Modified iterative aggregation algorithms

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The iterative aggregation methods were developed in 1960s due to the necessity of practical solution of problems in mathematical economy; so they admit an economical interpretation. These methods are still insufficiently investigated from the mathematical point of view and not well-known. The development of the iterative aggregation methods, as well as the term, is connected with the works of L. M. Dudkin and E. B. Ershov (see [1], pp. 155–158). The following view on these methods is very significant ([1], p. 158): “Since the theory of the method is developed insufficiently and no convergence conditions are obtained for it, many numerical tests were performed; in many cases the method proved to be efficient”. An important result for an one-parameter case is established in [1]. As applied to a system of linear algebraic equations of the form x = Ax+ b, the convergence conditions for this method adduced in [1] (p. 156) require, in particular, that the elements aij (i = 1, . . . , n; j = 1, . . . ,m) of the matrix A be positive, the components bi of the vector of free terms be nonnegative, and the spectral radius of the matrix Ameets the inequality ρ(A) < 1. In this paper we study several modifications of the iterative aggregation methods for the linear equation