A Note on Comparison Theorems of Splittings and Multisplittings

We discuss iterative methods for the solution of the linear system Ax = b, which are based on a single splitting or a multisplitting of A. In order to compare diierent methods , it is common to compare the spectral radius of the iterative matrix. For M{matrices A and weak regular splittings there exist well-known comparison theorems. Here, we give a comparison theorem for splittings of Hermitian positive deenite matrices. Furthermore, we establish a comparison theorem for multisplittings of a Hermitian positive deenite matrix. x1 Introduction and Deenitions If we consider linear systems of equations Ax = b, A 2 I C n;n ; x; b 2 I C n , that occur in the numerical analysis, in many cases the matrix A = a ij ] is an M{matrix, i.e. A 2 IR n;n ; a ij 0 for i 6 = j; A ?1 exists and is nonnegative, or A is a Hermitian positive deenite matrix. Stimulated by O. Taussky in 1958 T], a lot of common properties of these two classes of matrices were found, as for instance, determinant inequalities for principal submatrices of A. Here, we consider iterative methods for approximating the solution of the linear system Ax = b. We establish results for Hermitian positive deenite matrices, which corresponds with well-known results for M{matrices. It is well-known that (1:1) converges for all x 0 if and only if