Abstract
In this paper, theidentification problem, thetolerance problem, and thecontrol problem are treated for the interval linear equation Ax=b. These problems require computing an inner approximation of theunited solution set Σ∃∃(A, b)={x ∈ ℝn | (∃A ∈ A)(Ax ∈ b)}, of thetolerable solution set Σ∀∃(A, b)={x ∈ ℝn | (∀A ∈ A)(Ax ∈ b)}, and of thecontrollable solution set Σ∃∀(A, b)={x ∈ ℝn | (∀b ∈ b)(Ax ∈b)} respectively. Analgebraic approach to their solution is developed in which the initial problem is replaced by that of finding analgebraic solution of some auxiliary interval linear system in Kaucher extended interval arithmetic. The algebraic approach is proved almost always to give inclusion-maximal inner interval estimates of the solutionsets considered. We investigate basic properties of the algebraic solutions to the interval linear systems and propose a number of numerical methods to compute them. In particular, we present the simple and fastsubdifferential Newton method, prove its convergence and discuss numerical experiments.
Abstract
Б этой работе рассматриваютсяэa¶rt;aчa u¶rt;eнmuфuxaцuu, эa¶rt;aчa o ¶rt;onyckax н эa¶rt;aчa o¶rt; ynpaвlenuu для интервальной линейной системы Ax=b, требующие нахожления внутренней оценки дляоб Σ∃∃(A, b)={x ∈ ℝn | (∃A ∈ A)(Ax ∈ b)}, Σ∀∃(A, b)={x ∈ ℝn | (∀A ∈ A)(Ax ∈ b)}, и Σ∃∀(A, b)={x ∈ ℝn | (∀b ∈ b)(Ax ∈b)} соответственно. Развиваетсян к их решению, при котором исходная задача заменяется задачей отысканиян для некоторой вспомогательной интервальной линейной системы в расширенной интервальной арифметике Каухера. Показано, что алгебраический нодход почти всегда дает максимальные по включению внутренние оценки для рассматриваемых множеств решений. Исследуются основные свойства алгебраическнх решений интервальных систем, обсужлаются численные метолы для их нахожления. Б частности, мы предалагаем простой и быстрыйн, доказываем его сходимость и приводим результаты численных экспериментов с ним.
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Shary, S.P. Algebraic approach to the interval linear static identification, tolerance, and control problems, or one more application of kaucher arithmetic. Reliable Comput 2, 3–33 (1996). https://doi.org/10.1007/BF02388185
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DOI: https://doi.org/10.1007/BF02388185