阶梯矩阵及其一般化在迭代法中的应用

基金项目: 辽宁省自然科学基金资助项目(20022021)

Stair Matrices and Their Generalizations With Applications to Iterative Methods

Department of Mathematics, Northeastern University, Shenyang 110004, P. R. China

摘要: Lu Hao首先给出了阶梯矩阵及其一般性的定义和性质．这类矩阵为迭代法提供了新矩阵分裂的基础．基于此新矩阵类的迭代方法的显著特征是它对于并行计算很容易被实现．应用这一新的分解方法，给出了一般的加速松弛方法（GAOR），而关于AOR方法的一些性质可以被延伸到该新方法中，并针对Hermite正定矩阵进行了新方法收敛性的分析．最后，给出了一些例子来表明新方法的优越性．
- 阶梯矩阵 /
- 迭代法 /
- 平行计算 /
- 一般加速松弛方法（GAOR）
Abstract: Stair matrices and their generalizations are introduced.The definitions and some properties of the matrices were first given by Lu Hao.This class of matrices provided bases of matrix splittings for iterative methods.The remarkable feature of iterative methods based on the new class of matrices is that the methods were easily implemented for parallel computation.In particular,a generalization of the accelerated overrelaxation method(GAOR) was introduced.Some theories of the AOR method were extended to the generalized method to include a wide class of matrices.The convergence of the new method was derived for Hermitian positive definite matrices.Finally,some examples are given in order to show the superiority of the new method.
- stair matrices /
- iterative method /
- parallel computation /
- generalization of the AOR method

[1]	Hadjidimos A.Accelerated overrelaxation method[J].Math Comp,1978,32(2):149—157. doi: 10.1090/S0025-5718-1978-0483340-6
[2]	Varga R S.Matrix Iterative Analysis[M].Englewood Cliffs,NJ:Prentice-Hall,1962,25—132.
[3]	Young D M.Iterative Solution for Large Systems[M].New York:Academic Press,1971，102—145.
[4]	LU Hao.Stair matrices and their generalizations with applications to iterative methods(Ⅰ)—A generalization of the successive overrelaxation method[J]. SIAM J Numer Anal,1999,37(1):1—17. doi: 10.1137/S0036142998343294
[5]	Li C,Li B,Evans D J.A generalized successive overrelaxation method for least squares problems[J].BIT,1998,38(2):347—356. doi: 10.1007/BF02512371
[6]	Varga R S.Extensions of the Successive Overrelaxation Theory With Applications to Finite Element Approximations,in Topics in Numerical Analysis[M].New York:Academic Press,1973,329—343.
[7]	Wild P,Niethammer W.Over- and under-relaxation for linear systems with weakly cyclic Jacobi matrices of index p[J].Linear Algebra Appl,1987,91(1):29—52. doi: 10.1016/0024-3795(87)90058-9