\documentstyle[IEEEtran]{article}
\begin{document}
\tolerance 10000
\title{If Input Intervals Are Small Enough, Then Interval Computations
Are Almost Always Easy}
\author{A. V. Lakeyev$^1$ and V. Kreinovich$^2$}

\pagestyle{myheadings}

\markright{APIC'95, El Paso,
Extended Abstracts,
A Supplement to the international journal of {\rm Reliable
Computing}\ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ }

\maketitle

\begin{abstract}
It is know that many problems in interval computations are NP-hard,
including the problem of computing the range $f({\bf x}_1,...,{\bf
x}_n)$ of a given polynomial $f$ on given intervals ${\bf x}_i$, and a
problem of finding the exact interval solution of the system of linear
interval equations.

NP-hard means, crudely speaking, 
that for every algorithm that produces the exact
solution to these problems, there are hard cases for which this
algorithm would take exponentially long time to finish its
computations.  Therefore, if we have a feasible algorithm that finds
an interval {\it enclosure} for the desired solution, this feasible
algorithm will inevitably {\it overestimate} for some (hard) cases.

In this paper, we show that if the input intervals are small
enough, then hard cases are {\it rare} in the following 
sense: there is a feasible
algorithm that computes the precise interval for almost all inputs
(``almost all'' in a natural sense).
\end{abstract}

\auffil{The authors are with 
$^1$Irkutsk Computing Center, Russian Academy of Science, 
Siberian Division, Lermontov Str. 134, Irkutsk 664033, Russia,
email lakeyev@icc.ccsoan.irkutsk.su and with 
$^2$Department of Computer Science,
University of Texas at El Paso, El Paso, TX, 79968, email
vladik@cs.utep.edu.
This work was partially 
supported by NSF grant No. CDA-9015006 and NASA Research Grant No. NAG
9-757. We are thankful to J. Rohn, S. Shary, and A. Neumaier for the
encouragement.}

\section{The Basic Problem Of Interval Computations Is Known To Be 
NP-Hard}
\noindent{\bf Definition 1.} {\sl By a 
{\it basic problem} of interval computations, we mean the following
problem:
\begin{itemize}
\item {\it Given:}
\begin{itemize}
\item[$\bullet$]$n$ intervals 
${\bf x}_1=[\tilde x_1-\Delta_1,\tilde x_1+\Delta_1],$ ...,
${\bf x}_n=[\tilde x_1-\Delta_1,\tilde x_1+\Delta_1],$ and
\item[$\bullet$]a program that computes a function $f(x_1,...,x_n)$.
\end{itemize}
\item{\it To compute:} the range of $f$, i.e., the endpoints of the
interval 
$${\bf y}=f({\bf x}_1,...,{\bf x}_n)=$$ $$\{f(x_1,...,x_n)\,|\,x_1\in {\bf
x}_1,...,x_n\in{\bf x}_n\}.$$
\end{itemize}}

It is known that this problem is {\it NP-hard} even for polynomial
functions $f$
\cite{Gaganov 1985}. 

NP-hard means, crudely speaking, that if we believe
that not all problems are easy (the formalization of 
this natural belief is usually denoted by P$\ne$NP), then
for every algorithm that solves this problem, there will be cases on
which the running time $t$ grows exponentially with $n$. So, for some
cases of reasonable size (e.g., $n\approx 300$), the running time of
this algorithm will exceed the lifetime of the Universe (for precise
definitions, see, e.g., \cite{Garey}).

\section{Several Other Problems Of Interval Computations Are
Also Known To Be NP-Hard}
Several other problems of interval computations are also known to be
NP-hard, for example, the problem of solving a system of interval
linear equations:
\smallskip

\noindent{\bf Definition 2.} 
{\sl \begin{itemize}
\item By an {\it interval vector} $\bf b$, we mean a
vector $({\bf b}_1,...,{\bf b}_n)$ whose components are intervals. 
\item If
$b=(b_1,...,b_n)$ is a vector, we say that $b\in {\bf b}$ if $b_i\in
{\bf b}_i$ for all $i$.
\item By
an {\it interval matrix} $\bf A$, we mean a matrix $${\bf A}=\pmatrix{
{\bf A}_{11} & ...& {\bf A}_{1n}\cr
& ...& \cr
{\bf A}_{m1} & ...& {\bf A}_{mn}}.$$ 
\item We say that $A\in {\bf A}$ if
$A_{ij}\in {\bf A}_{ij}$ for all $i$ and $j$.
\end{itemize}}

