Stair Matrices and Their Generalizations with Applications to Iterative Methods Ii: Iteration Arithmetic and Preconditionings

Iteration arithmetic is formally introduced based on iteration multiplication and αaddition which is a special multisplitting. This part focuses on construction of convergent splittings and approximate inverses for Hermitian positive definite matrices by applying stair matrices, their generalizations and iteration arithmetic. Analysis of the splittings and the approximate inverses is also presented. Application of some of the results extends the classical convergence result of the SSOR method. In particular, multiplication symmetrization and addition symmetrization are introduced, which produce Hermitian positive definite approximations for the inverse of an Hermitian positive definite matrix. Furthermore, preconditioning average is introduced to improve some preconditioning methods. Numerical results show a significant improvement of preconditioning average to the approximate inverse preconditionings if an anisotropic elliptic equation is solved.