS迭代方法构建的．本文中证明当C和S都是正定矩阵时，对于适当的m，预处理矩阵（Pm＊A）·＊（Pm＊A）的谱半径聚集于1．实验结果表明，对于适当的m，本文提出的预处理子优于T—Chan预处理子[3]．
语种：
中文
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