NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS Numer. Linear Algebra Appl. 2010; 17:3–15

Block (including s-step) iterative methods for (non)symmetric linear systems have been studied and implemented in the past. In this article we present a (combined) block s-step Krylov iterative method for nonsymmetric linear systems. We then consider the problem of applying any block iterative method to solve a linear system with one right-hand side using many linearly independent initial residual vectors. We present a new algorithm which combines the many solutions obtained (by any block iterative method) into a single solution to the linear system. This approach of using block methods in order to increase the parallelism of Krylov methods is very useful in parallel systems. We implemented the new method on a parallel computer and we ran tests to validate the accuracy and the performance of the proposed methods. It is expected that the block s-step methods performance will scale well on other parallel systems because of their efficient use of memory hierarchies and their reduction of the number of global communicatio