Some Investigation Of Nonsingular H-Matrices And Estimate For The Spectral Radius Of Iterative Matrices

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,thenρ(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]).