A general Krylov method for solving symmetric systems of linear equations

Krylov subspace methods are used for solving systems of linear equations Hx+ c = 0. We present a general Krylov subspace method that can be applied to a symmetric system of linear equations, i.e., for a system in which H = H . In each iteration, we have a choice of scaling of the orthogonal vectors that successively make the Krylov subspaces available. We define an extended representation of each such Krylov vector, so that the Krylov vector is represented by a triple. We show that our Krylov subspace method is able to determine if the system of equations is compatible. If so, a solution is computed. If not, a certificate of incompatibility is computed. The method of conjugate gradients is obtained as a special case of our general method. Our framework gives a way to understand the method of conjugate gradients, in particular when H is not (positive) definite. Finally, we derive a minimum-residual method based on our framework and show how the iterates may be updated explicitly based on the triples.