\noindent{\bf Definition 3.}
{\sl Assume that we have an interval matrix $\bf A$ and an interval vector
$\bf b$. 
\begin{itemize}
\item We say that a real-valued vector 
$x=(x_1,...,x_n)$ is a {\it possible solution} of the system ${\bf
A}x={\bf b}$, if $Ax=b$ for some $A\in {\bf A}$ and $b\in {\bf b}$. 
\item We say that $x_i$ is a {\it possible value of $i-$th component}
$x_i$ of the solution 
if there exists a possible solution with this very value of $x_i$.
\item By an {\it interval of possible values of $x_i$}, we mean the
smallest interval that contains all possible values of $x_i$.
\end{itemize}}

\noindent{\bf Definition 4.} {\sl By the {\it problem of solving a system
of interval linear equations}, we mean the following problem:
\begin{itemize}
\item {\it Given:}
\begin{itemize}
\item[$\bullet$]an interval matrix $\bf A$;
\item[$\bullet$]an interval vector $\bf b$;
\item[$\bullet$]an integer $i$;
\end{itemize}
\item{\it To compute:} the interval of possible values of $x_i$.
\end{itemize}}

This problem is also know to be NP-hard, even for the case when we
only allow square matrices (with $m=n$) \cite{Rohn}.

\section{The Usual Interpretation Of These NP-Hardness Results: Every
Feasible Enclosure Algorithm Sometimes Overestimates}
The usual interpretation of these results is as follows: 
We have no chance to
design a {\it feasible} algorithm that 
always finds the exact interval: be it the exact range of the
function, or the exact interval of possible values of $x_i$
(in theory of computation, an algorithm is usually described as 
feasible if its running time is bounded by a {\it
polynomial} $P(n)$ of the length $n$ of the input 
\cite{Lewis 1981}, Section 7.4; \cite{M91}, Ch. 23). 
Therefore,
we must aim at a feasible algorithm that finds an {\it enclosure} (i.e., an
interval that contains the desired algorithm $\bf y$).
 
For such algorithms, NP-hardness can be, crudely speaking,
reformulated as follows: if we have a feasible algorithm that always
produces an enclosure of the desired interval, then there are cases in
which this algorithm will overestimate.

\section{The Existing Interval Enclosure Algorithms Overestimate For
Almost All Practical Cases}
The existing enclosure methods, starting from naive 
interval computations \cite{Moore 1966}, do overestimate.
But the bad thing is that they overestimate not just in a few cases, but
in {\it almost all} practical cases. 

\section{We Would Like To Have An Algorithm That Overestimates As Rarely
As Possible}
Due to NP-hardness results, we cannot hope to improve these methods to
the point when they would never overestimate, but it would be nice to
have a feasible interval computation
algorithm that overestimates as rarely as possible.

\section{Our Main Result: Informal Description}
In this paper, such an algorithm is presented for the case when the
input intervals are small enough (i.e., when the half-widths
$\Delta_i$ of these intervals are small enough).

We will show that in this case, we can have an algorithm that
almost always (in some reasonable sense) produces the exact
interval answer.

\section{Definitions And The Main Result} 
\noindent{\bf Definition 5.} {\sl Let $\delta>0$ be  a real number. We
say that an interval $[\tilde x-\Delta,\tilde x+\Delta]$ is
{\it $\delta-$small} if $\Delta\le\delta$.}
\smallskip

\noindent{\bf Definition 6.} {\sl Let the following be given:
\begin{itemize}
\item a real number $\varepsilon>0$;
\item a compact domain $D\subseteq R^n$ whose Lebesgue measure is
positive: $\mu(D)>0$ (for polytopes, Lebesgue measure is a standard volume);
\item a property $P(x)$ (that may be true for some
elements $x\in D$ and false for some other $x\in D$). 
\end{itemize}
We say that the property $P$ is true for 
{\it $(D,\varepsilon)-$almost
all} $x$ if $\mu(\{x\in D\,|\,\neg P(x)\})\le \varepsilon\cdot \mu(D)$.}
\smallskip

\noindent{\it Comments.} 

1. In other words, a property is true for
$(D,\varepsilon)-$almost all $x$ if the portion of the domain $D$ 
for which this property is false does not exceed $\varepsilon$.

