二次PE方法的研究
|
摘要
在工程领域，如网络，控制等方面，许多实际问题都归结为解一
个系数矩阵A为大型分块三对角矩阵的线性方程组
AX＝f
解这类方程组通常用迭代法来求解；本文对这种类型系数矩阵的方程
给出了一种迭代解法，主要讨论了如下几个问题：
（1）在对系数矩阵实行不完全LU分解的基础上，导出了二次
PE方法，对系数矩阵A为正定矩阵，对角优势L-矩阵，
证明了二次PE方法的收敛性，并且在最后通过实例计算
说明了二次PE方法是一种收敛性比较好的方法。
（2）在二次PE方法的基础上，通过引入参数k，导出了二次
PE_k方法，并且对系数矩阵A为Hermite矩阵，M-矩阵及
H-矩阵证明了二次PE_k方法的收敛性；最后通过实例计算
说明了对适当的参数k，二次PE_k方法比PE方法及二次PE
方法收敛性都好。
In the networks and controlling fields of engineering, many problenis can be described as solving a system of linear algebraic equations
AX=fWhose matrix of coefficients is large tridiagonal blocked matrix The usual method of solving this kind of equations is the iterative method Iii this paper, a new iterative method will be given. It is to be discussec as follows:
(1) The quadratic PE method is derived on the base of incompletely LU triangular decomposition. Based on the results, the convergence of the quadratic PE method in positive definite matrix and diagomdlv dominant L matrix is proved, and the availability of the method is ~iIso illustrated .ln the end, some examples are given to illustrate the method
(2) On the results of quadratic PE method, the parameter k is introduced and the convergence of the quadratic PE method is impro\ ed. The convergence of the quadratic PE k method is proved about the Hermite matrix, M-matrix and H-matrix. In chapter four, some examples are given to illustrate that the convergence of quadratic PEk method is better than the quadratic PE method to proper parameter k.
个系数矩阵A为大型分块三对角矩阵的线性方程组
AX＝f
解这类方程组通常用迭代法来求解；本文对这种类型系数矩阵的方程
给出了一种迭代解法，主要讨论了如下几个问题：
（1）在对系数矩阵实行不完全LU分解的基础上，导出了二次
PE方法，对系数矩阵A为正定矩阵，对角优势L-矩阵，
证明了二次PE方法的收敛性，并且在最后通过实例计算
说明了二次PE方法是一种收敛性比较好的方法。
（2）在二次PE方法的基础上，通过引入参数k，导出了二次
PE_k方法，并且对系数矩阵A为Hermite矩阵，M-矩阵及
H-矩阵证明了二次PE_k方法的收敛性；最后通过实例计算
说明了对适当的参数k，二次PE_k方法比PE方法及二次PE
方法收敛性都好。
In the networks and controlling fields of engineering, many problenis can be described as solving a system of linear algebraic equations
AX=fWhose matrix of coefficients is large tridiagonal blocked matrix The usual method of solving this kind of equations is the iterative method Iii this paper, a new iterative method will be given. It is to be discussec as follows:
(1) The quadratic PE method is derived on the base of incompletely LU triangular decomposition. Based on the results, the convergence of the quadratic PE method in positive definite matrix and diagomdlv dominant L matrix is proved, and the availability of the method is ~iIso illustrated .ln the end, some examples are given to illustrate the method
(2) On the results of quadratic PE method, the parameter k is introduced and the convergence of the quadratic PE method is impro\ ed. The convergence of the quadratic PE k method is proved about the Hermite matrix, M-matrix and H-matrix. In chapter four, some examples are given to illustrate that the convergence of quadratic PEk method is better than the quadratic PE method to proper parameter k.
