On solving symmetric systems of linear equations in an unnormalized Krylov subspace framework

In an unnormalized Krylov subspace framework for solving symmetric systems of linear equations, the orthogonal vectors that are generated by a Lanczos process are not necessarily on the form of gradients. Associating each orthogonal vector with a triple, and using only the three-term recurrences of the triples, we give conditions on whether a symmetric system of linear equations is compatible or incompatible. In the compatible case, a solution is given and in the incompatible case, a certificate of incompatibility is obtained. In particular, the case when the matrix is singular is handled. We also derive a minimum-residual method based on this framework and show how the iterates may be updated explicitly based on the triples, and in the incompatible case a minimum-residual solution of minimum Euclidean norm is obtained.