2. We want to apply interval estimation algorithms to {\it computable}
functions $f$, i.e., to the functions that can be computed by a
computer program.
How to describe such functions?
\begin{itemize}
\item Inside the computer, every function $f$ is described as a sequence of
arithmetic operations. Everything that we can obtain from the
variables $x_1,...,x_n$ by applying four arithmetic operations is a
rational function. So, as a result, what the computer actually
computes is always a {\it rational} function. 
\begin{itemize}
\item[]Even when we write
$\sin(x)$, or $\exp(x)$ in a programming language, then, when the
compiler compiles this expression, it translates it into a sequence of
hardware supported operations, i.e., arithmetic operations. As a
result, what actually is computed is a {\it rational} function that is an
{\it approximation} to the desired one.
\end{itemize}
\item Similarly, when a number
is represented in a computer, this number is represented by its finite 
binary expansion. This expansion is a rational number. So, the
rational function $f$ must have rational coefficients.
\end{itemize}
So, if we are
interested in solving the basic problem of interval computations for
computable functions, it is sufficient to consider the case of
{\it rational functions $f$ with rational coefficients}.
\newpage

\noindent{\bf THEOREM 1.} {\sl There exists a feasible algorithm
$\cal U$ with the following two properties:
\begin{itemize}
\item If we are given:
\begin{itemize}
\item[$\bullet$]$n$ intervals 
${\bf x}_i=[\tilde x_i-\Delta_i,\tilde x_i+\Delta_i],$ and 
\item[$\bullet$]a rational function $f$ with rational 
coefficients, 
\end{itemize}
then this algorithm returns an interval (maybe, an infinite one) 
that {\it encloses} the range 
${\bf y}=f({\bf
x}_1,...,{\bf x_n})$ of the function $f$. 
\item Suppose that we are given:
\begin{itemize}
\item[$\bullet$]a compact domain $D$ of positive Lebesgue measure;
\item[$\bullet$]a rational function $f$ with rational 
coefficients that is finite in some open neighborhood $N\supseteq D$; and
\item[$\bullet$]a real number $\varepsilon>0$.
\end{itemize}
Then, there exists a number $\delta>0$ for which the following is
true:
\begin{itemize}
\item[]For {\it $(D,\varepsilon)-$almost all} $\tilde x=(\tilde
x_1,...,\tilde x_n)$, 
if all $n$ intervals 
${\bf x}_i=[\tilde x_i-\Delta_i,\tilde x_i+\Delta_i]$ are
$\delta-$small, 
then 
this algorithm $\cal U$ returns the {\it exact value of the range} 
$f({\bf x}_1,...,{\bf x}_n)$.
\end{itemize}
\end{itemize}}

\noindent{\bf THEOREM 2.} {\sl There exists a feasible algorithm
$\cal U$ with the following two properties:
\begin{itemize}
\item If we are given:
\begin{itemize}
\item[$\bullet$]a square $n\times n$ interval matrix ${\bf A}$ with elements
${\bf A}_{ij}=[\tilde A_{ij}-\Delta_{ij},\tilde A_{ij}+\Delta_{ij}];$ 
\item[$\bullet$]an $n-$dimensional interval vector ${\bf b}$ with
elements 
${\bf b}_{i}=[\tilde b_{i}-\Delta_{i},\tilde b_{i}+\Delta_{i}],$ and
\item[$\bullet$] an integer $i\le n$;
\end{itemize}
then this algorithm returns an interval (maybe, an infinite one) 
that {\it encloses} the
interval of possible values of $x_i$.  
\item Suppose that we are given:
\begin{itemize}
\item[$\bullet$]a compact domain $D$ of positive $(n\times
n)+n-$dimensional Lebesgue measure, and an open neighborhood $N\supseteq
D$ such that if $(A,b)\in N$, then
the matrix $A$ is non-degenerate (i.e., invertible). 
\item[$\bullet$]a positive real number $\varepsilon>0$. 
\end{itemize}
Then, there exists a number $\delta>0$ for which the following is
true:
\begin{itemize}
\item[]For {\it $(D,\varepsilon)-$almost all} pairs 
$(\tilde A,\tilde b)=(\tilde
A_{11},\tilde A_{12},...,\tilde A_{nn},\tilde b_1,...,\tilde b_n)$,  
if all intervals ${\bf A}_{ij}=[\tilde A_{ij}-\Delta_{ij},\tilde
A_{ij}+\Delta_{ij}],$ and 
${\bf b}_{i}=[\tilde b_{i}-\Delta_{i},\tilde b_{i}+\Delta_{i}]$ are
$\delta-$small, then 
this algorithm $\cal U$ returns the {\it exact} interval of possible
values of $x_i$.
\end{itemize}
\end{itemize}}

