On the parameter selection in the transformed matrix iteration method

Two Block Splitting Iteration Methods for Solving Complex Symmetric Linear Systems from Complex Helmholtz Equation

“…In each iteration of AIBS, IBS, NBS, PBS, AGSOR, and PMHSS methods, we use the Cholesky factorization to solve the subsystems. In addition, we use the preconditioned restarted GMRES (20) in our numerical tests. The iteration number k of GMRES means k = (i − 1) × 20 + p, where i is the number of restarting, and p is the iteration number of the last restarting, respectively.…”

“…In 2019, Axelsson and Salkuyeh [19] derived the transformed matrix iteration (TMIT) method and gave the convergence conditions; they also determined that the optimal parameter of TMIT method was contained in the interval (0, 1). In 2021, Siahkolaei et al [20] gave an upper bound for the spectral radius of the iteration matrix of the TMIT method and then obtained the parameter which minimizes this upper bound.…”

A new double-step splitting iteration method for certain block two-by-two linear systems

“…Deriving the efficient selection strategy of parameters for the NDSS method is significant to obtain fast convergence rate of it. Based on Theorems 2.3-2.4 and the methods used in Chen (2015) and Siahkolaei and Salkuyeh (2020), we have introduced three practical ways for the choice of iteration parameters of the NDSS iteration method as follows:…”

“…2020presented a novel method to derive the parameter for the TMIT method, which minimizes the upper bound of the spectral radius of this method. Inspired by the idea of Siahkolaei and Salkuyeh (2020), we use the technique applied in Siahkolaei and Salkuyeh (2020) to deduce the parameters to minimize the upper bound of the spectral radius of the proposed NDSS method. From 29, we have…”

“…We assume that μ min = μ 1 ≤ μ 2 ≤ • • • ≤ μ n = μ max are the eigenvalues of the matrix W − 1 2 T W − 1 2 . From Theorem 1 of Siahkolaei and Salkuyeh (2020), the parameter α minimizing the upper bound of σ (α) is…”