• Journal of applied mathematics & informatics
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• 제6권3호
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• pp.757-770
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• 1999

We analyse in this paper the possibility of using preconditioning techniques as for square non-singular systems, also in the case of inconsistent least-squares problems. We find conditions in which the minimal norm solution of the preconditioned least-wquares problem equals that of the original prblem. We also find conditions such that thd Kaczmarz-Extendid algorithm with relaxation parameters (analysed by the author in [4]), cna be adapted to the preconditioned least-squares problem. In the last section of the paper we present numerical experiments, with two variants of preconditioning, applied to an inconsistent linear least-squares model probelm.

• Journal of applied mathematics & informatics
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• 제6권1호
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• pp.51-64
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• 1999

We give a new characterization of the solutions set of the general (inconsistent) linear least-squares problem using the set of linit-points of an extended version of the classical Daczmarz's pro-jections method. We also obtain a "step error reduction formula" which in some cases can give us apriori information about the con-vergence properties of the algorithm. Some numerical experiments with our algorithm and comparisons between it and others existent in the literature are made in the last section of the paper.

• Journal of applied mathematics & informatics
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• 제8권1호
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• pp.9-26
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• 2001

In some previous papers the author extended two algorithms proposed by Z. Kovarik for approximate orthogonalization of a finite set of linearly independent vectors from a Hibert space, to the case when the vectors are rows (not necessary linearly independent) of an arbitrary rectangular matrix. In this paper we describe combinations between these two methods and the classical Kaczmarz’s iteration. We prove that, in the case of a consistent least-squares problem, the new algorithms so obtained converge ti any of its solutions (depending on the initial approximation). The numerical experiments described in the last section of the paper on a problem obtained after the discretization of a first kind integral equation ilustrate the fast convergence of the new algorithms. AMS Mathematics Subject Classification : 65F10, 65F20.