Article information

1998 , Volume 3, ¹ 2, p.67-114

Shary S.P.

Algebraic approach in the `outer problem` for interval linear equations

The subject of our work is the classical ``outer`` problem for the interval linear algebraic system Ax = b with the interval [Java Applet]-matrix A and interval right-hand side n-vector b: find ``outer`` component-wise estimates of the solution set $\Sigma\!=\!\{x\in\Bbb{R}^n\mid(\exists A\in\bold{A})\linebreak(\exists b\in \bold{b})(Ax = b)\,\}$ formed by all solutions to the point systems Ax = b with A$\in$A and b $\in$b, that is, evaluate $\min\{\,x_k \mid x\in\Sigma\,\}$ from below and $\max\{\,x_k \mid x\in\Sigma\,\}$ from above, $k = 1,2, \ldots, n$. The purpose of this work is to advance a new algebraic approach to the problem, in which it reduces to computing the algebraic solution to an auxiliary system in Kaucher complete interval arithmetic, or, what is equivalent, to solving one noninterval (point) equation in the Euclidean space of the double dimension R2n. We construct a specialized algorithm - subdifferential Newton method - that implements the new approach, present results of numerical testing that demonstrate its exclusive computational efficacy with high quality enclosures of the solution set.

[full text] Classificator Msc2000:
*65F30 Other matrix algorithms
65G30 Interval and finite arithmetic
Classificator Computer Science:
*G.1.0 General (Numerical Analysis)
G.1.3 Numerical Linear Algebra

Keywords: interval linear systems,`outer` problem,algebraic approach,Kaucher complete interval arithmetic, interval analysis, linear system, interval arithmetic, numerical examples, algorithm, subdifferential Newton method, computational efficiency

Author(s):
Shary Sergey Petrovich
Dr. , Senior Scientist
Position: Leading research officer
Office: Federal Research Center for Information and Computational Technologies
Address: 630090, Russia, Novosibirsk, Ac. Lavrentiev ave, 6
Phone Office: (3832) 30 86 56
E-mail: shary@ict.nsc.ru
SPIN-code: 9938-9344


Bibliography link:
Shary S.P. Algebraic approach in the `outer problem` for interval linear equations // Computational technologies. 1998. V. 3. ¹ 2. P. 67-114
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