Large linear system solution, with iterative methods

Large Linear System Solution with Iterative Methods... 

It may appear as if this is no great improvement, since finding a solution to a linear equation system with direct methods requires about n3 operations, about half as many as the inversion. However, the solution of the linear equation system can be accomplished by iterative methods where, in each step, some product jv is formed. Superficially, this cuts down, the number of operations, but still requires the Jacobian to be computed and stored. However, for a very large class of important problems, such a product can be efficiently computed without the need of precalculating or storing the Jacobian. 

The round-off error propagation associated with the use of Shacham and Kehat s direct method for the solution of large sparse systems of linear equations is investigated. A reordering scheme for reducing error propagation is proposed as well as a method for iterative refinement of the solution. Accurate solutions for linear systems, which contain up to 500 equations, have been obtained using the proposed method, in very short computer times. 

With these basic definitions in hand, we now begin to consider the solution of the linear system Ax = b, in which x, b and is an A x A/ real matrix. We consider here elimination methods in which we convert the linear system into an equivalent one that is easier to solve. These methods are straightforward to implement and work generally for any linear system that has a unique solution however, they can be quite costly (perhaps prohibitively so) for large systems. Later, we consider iterative methods that are more effective for certain classes of large systems. 

---