New Convergence Analysis of GMRES with Weighted Norms, Preconditioning and Deflation, Leading to a New Deflation Space *

New convergence bounds are presented for weighted, preconditioned, and deflated GMRES for the solution of large, sparse, nonsymmetric linear systems, where it is assumed that the symmetric part of the coefficient matrix is positive definite. The new bounds are sufficiently explicit to indicate how to choose the preconditioner and the deflation space to accelerate the convergence. One such choice of deflating space is presented, and numerical experiments illustrate the effectiveness of such space.