Componentwise error analysis for stationary iterative methods

How small can a stationary iterative method for solving a linear system Ax = b make the error and the residual in the presence of rounding errors? We give a componentwise error analysis that provides an answer to this question and we examine the implications for numerical stability. The Jacobi, Gauss-Seidel and successive over-relaxation methods are all found to be forward stable in a componentwise sense and backward stable in a normwise sense, provided certain conditions are satisfied that involve the matrix, its splitting, and the computed iterates. We show that the stronger property of componentwise backward stability can be achieved using one step of iterative refinement in fixed precision, under suitable assumptions.