2-循环相容次序阵的AOR迭代的收敛域

Convergence Region for the AOR Iteration Matrix of an 2-Cyclic Consistently Ordered Matrix

设A∈Cn×n是2-循环相容次序阵,其Jacobi阵J的非零特征值均为纯虚数.记α=ρ(J).本文证明了A的AOR迭代阵Lr,ω(约定ω>0,r≠0)收敛当且仅当参数ω,r满足条件0<ω<21+α2,ω+ωα-22<r<12ω+(2ω-αω2)2,r≠0,或等价地,r≥rb,0<ω<2+rα2-α1+αr22α2+4r-4;rb≥r>-α22,r≠0,0<ω<21++rαα22,其中rb=1+21+α2.这一结果纠正了薛秋芳文给出的相应结果,并指出了其中的3个问题.

Suppose A∈C^n×n is an 2 - cyclic consistently ordered matrix and its non-zero eigenvalues of Jacobi matrix J has only pure imaginary. Set α=ρ（J）. In this paper we prove that the AOR iteration matrix Br,ω,（with ω 〉0 and r≠O） of A converges iff 0〈ω〈21＋α2,ω＋ωα-22-α22,r≠0 or equivalently,{r≥rb,0〈ω〈2＋rα^2-α√r^2α^2＋4r-4/ 1＋α^2 rb≥r〉-2/α^2,r≠0,0〈ω〈2＋rα^2/1＋α^2,其中rb=1/1＋√1＋α2.This result corrects the corresponding one obtained by Xue.