CGS, A Fast Lanczos-Type Solver for Nonsymmetric Linear systems

A Lanczos-type method is presented for nonsymmetric sparse linear systems as arising from discretisations of elliptic partial differential equations. The method is based on a polynomial variant of the conjugate gradients algorithm. Although related to the so-called bi-conjugate gradients (Bi-CG) algorithm, it does not involve adjoint matrix-vector multiplications, and the expected convergence rate is about twice that of the Bi-CG algorithm. Numerical comparison is made with other solvers, testing the method on a family of convection diffusion equations, on various grids, and with the use of two different preconditioning methods. Upwind as well as central differencing is used in the experiments.