Consider the 'basic LUL factorization' of the matrices as the generalization of the LU factorization and the UL factorization, and using this LUL factorization of the matrices, we propose an "improved iterative method" such that the spectral radius of this iterative matrix is equal to zero, and this method converges at most n iterations. Our main concern is the necessary and sufficient conditions that the improved iterative matrix is equal to the iterative matrix of the improved SOR method with orderings. Concerning the tridiagonal matrices and the upper Hessenberg matrices, this method becomes the improved SOR method with orderings, and we give n selections of the multiple relaxation parameters such that the spectral radii of the corresponding improved SOR matrices are 0. We extend these results to a class of $n \times n$ matrices. We also consider the basic LUL factorization and improved iterated method 'corresponding to permutation matrices'.
['Yoshiaki Muroya', 'Emiko Ishiwata']
Basic LUL factorization and improved iterative method with orderings