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.
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