Информационная система "Конференции"

Abstracts

Computational Algebra

The subject matter of our work is the problem of outer interval estimation of the solution set to an interval linear system $$ {bf A} x = {bf b} $$ with a band interval $ntimes n$-matrix ${bf A}$ and interval right-hand side $n$-vector ${bf b}$. The {em solution set} of the interval linear system will be referred to as the set $$ Xi({bf A,b}) = bigl{,xin{{rm I}hspace{-0.49ex}{rm R}}^n mid (exists Ain{bf A})(exists bin{bf b})(,Ax= b,),bigr}, $$ formed by solutions to all the point systems $Ax = b$ with $Ain{bf A}$ and $bin{bf b}$. An exact description of the solution set is practically impossible for the dimensions $n$ larger than several tens, since its complexity grows exponentially with $n$. On the other hand, such an exact description is not really necessary in most cases. The users traditionally confine themselves to the problems of computing some estimates, in a prescribed sense, of the solution set, and we are going to solve the problem of outer (by supersets) interval estimation: $$ mbox{sl begin{tabular}{c} For, an, interval, system, of, linear, equations, ${bf A}x = {bf b},$ find an interval enclosure of the solution set $,Xi({bf A,b})$. end{tabular}} $$ The above problem is one of the historically first and most popular in modern interval analysis, and we advance an efficient adaptive numerical technique for computing {em suboptimal} (nearly exact) outer estimates of the solution sets for band interval linear systems, discuss its advantages and shortcomings. In particular, the distinctive feature of the new approach is its applicability for interval linear systems whose matrices are not {em strongly regular}, the latter requirement being traditional for most of the existing interval analysis methods.

Note. Abstracts are published in author's edition

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