Round-off error analysis of iterations for large linear systems

Abstract

We deal with the rounding error analysis of successive approximation iterations for the solution of large linear systemsA x =b. We prove that Jacobi, Richardson, Gauss-Seidel and SOR iterations arenumerically stable wheneverA=A *>0 andA has PropertyA. This means that the computed resultx k approximates the exact solution ? with relative error of order ? ?A?·?A ?1? where ? is the relative computer precision. However with the exception of Gauss-Seidel iteration the residual vector ?Ax k ?b? is of order ? ?A?2 ?A ?1? ??? and hence the remaining three iterations arenot well-behaved.