  A new approach is discussed for solving large nonsymmetric systems of linear
equations with multiple right-hand sides. The first system is solved with a
deflated GMRES method that generates eigenvector information at the same time
that the linear equations are solved. Subsequent systems are solved by
combining restarted GMRES with a projection over the previously determined
eigenvectors. This approach offers an alternative to block methods, and it can
also be combined with a block method. It is useful when there are a limited
number of small eigenvalues that slow the convergence. An example is given
showing significant improvement for a problem from quantum chromodynamics. The
second and subsequent right-hand sides are solved much quicker than without the
deflation. This new approach is relatively simple to implement and is very
efficient compared to other deflation methods.