引文
[1] 胡家赣·解线性代数方程组PE方法·计算 数学，1982，4(2) ：151-157
[2] 胡家赣·解线性代数方程组的PER方法计算 数学，1993． 4(2) ：146-156
[3] 胡家赣，刘兴平·线性代数议程组选代解法的 收敛性·数值计算与计算机应用，1991·12(3) ：165-174
[4] 胡家赣·予条件共轭梯渡法的收敛性，计算 物理，1986·3(4) ：487-495
[5] 胡家赣，王邦荣·严抟对角优势短阵PE方法的 收敛性·数值计算机应用，1986·12(4) ：206-217
[6] 胡家赣，刘兴平·EPER方法和可E定化矩阵·数值 计算与计算机应用，1997·3(1) ：30-39
[7] Helliwell,A Fast Implicit Namerical Metled for solving Multidimensional patial Diferential Equations,A Collection of Techincal Papers AIAA 3rd Conputational Flaid Dynamics ,1977
[8] H.L.Stone,Iterative Solustion of Impluit Appronimcdions of Multidimensionld Partial Diyerential Equations, SIAM J.Namer.And.Vol.5 No.3,1968,PP.530-558
[9] David S. Kershaw, The Incomplete Cholesky Coniugate Gradient Methed ofr the Iterative Solution of LMear Equations，J.Canipia. Physics 26，1978,pp.43-65
[10] A．hadjidimos, A.Yeyos ,A. Yeyios, The Principal of Ex-trapolation in Connection with the Accelerated Over-relaxation Metlool, Lin. Alg. Appl．, 30,1980,115-128
[11] 张凯院，徐仲数值代数·西安：西北工业大学出版 社，2000
[12] 胡家赣·线性代数方程组的迭代解法，科学 出版社，1997
[13] 蔡大用·数值代数·清华大学出版社，1987
[14] 罗家洪·矩阵分析引论，华南理工大学出版社，1997
[15] 程云鹏·矩阵论(第2版)·西安：西北工业大学出版 社，1999
[16] 王松桂，贾忠贞·矩阵论中不寻式·安徽教育出版 社，1993
[17] 薛定宇·科学运算语言Matlab 5． 3程序设计与应用· 清华大学出版社，2000
[18] Louis A.Hageman,David M.Young,Applied Iterative Method，Academic Press,1981
[19] David M.Young, Iterative Solution of Large Linear System, Academic press. Inc., New York,1971
[20] Richard S. Varga, Matrix Iterative Analysis, Prentice-Hall,Inc.,Engleusood Cliff, N.J. 1962
[2] 胡家赣·解线性代数方程组的PER方法计算 数学，1993． 4(2) ：146-156
[3] 胡家赣，刘兴平·线性代数议程组选代解法的 收敛性·数值计算与计算机应用，1991·12(3) ：165-174
[4] 胡家赣·予条件共轭梯渡法的收敛性，计算 物理，1986·3(4) ：487-495
[5] 胡家赣，王邦荣·严抟对角优势短阵PE方法的 收敛性·数值计算机应用，1986·12(4) ：206-217
[6] 胡家赣，刘兴平·EPER方法和可E定化矩阵·数值 计算与计算机应用，1997·3(1) ：30-39
[7] Helliwell,A Fast Implicit Namerical Metled for solving Multidimensional patial Diferential Equations,A Collection of Techincal Papers AIAA 3rd Conputational Flaid Dynamics ,1977
[8] H.L.Stone,Iterative Solustion of Impluit Appronimcdions of Multidimensionld Partial Diyerential Equations, SIAM J.Namer.And.Vol.5 No.3,1968,PP.530-558
[9] David S. Kershaw, The Incomplete Cholesky Coniugate Gradient Methed ofr the Iterative Solution of LMear Equations，J.Canipia. Physics 26，1978,pp.43-65
[10] A．hadjidimos, A.Yeyos ,A. Yeyios, The Principal of Ex-trapolation in Connection with the Accelerated Over-relaxation Metlool, Lin. Alg. Appl．, 30,1980,115-128
[11] 张凯院，徐仲数值代数·西安：西北工业大学出版 社，2000
[12] 胡家赣·线性代数方程组的迭代解法，科学 出版社，1997
[13] 蔡大用·数值代数·清华大学出版社，1987
[14] 罗家洪·矩阵分析引论，华南理工大学出版社，1997
[15] 程云鹏·矩阵论(第2版)·西安：西北工业大学出版 社，1999
[16] 王松桂，贾忠贞·矩阵论中不寻式·安徽教育出版 社，1993
[17] 薛定宇·科学运算语言Matlab 5． 3程序设计与应用· 清华大学出版社，2000
[18] Louis A.Hageman,David M.Young,Applied Iterative Method，Academic Press,1981
[19] David M.Young, Iterative Solution of Large Linear System, Academic press. Inc., New York,1971
[20] Richard S. Varga, Matrix Iterative Analysis, Prentice-Hall,Inc.,Engleusood Cliff, N.J. 1962