【摘要】 设A是2-循环相容次序阵,其Jacobi阵J的特征值均为实数,记α=ρ(J)>0.本文证明了两个主要结论:(1)SAOR迭代收敛α<1且参数ω与γ满足条件0<ω<2,ω-2-ω/α<γ<ω+2-ω/α,或等价地,2≤γ<2/α,0<ω<2-γα/1-α;-2/α<γ≤2,0<ω<2+γα/1+α.(2)以Sγ,ω表示SAOR迭代阵,则:当ω≠1时,ρ(Sγ,ω)>α2;当ω=1时,ρ(Sγ,1)=α2,若γ∈[0,2];(γ-1)2α2>α2,若γ<0或γ>2.这表明:SAOR迭代的渐近收敛因子是α2,最优参数是ω=1与γ∈[0,2].本文的结果改进了张引的两个相关结论.更多还原

【Abstract】 Suppose A is an 2-cyclic consistently ordered matrix and its Jacobi matrix has only real eigenvalues.Set α=ρ(J)>0.In this paper we prove the main conclusions: (1) The SAOR iteration converges α<1 and the parameters ω and γ satisfy 0<ω<2,ω-2-ω/α<γ<ω+2-ω/α,or equivalently,{2≤γ<2/α,0<ω<2-γα/1-α;-2/α<γ≤2,0<ω<2+γα/1+α.(2) Let Sγ,ω be the SAOR iterative matrix.Then ρ(Sγ,ω)>α2,if ω≠1;ρ(Sγ,1)=(γ-1)2α2>α2,if γ<0 or γ>2;ρ(Sγ,1)=α2,if γ∈.Our results improve the two related conclusions due to Zhang Yin.更多还原