On the parameter selection in the transformed matrix iteration method

Recently, Axelsson and Salkuyeh in (BIT Numerical Mathematics, 59 (2019) 321–342) proposed the transformed matrix iteration (TMIT) method for solving a certain two-by-two block matrices with square blocks. However, they did not present any formula for the optimal parameter of the method which minimizes the spectral radius of the iteration matrix. In this work, we give an upper bound for the spectral radius of the iteration matrix of the method and then compute the parameter which minimizes this upper bound. Numerical results are presented to demonstrate the theoretical results.