The acceleration of the original projective iterative methods of multiplicative or additive type for solving systems of linear algebraic equations (SLAEs) by means of conjugate direction approaches is con-sidered. The orthogonal and varitional properties of the preconditioned conjugate gradient, conjugate residual and semi-conjugate residual algo-rithms, as well as estimations of the number of iterations, are presented. Similar results were obtained for the dynamically preconditioned itera-tive process in Krylov subspaces. Application of discussed techniques for domain decomposition, Kaczmarz, and Cimmino methods is proposed.
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