Fast secant methods for the iterative solution of large nonsymmetric linear systems

A family of secant methods based on general rank-1 updates has been revisited in view of the construction of iterative solvers for large non-Hermitian linear systems. As it turns out, both Broy-den's "good" and "bad" update techniques play a special role — but should be associated with two different line search principles. For Broyden's "bad" update technique, a minimum residual principle is natural — thus making it theoretically comparable with a series of well-known algorithms like GMRES. Broyden's "good" update technique, however, is shown to be naturally linked with a minimum "next correction" principle — which asymptotically mimics a minimum error principle. The two minimization principles differ significantly for sufficiently large system dimension. Numerical experiments on discretized PDE's of convection diffusion type in 2-D with internal layers give a first impression of the possible power of the derived "good" Broyden variant.