Multi-iterative methods

A general class of iterative methods is introduced for solving positive definite linear systems Ax = b. These methods use two or more different iterative techniques and each of them reduces the error by a constant factor in a different subspace of Rn. The Multigrid, as an example of a multi-iterative method, combines these partial results and we reach a reduction of the error by θν at each step, where ν is the number of applications of the first scheme and θ \gt 1 is a constant independent of n. In the second part, we consider the "prolongation" operator p with a special "antidiagonal" structure and we show that the multigrid method for Toeplitz matrices [1] is an example of a method of this class. In a third part, we extend the multigrid method to block-Toeplitz matrices, obtaining a "block-antidiagonal" structure; for Toeplitz and block-Toeplitz cases, the choice of p which maximizes convergence speed simply depends on the coefficients of matrix A. The numerical results on 1-D and 2-D elliptic differential problems confirm the effectiveness of this multi-iterative method; in fact its spectral radius is very small and independent of the mesh spacing h.