Affiliation:
1. Chuvash State University
2. Flinders University of South Australia; Lomonosov Moscow State University
Abstract
The research purpose is to develop and fully mathematically justify a stable method for finding a normal D-pseudosolution of inconsistency systems of linear algebraic equations with approximate data.
Materials and methods. The paper uses an analogue of the Weirstrass theorem from the theory of optimization methods and the concept of norms in finite-dimensional spaces and extended version of Hoffman’s lemma to determine the distance from an arbitrary point to a polyhedron.
Research results. The article proposes an ideologically simple, reliable and stable method – the pointwise residual method for finding D-pseudosolutions and measures of inconsistency of systems of linear algebraic equations, obtained during the approximation of Fredholm integral equations of the first kind, which describe a number of engineering tasks. To use this method, it is enough to know information of approximate data and estimates of their error. The convergence theorem of the method is proved and estimate of the convergence rate of the method of the same order as that of setting errors in the initial data is obtained. The method is optimal in order.
Conclusions. A new stable method for numerically finding a normal D-pseudosolution of systems of linear algebraic equations with approximate data in the absence of information about their solvability is proposed. This method is nonparametric and requires one time solving an optimization problem with piecewise linear constraints, and in some cases solving a quadratic programming problem.
Publisher
I.N. Ulianov Chuvash State University
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