【作者】 冉瑞生；

【导师】 黄廷祝；

【作者基本信息】 电子科技大学 ， 计算数学， 2004， 硕士

【摘要】 本文主要分四部分: 1. 给出了非奇H矩阵的简洁判据, 推广了文[9]的结果,使得判别条件适用条件更广。如设A=（a_v）∈M_N（C）,A是不可约矩阵。若|a_ü|≥（?）|a_it|+（?）|a_it|/|a_tt|R_t（A）,（?）i=N₁,且至少有一个严格不等号成立,则A为非奇H阵。2. 研究了非奇H矩阵的谱半径估计,即设A=(a_ij)∈M_n（c）,若A是非奇H矩阵,则ρ（A）≤2max|a_ij|. 该结果实用可行,且在一定程度上比用Frobenius不等式判别非负矩阵的谱半径优越。3. 迭代矩阵的谱半径估计。在M为C_r对角占优和α-对角占优矩阵的条件下分别研究了迭代阵M^-1N的谱半径估计。该估计在一定程度上比M为严格对角占优和Nekrasov矩阵得到的M^-1N的谱半径估计优越。3.1节得到当M∈C_r时,ρ（M-1N）≤max(|n_u|+r_i（?）|n_ij|)/|m_ij-r_i（?）|m_ij||又如果对某α∈[0,1]有M∈D^α,且（?）R_i^1-α/(R_t^1-α+S_i^1-α)≤1,则ρ(M^-1N)≤max(|n_ij|+（S_i/R_i）^1-α（?）|n_ij|)/(|M_ij|-R_i^αS_i^1α) 4. 将上述结果应用于著名的迭代法如:Gauss-Seidel、JOR、SOR、AOR、MSOR等,得出了比较好的谱半径估计。并对JOR和SOR迭代法作了收敛性分析。3.2节讨论了JOR和SOR迭代法的迭代阵的谱半径估计及收敛性分析,<WP=5>如设A∈C_r,则当参数0<λ<2/(1+r_imax(|l_ij||u_ij|))时,JOR迭代法收敛。 3.3、3.4节分别讨论了M为C_r对角占优和α-对角占优矩阵的时AOR、MSOR的迭代矩阵的谱半径估计。更多还原

【Abstract】 This paper mainly includes four parts:1. I present the criteria of nonsingular H-matrices, which extend the results in [9] and are applicable to more matrices. For example, let A=（a_v）∈M_N（C）,and A be irreducible If |a_ü|≥（?）|a_it|+（?）|a_it|/|a_tt|R_t（A）,（?）i=N₁,and there exist a “>” at least. Then A is nonsingular H-matrix.2. I study the estimate for the spectral radius of nonsingular H-matrices. Let A=(a_ij)∈M_n（c） ,if A is nonsingular H-matrix, then ρ（A）≤2max|a_ij|.The estimate is practical and very simple. The estimate is superior to the Frobenius inequality, which is applicable to the estimate for the spectral radius of nonnegative matrices. 3. Estimate for the spectral radius of iterative matrices. when M∈C_r or M∈C^α, we study the the estimate for the spectral radius of the iterative matrix M^-1N, which is superior to the estimate when M is strictly diagonal dominant or Nekrasov matrix in a way. In 3.1, when M∈C_r,ρ（M-1N）≤max(|n_u|+r_i（?）|n_ij|)/|m_ij-r_i（?）|m_ij|| If M∈D^α(α∈[0,1]有M∈D^α,and D（?）R_i^1-α/(R_t^1-α+S_i^1-α)≤1,then<WP=7>ρ(M^-1N)≤max(|n_ij|+（S_i/R_i）^1-α（?）|n_ij|)/(|M_ij|-R_i^αS_i^1α)4. Applying the above results to the famous iterations, such as Gauss-Seidel、JOR、SOR、AOR、MSOR, etc., I obtain more accurate results and analysis the convergence of JOR and SOR. In 3.2, I discuss the estimate of the spectral radius of iterative matrices of JOR and SOR and analysis their convergence. For example, let A∈C_r, JOR is convergent when 0<λ<2/(1+r_imax(|l_ij||u_ij|))In 3.3 and 3.4, I discuss the the estimate of the spectral radius of iterative matrices of AOR and MSOR when M∈C_r or M∈D^α（α∈[0,1]）.更多还原