\noindent{\it Comment.} In other words, if the intervals are small
enough, then these algorithms give {\it exact} intervals in almost all
cases.

\section{Proofs}

\subsection{Proof of Theorem 1}
\subsubsection{A comment on the proof}
We will prove that this theorem is true for a quite reasonable
algorithm $\cal U$ (that we will describe below). This algorithm has,
in effect, been used many times before, and it is
not our main achievement. Our main achievement is the observation that this
algorithm gives an exact estimate in almost all cases.

\subsubsection{Description of the algorithm. Stage 1: simplifying $f$}
First, we must check whether the given function $f$ actually depends
on all of its variables. 

To do that, we will:
\begin{itemize}
\item apply analytical
differentiation formulas to $f$ (see, e.g., \cite{Rall}),
and 
\item check whether the resulting derivatives $$f_{,i}={\partial
f\over\partial x_i}$$
are identically 0 or not. 
\end{itemize}
The derivative of a rational function is also
rational. Therefore, this rational function can be represented as a
ratio of two polynomials $f_{,i}=P/Q$. This ratio is identically equal
to 0 iff the polynomial $P$ is identically equal to 0, and the
polynomial is identically 0 iff all its coefficients are zeros. Since
all the coefficients of $f$ are rational numbers, the coefficients of
$f_{,i}$ (and thus, of $P$) are also rational numbers, and for
rational numbers, equality to 0 is easy to check.

If for some $i$, it turns out that the partial derivative $f_{,i}$ 
is identically 0, then $f$ does not depend on this variable $x_i$ at
all. So, we can substitute an arbitrary value of $x_i$ into the
original formula $f$, and get a new expression with one fewer
variable. 

We can repeat this procedure for all the variables $x_i$ 
for which $f_{,i}$ is identically 0. As a result, we get a new
expression for $f$ that only contains the variables on which $f$
essentially depend (i.e.g, for which the derivative $f_{,i}$ is not
identically 0). 
\smallskip

\noindent{\it Comment.} To explain why this procedure is so
complicated, let us give an example of a rational 
function $f$ that does not depend on $x_3$ but whose expression
explicitly contains $x_3$:
$$f(x_1,x_2,x_3)={x_1^2+x_1x_2+x_1x_3+x_2x_3\over x_1^2+x_1x_3}.$$
If we factorize the numerator and the denominator, we will see that
this function $f$, indeed, does not depend on $x_3$:
$$f={(x_1+x_2)(x_1+x_3)\over x_1(x_1+x_3)}={x_1+x_2\over x_1}.$$
We do not want to factorize, because factorization can be a very
time-consuming procedure. Instead, we differentiate with respect to
all three variables, and indeed get $f_{,3}=0$. As a result, we
substitute an arbitrary constant instead of $x_3$, and get an
equivalent expression for $f$ that contains only two variables. For
example, if we
substitute $x_3=0$, we get the following expression:
$$f(x_1,x_2)={x_1^2+x_1x_2\over x_1^2}.$$

