System of linear algebraic equations with a symmetric and positive definite matrix arise from many areas in science and engineering Although, direct solvers may deliver robust solution, iterative solver based on the conjugate gradient method ( is often method of choice Efficiency of CG strongly depends on preconditioner, i e on our ability to approximate the inverse of the system matrix It can be provided by incomplete algorithms Well known are the variants of the incomplete Cholesky factorization, on the other hand backward/forward solve steps in every iteration may significantly limit performance of CG The inverse preconditioners represent a counterpart to the direct ones Their computation is in general more expensive, on the other hand, especially in parallel environment, their application can be very fast.

We deal with incomplete algorithms based on the Gram Schmidt orthogonalization with respect to non-standard inner product (induced by the system matrix) Incomplete algorithms employ techniques to preserve sparsity of the computed matrices We will discuss how to exploit theoretical results to construct such techniques In addition, numerical aspects will be accompanied by test problems.

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