The Generalized Minimal Residual method (GMRES) seeks optimal approximate solutions of linear system Ax = b from Krylov subspaces by minimizing the residual norm ‖Ax − b‖2 over all x in the subspaces. Its main cost is computing and storing basis vectors of the subspaces. For difficult systems, Krylov subspaces of very high dimensions are necessary for obtaining approximate solutions with desire...

GMRES is an iterative method that provides better solutions when dealing with larg linear systems of equations with unsymmetric coefficient matrix. By shifting the Arnoldi process to begin with Ar0 instead of r0, simpler GMRES implementation, proposed by Walker and Zhou in 1994, is obtained that in this method, an upper triangular problem is solved instead of hessenberg least square problem. Th...

When solving large nonsymmetric systems of linear equations with the restarted GMRES algorithm, one is inclined to select a relatively large restart parameter in the hope of mimicking the full GMRES process. Surprisingly, cases exist where small values of the restart parameter yield convergence in fewer iterations than larger values. Here, two simple examples are presented where GMRES(1) conver...

Globally convergent homotopy methods are used to solve difficult nonlinear systems of equations by tracking the zero curve of a homotopy map. Homotopy curve tracking involves solving a sequence of linear systems, which often vary greatly in difficulty. In this research, a popular iterative solution tool, GMRES(k), is adapted to deal with the sequence of such systems. The proposed adaptive strat...

We introduce an iterative method named Gpmr (general partitioned minimum residual) for solving block unsymmetric linear systems. is based on a new process that simultaneously reduces two rectangular matrices to upper Hessenberg form and closely related the block-Arnoldi process. tantamount Block-Gmres with right-hand sides in which approximate solutions are summed at each iteration, but its sto...