\subsubsection{Description of the algorithm. Stage 2}
After stage 1, we have a rational expression $f(x_1,...,x_m)$ for which
for all $i\le m$, the derivative $f_{,i}$  is not identically 0.
For this expression, we do the following:
\begin{itemize}
\item First, for each $i$ from 1 to $m$, 
we apply an interval centered form \cite{Center} 
to compute the interval enclosure 
${\bf e}_i=[e_i^-,e^+_i]$ for $f_{,i}({\bf x}_1,...,{\bf x}_m)$. 
\item If for some $i$, the interval ${\bf e}_i$ contains 0, then we
apply the same interval center form (or any other interval computation
technique) to compute an enclosure ${\bf e}$ for the desired range
$f({\bf x}_1,...,{\bf x}_m)$. 
\item If for all $i$, the interval ${\bf e}_i$ does not contain 0,
then we compute $s_i={\rm sign}(e^+_i)$ (where ${\rm sign}(a)=1$ for
$a>0$ and ${\rm sign}(a)=-1$ if $a<0$), and return the interval
$[y^-,y^+]$, where
$$y^-=f(\tilde x_1-s_1\Delta_1,...,\tilde x_m-s_m\Delta_m),$$
$$y^+=f(\tilde x_1+s_1\Delta_1,...,\tilde x_m+s_m\Delta_m),$$ as the exact
value of the range.
\end{itemize}

\subsubsection{Proof that this algorithm always computes the enclosure}
If $0\in {\bf e}_i$, then the fact that the above-described algorithm
returns an enclosure follows from the properties of interval
computation methods.
\smallskip

If $0\not\in{\bf e}_i$ for all $i$, this means that for every $i$,
this means that for $(x_1,...,x_m)\in {\bf x}_1\times ...\times {\bf
x}_m$, the function $f$ is {\it monotonic} with respect to each of its
variables:
\begin{itemize}
\item If ${\bf e}_i>0$, then $f_{,i}>0$, and $f$ is {\it increasing}
w.r.t. $x_i$.
\item If ${\bf e}_i<0$, then $f_{,i}<0$, and $f$ is {\it decreasing}
w.r.t. $x_i$. 
\end{itemize}

If a function $f$ is increasing w.r.t. $x_i$, then its maximum is
attained when $x_i$ takes the largest possible values. On the interval
$[\tilde x_i-\Delta_i,\tilde x_i+\Delta_i]$, the largest possible
value is $x_i=\tilde x_i+\Delta_i$. 

If a function $f$ is decreasing w.r.t. $x_i$, then its maximum is
attained when $x_i$ takes the smallest possible values. On the interval
$[\tilde x_i-\Delta_i,\tilde x_i+\Delta_i]$, the smallest possible
value is $x_i=\tilde x_i-\Delta_i$. 

We can combine these two cases by saying that if $f$ is monotone
w.r.t. $x_i$, then the maximum of $f$ is attained for 
$x_i=\tilde x_i+s_i\Delta_i$.

In our case, $f$ is monotone w.r.t. all its variables, so, its maximum
is attained when each of its variables $x_i$ takes the value
$x_i=\tilde x_i+s_i\Delta_i$. This maximum is therefore equal to 
$$f(\tilde x_1+s_1\Delta_1,...,\tilde x_m+s_m\Delta_m).$$

Similarly, one can prove that the minimum of the set of possible
values of $f$ (i.e., the lower endpoint of the range interval $f({\bf
x}_1,...,{\bf x}_m)$ is equal to 
$$f(\tilde x_1-s_1\Delta_1,...,\tilde x_m-s_m\Delta_m).$$

So, in this case, the result of our algorithm not only {\it encloses}
the desired range, it {\it coincides} with this range.
\smallskip

\noindent{\it Comments.}

1. As we have already mentioned, the use of monotonicity is not a
novel idea. As examples of successful applications of this idea, we
can cite 
\cite{Collatz 1952,Collatz 1960,Lakshmikantham 1969,Walter 1970,Harrison 1977,Moore 1979,Schroder 1980,Rall 1983,Mannshardt 1990,Dimitrova 1991,Markov 1992}.

