\documentclass{article}

\title{On the Algebraic Solution to Interval Equations}
\author{Svetoslav Markov
        \thanks{Institute of Mathematics and Informatics,
                Bulgarian Academy of Sciences,
                Acad.~G.~Bonchev str., block 8,
                BG-1113 Sofia, Bulgaria,
                E-mail : smarkov@iph.bio.acad.bg
                } }
\date{}
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\begin{document}

\maketitle

\thispagestyle{empty}

Consider a linear algebraic system ${\bf Ax}={\bf b}$ involving
intervals in the $(n\times n)$-matrix ${\bf A}$ and in the right-hand
side $n$-vector ${\bf b}$.  Here we shall be concerned with the {\em
(interval) algebraic solution} which is an interval $n$-vector ${\bf
x}$ satisfying the system whenever the arithmetic operations are
performed in interval arithmetic.  We shall write this problem in the
form
\begin{equation}
\label{Isolution} {\bf A}\times {\bf x}={\bf b}, \end{equation}
to emphasize that the symbol ``$\times$'' means {\em interval
multiplication} (in the sense of \cite{GardenesTrepat80},
\cite{Kaucher80}, \cite{Ortolf69}) of the interval matrix ${\bf A}$
by the solution vector ${\bf x}$; the latter being generally an
interval vector.

The notation used for the interval arithmetic operations is
convenient for the formulation of new computational rules and for
algebraic transformations.  Using such symbolic notation based on
binary
variables taking values from the set $\{ +, - \}$, we formulate new
distributivity relations and simple rules for the transformation of
algebraic expressions and equations \cite{Markov92}--\cite{Markov95}.
This leads to a powerful complete interval algebraic structure,
called {\em directed interval arithmetic}.  The latter unifies both
Kaucher's extended interval arithmetic and the extended arithmetic
for normal intervals using {\em inner} ({\em nonstandard}) operations
developed earlier by the author \cite{Markov80}, \cite{Markov92}.  In
particular it has been proved that extended interval arithmetic with
inner operations:  i) is a ``projection'' of the directed interval
arithmetic system on the system of normal intervals, and, ii) can be
isomorphically embedded in the directed interval algebraic system
\cite{Markov95}, \cite{Markov96}.

Directed interval arithmetic provides a simple general framework
which enables us to pass from directed (proper and improper)
intervals to proper intervals (with inner operations), and vice
versa.  Briefly, directed interval arithmetic is the interval
arithmetic described in \cite{GardenesTrepat80}, \cite{Kaucher80},
\cite{Ortolf69}, which:  1) incorporates the exended interval
arithmetic with inner operations (as a result of applying familiar
algebraic constructions), and 2) is equipped by a set of new
relations and computational rules (formulated in terms of our
``plus-minus'' notations), necessary for the straightforward symbolic
transformation of algebraic expressions and equations.

It has been noticed that the interval algebraic solution to
(\ref{Isolution}) is closely related to the solutions of three
practically significant linear algebraic problems involving interval
coefficients:  the united, the controlled and the tolerable solutions
\cite{GardenesTrepat80}--\cite{Kupriyanova94},
\cite{Shary93}--\cite{Shary96}.  This shows the importance of a
selfcontained study of the algebraic solution to (\ref{Isolution}).

In this work we propose some algebraic rules and methods for
finding the algebraic solution of the interval system
(\ref{Isolution})
using  directed interval arithmetic.

\begin{thebibliography}{99}

\bibitem{GardenesTrepat80}  Garde\~{n}es, E., A. Trepat,         {\em
Fundamentals of SIGLA, an Interval Computing System over the
Completed
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\bibitem{Kaucher80} Kaucher, E., {\em Interval Analysis in the
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\bibitem{Kupriyanova94} Kupriyanova, L., {\em Inner Estimation of the
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\bibitem{Markov80} Markov, S.,  {\em Some Applications of the
Extended Interval Arithmetic to Interval Iterations}, Computing
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{\sl 2}, 69--84 (1980).

\bibitem{Markov92} Markov, S.,  {\em Extended Interval Arithmetic
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{\sl 6}, 3,  269--304 (1992).

\bibitem{Markov93} Markov, S., {\em Some Interpolation Problems
Involving Interval Data}, Interval Computations, 3, 164--182 (1993).

\bibitem{Markov95} Markov, S., {\em On Directed Interval Arithmetic
and
    its Applications}, J. UCS, 1, 7, 510--521 (1995).

\bibitem{Markov96} Markov, S., {\em Isomorphic Embeddings of Interval
  Systems}, Reliable Computing, vol.~3, 3, 199--207 (1997).

\bibitem{Ortolf69} Ortolf, H.-J., Eine Verallgemeinerung der
Intervallarithmetik, Berichte der GMD, Bonn, 11, 1969.

\bibitem{Shary93} Shary, S.,  {\em  Controllable Solution Set to
Interval Static Systems}, Applied Mathematics and Computations,
86/2--3,
185--196 (1997);
see also: Shary, S.,  {\em On Controlled Solution Set of
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\bibitem{Shary95} Shary, S.,  {\em Solving the Linear Interval
Tolerance
Problem}, Mathematics and Computers in Simulation, 39, 53--85,
(1995);
see also: Shary, S.,  {\em Solving the Tolerance Problem
for Interval Linear Equations}, Interval Computations, 2,
6--26 (1994).

\bibitem{Shary96} Shary, S.,  {\em Algebraic Approach to the
Interval Linear Static Identification, Tolerance and Control
Problems,
or One More Application of Kaucher Arithmetic}, Reliable Computing,
2,
 1, 1996, 1--32.

\end{thebibliography}

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