A Hybrid Algorithm based on Stationary and Krylov methods for Nonsy- mmetric Linear Systems

Resumo

Iterative Krylov methods, like Generalized Minimal Residual (GMRES) and Full Orthogonalization

Method (FOM), are normally used for the solution of sparse and nonsymmetric linear systems from Computa-
tional Mechanics problems. In practice, restarted versions, are used to reduce storage and orthogonalization costs.

However, numerical experience shows that these methods may present stagnation or slow convergence. The Sta-
tionary method is older, simpler to understand and implement, but usually not completely effective. Contrarily, the

Krylov method has a more recent development and is more effective than the former, but the analysis is usually
harder to understand with difficulties in selecting its parameters. A cycle of a proposed hybrid method consists
of n Stationary iterations of Richardson followed by m × k iterations of the restarted GMRES, where n, m and
k are values much smaller than the dimension of the non-symmetric matrix. Such cycles can be repeated until
convergence is achieved. The advantage of this approach is in the opportunity to allow better performance of its
individual properties. This combination of methods is competitive from the point of view of helping to accelerate

convergence with respect to the number of iterations for some linear problems. We are going to present compu-
tational experiments to show the advantages and the main problems raised from the perspective of the proposed

hybrid method.