Toward a Memory-efficient Linear Solver

We present an alternative to the standard restarted GMRES (generalized minimum residual) algorithm for solving a single right-hand side linear system Ax=b based on solving the block linear system AX=B. Additional starting vectors and right-hand sides are chosen to accelerate convergence. Algorithm performance, i.e. time to solution, is improved by using the matrix A in block operations with groups of vectors, reducing the movement of A through memory. The efficient implementation of our method depends on a fast matrix-multivector multiply routine. We discuss the new block algorithm and present numerical results.