Generalized conjugate gradient squared

The Conjugate Gradient Squared (CGS) is an iterative method for solving nonsymmetric linear systems of equations. However, during the iteration large residual norms may appear, which may lead to inaccurate approximate solutions or may even deteriorate the convergence rate. Instead of squaring the Bi-CG polynomial as in CGS, we propose to consider products of two nearby Bi-CG polynomials which leads to generalized CGS methods, of which CGS is just a particular case. This approach allows the construction of methods that converge less irregularly than CGS and that improve on other convergence properties as well. Here, we are interested in a property that got less attention in literature: we concentrate on retaining the excellent approximation qualities of CGS with respect to components of the solution in the direction of eigenvectors associated with extreme eigenvalues. This property seems to be important in connection with Newtonʹs scheme for nonlinear equations: our numerical experiments show that the number of Newton steps may decrease significantly when using a generalized CGS method as linear solver for the Newton correction equations.