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We consider a flexible sequential block iterative method for the solution of consistent linear systems of equations and give its convergence analysis. The method is able to use weight matrices and relaxation parameters which can be updated in each iteration whereas the most of previous studies on sequential block iterative methods considered a finite number of weight matrices. Furthermore, we consider the constraint version of the method and give its convergence analysis for the special case of relaxation parameters and weight matrices. We report on some numerical tests with examples taken from the field of image reconstruction from projections. Our numerical results show considerable improvement, specially on noisy data, compared to the other methods which use finite number of weight matrices.
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