Two algorithms characterized by a short recurrence as well as a minimization property in the energy norm are derived by choosing suitable preconditioning matrices for certain step-dependent preconditioned conjugate Krylov subspace (CKS) algorithms. It is shown that the two methods are also equivalent to two special truncated generalized CG algorithms. Therefore, it is possible to state a minimization property for truncated generalized CG algorithms in a k-dimensional space. Numerical comparison with other generalized CG algorithms requiring the same amount of storage shows that the derived methods are competitive and for certain problems faster.
Wagner, B., & Weiss, R. (1999). Minimization properties and short recurrences. Applied Numerical Mathematics, 30(2), 175–190. https://doi.org/10.1016/S0168-9274(98)00109-3