Instead of the standard estimate in terms of the spectral condition number we develop a new CG iteration number estimate depending on the quantity B = AtrM/(detM)'/", where M is an n x n preconditioned matrix. A new family of iterative methods for solving symmetric positive definite systems based on Breducing strategies is described. Numerical results are presented for the new algorithms and compared with several well-known preconditioned CG methods.

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Some techniques suitable to the control of the solution error in the Preconditioned Conjugate Method are considered and compared. The estimation can be performed both in the course ot the iterations and after their termination. The importance of such techniques follows from the non-existence of some reasonable a priori error estimate for very ill-conditioned linear systems when information about the right-hand side vector is lacking. Hence, some a posteriori estimates are required, which make it possible to verify the quality of the solution obtained for a prescribed right hand side. The performance of the considered error control procedures is demonstrated using real-world large-scale linear systems arising in computational mechanics.

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