正负定矩阵下GAOR迭代法的收敛性
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摘要
为了研究GAOR迭代法在线性方程组系数矩阵分别为Hermite正定矩阵和负定矩阵两种情况下的收敛性,将Householder-John定理推广到负定情况下,并给出负定条件下GAOR迭代法收敛的充要条件.利用Householder-John定理,完善GAOR迭代法的收敛性结论.最后借助推广的Householder-John定理,分析GAOR迭代法在线性方程组系数矩阵为Hermite负定矩阵条件下的收敛性.
In order to study the convergence of GAOR iterative method on the basis of Hermitian positive and negative definite matrices,firstly the Householder-John theorem is introduced and generalized to the case of negative definite matrices.Then a sufficient and necessary condition for the convergence of GAOR iterative method is given under the negative definite condition.By using the Housholder-John theorem,the convergent conclusion of GAOR iterative method is improved.Finally,the convergence of GAOR iterative method under the Hermitian negative definite condition is analyzed through the generalized Householder-John theorem.
In order to study the convergence of GAOR iterative method on the basis of Hermitian positive and negative definite matrices,firstly the Householder-John theorem is introduced and generalized to the case of negative definite matrices.Then a sufficient and necessary condition for the convergence of GAOR iterative method is given under the negative definite condition.By using the Housholder-John theorem,the convergent conclusion of GAOR iterative method is improved.Finally,the convergence of GAOR iterative method under the Hermitian negative definite condition is analyzed through the generalized Householder-John theorem.
引文
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[11]TIAN G X,HUANG T Z,CUI S Y.Convergence of generalized AOR iterative method for linear systems with strictly diagonally dominant matrices[J].Journal of Computational and Applied Mathematics,2008,213(1):240-247.
[12]YUAN J Y,JIN X Q.Convergence of the generalized AOR method[J].Applied Mathematics and Computation,1999,99(1):35-46.
[13]LI Y T,DAI P F.Generalized AOR methods for linear complementarity problem[J].Applied Mathematics and Computation,2007,138(1):7-18.
[14]WANG G B,WEN H,LI L L,et al.Convergence of GAOR method for doubly diagonally dominant matrices[J].Applied Mathematics and Computation,2011,217(18):7509-7514.
[15]WANG X M.Convergence theory for the general GAOR type iterative method and the MSOR iterative method applied to H-matrices[J].Linear Algebra and its Applications,1997,250:1-19.
[16]WANG X M.Convergence for a general form of the GAOR method and its application to the MSOR method[J].Linear Algebra and its Application,1994,196:105-123.
[17]HADJIDIMOS A.The matrix analogue of the scalar AOR iterative method[J].Journal of Computational and Applied Mathematics,2015,288:366-378.
[2]VARGA R S.Matrix iterative analysis[M].Second Revised and Expanded Edition.Berlin:Springer-Verlag,2000:81-87.
[3]郭煜,畅大为.预条件P(S)=I+(S)下的USSOR迭代方法及其比较定理[J].西安工程大学学报,2012,26(3):370-373.GUO Y,CHANG D W.Comparison theorems for USSOR iterative methods with the precondition P(S)=I+(S)[J].Journal of Xi′an Polytechnic University,2012,26(3):370-373.
[4]ORTEGA J M,PLEMMONS R J.Extensions of the Ostrowski-Reich theorem for SOR iterations[J].Linear Algebra and its Applications,1979,28:177-191.
[5]叶绒绒,畅大为,韩俊佳.2-循环矩阵下MASOR迭代法的收敛性分析[J].纺织高校基础科学学报,2016,29(4):428-434.YE R R,CHANG D W,HAN J J.The convergence analyse of MASOR iteration method on the basis of 2-cylic matrix[J].Basic Sciences Journal of Textile Universities,2016,29(4):428-434.
[6]熊劲松,畅大为.2-循环系数矩阵对称MSOR法收敛的充分必要条件[J].纺织高校基础科学学报,2011,24(4):2-4.XIONG J S,CHANG D W.The necessary and sufficient condition for the convergence of the symmetric MSOR for 2-cylic coefficient matrices[J].Basic Sciences Journal of Textile Universities,2011,24(4):2-4.
[7]董瑾,畅大为,杨青青,等.一类2-循环系数矩阵对称MSOR法收敛的充分条件[J].纺织高校基础科学学报,2014,27(2):222-226.DONG J,CHANG D W,YANG Q Q,et al.The sufficient condition of symmetrical MSOR convergence for one type of2-cyclic coefficient matrices[J].Basic Sciences Journal of Textile Universities,2014,27(2):222-226.
[8]HADJIDIMOS A.Accelerated overrelaxation method[J].Mathematics of Computation,1978,32(141):149-157.
[9]高树玲,畅大为.相容次序矩阵AOR迭代收敛的充要条件[J].纺织高校基础科学学报,2009,22(2):229-231.GAO S L,CHANG D W.Sufficient and necessary conditions for compatible order matrix AOR[J].Basic Sciences Journal of Textile Universities,2009,22(2):229-231.
[10]宋永忠.块AOR迭代法的收敛性[J].应用数学,1993,6(1):39-45.SONG Y Z.The convergence of the block AOR iterative method[J].Mathematica Applicata,1993,6(1):39-45.
[11]TIAN G X,HUANG T Z,CUI S Y.Convergence of generalized AOR iterative method for linear systems with strictly diagonally dominant matrices[J].Journal of Computational and Applied Mathematics,2008,213(1):240-247.
[12]YUAN J Y,JIN X Q.Convergence of the generalized AOR method[J].Applied Mathematics and Computation,1999,99(1):35-46.
[13]LI Y T,DAI P F.Generalized AOR methods for linear complementarity problem[J].Applied Mathematics and Computation,2007,138(1):7-18.
[14]WANG G B,WEN H,LI L L,et al.Convergence of GAOR method for doubly diagonally dominant matrices[J].Applied Mathematics and Computation,2011,217(18):7509-7514.
[15]WANG X M.Convergence theory for the general GAOR type iterative method and the MSOR iterative method applied to H-matrices[J].Linear Algebra and its Applications,1997,250:1-19.
[16]WANG X M.Convergence for a general form of the GAOR method and its application to the MSOR method[J].Linear Algebra and its Application,1994,196:105-123.
[17]HADJIDIMOS A.The matrix analogue of the scalar AOR iterative method[J].Journal of Computational and Applied Mathematics,2015,288:366-378.