On the extrapolated Jacobi or simultaneous displacements method in the solution of matrix and operator equations

Abstract

(vi) Jw -(co 1) + c D-1Q}, where X is the relaxation parameter which is a real number. It is known [171 that for an arbitrary x(?) the error vectors e(m?+) of these iterative methods tend to the zero vector if and only if the Jm+1 tend to the zero matrix, or equivalently, if and only if the spectral radius r(J.) (-max,5i<n I Xs(J,,) , ) of J. is less than unity. However, since the evaluation of the eigenvalues of a general matrix is very complicated, it is difficult to guarantee in advance that the convergence conditions will be satisfied, and theoretical results for the Jacobi and extrapolated Jacobi methods, in spite of their being long in existence, are available only for very special classes of matrices. Furthermore, almost all of these results give only sufficient conditions for convergence. Thus, if the matrix A is strictly diagonally dominant,' von Mises