2. To complete our proof, we must show that the described algorithm
returns the exact range in almost all cases.

\subsubsection{Proof that 
this algorithm almost always computes the exact range}

After the first stage, we have a 
rational function $f(x_1,...,x_m)$ that depends on each of its $m$ variables. 
This rational function is finite on some open neighborhood $N$ of $D$.
Since $N$ is an open neighborhood, there exists a $\delta_0>0$ 
such that every point that is $\le\delta_0-$close 
to the set $D$ (in the $\sup$ metric $d(x,y)=\max|x_i-y_i|$) 
also belongs to $N$. Let us denote the set of all the
points that are $\delta_0-$close to $D$ (i.e., a $\delta_0-$neighborhood
of $D$) by $D_0$. This set $D_0$ is closed and bounded (since $D$ is
bounded), so, $D_0$ is a compact. 

A rational function $f$ is continuous in every point $x$ in which it is
finite. Since $f$ is finite on all points of $N$, it is also finite on
all points from $D_0\subseteq N$. Therefore, $f$ is continuous on every
point of $D_0$.

Since $f$ is rational, for 
every $i$, the partial derivative 
$f_{,i}$ is also a rational function,
i.e., a ratio of two polynomials: $f_{,i}=P_i/Q_i$. 
Therefore, the set $S_i$ of all points
$x\in D$ for which $f_{,i}(x)=0$ is a set of solutions of the equation
$P_i(x)=0$ for some polynomial $P_i$. From the topological viewpoint,
the structure of the set of all the roots of a polynomial (that is not
identically 0) is well known: it is 
either a manifold of dimension $\le m-1$, or a union of 
finitely many manifolds of dimension $\le m-1$. 
In both cases, the Lebesgue measure $\mu(S_i)$ of this set is equal to 0:
$$\mu(S_i)=\mu(\{x\,|\,f_{,i}(x)=0\})=0.$$

In our algorithm, we do not use the exact values of $f_{i,}$, we only
use an {\it estimate} (i.e., in general, an {\it approximation}) to
the range of $f_{,i}$. Therefore, to analyze the behavior of our
algorithm, we would like to have some measure estimate
that is based not on the exact value of $f_{,i}(x)$, but on the
interval of possible values. Such an estimate is easy to get:
Indeed, $S_i$ is an intersection of the sequence of monotone
decreasing sequence of sets:
$$S_i=\{x\,|\,f_{,i}(x)=0\}=\bigcap_{k=1}^\infty 
\{x\,|\,|f_{,i}(x)|\le 2^{-k}\}.$$ Therefore, 
$$\mu(\{x\,|\,|f_{,i}(x)|\le 2^{-k}\})\to \mu(S_i)=0$$
as $k\to\infty$. So, there exists a $k_i$ for which 
$$\mu(\{x\,|\,|f_{,i}(x)|\le 2^{-k_i}\})\le {\varepsilon\cdot\mu(D)\over m}.$$
If we denote by $k$ the largest of these $k_i$:
$$k=\max(k_1,...,k_m),$$ then we will conclude
that 
$$\mu(\{x\,|\,|f_{,i}(x)|\le 2^{-k}\})\le $$
$$\mu(\{x\,|\,|f_{,i}(x)|\le 2^{-k_i}\})\le 
{\varepsilon\cdot\mu(D)\over m}.$$
Let us denote the corresponding set 
$$\{x\,|\,|f_{,i}(x)|\le 2^{-k}\}$$
by $A_i$. Then, for a union $A$ of these sets
$$A=\bigcup_{i=1}^m A_i=\{x\,|\,|f_{,i}(x)|\le 2^{-k}{\rm\ for\ some\ }i\},$$
we have $$\mu(A)\le \sum_{i=1}^m \mu(A_i)\le \sum_{i=1}^m 
{\varepsilon\cdot\mu(D)\over m}=\varepsilon\cdot\mu(D).$$

We want to find $\delta$ for which the above-described algorithm 
computes the precise range for all $\delta-$small intervals whose
centers $\tilde x=(\tilde x_1,...,\tilde x_n)$ satisfy the condition  
$\tilde x\not\in A$. Let us find such a $\delta$.
For that, we will use the following two ideas:
\begin{itemize}
\item For every $i$ from 1 to $m$, the rational function $f_{,i}$ is
continuous on a compact $D_0$. Therefore, $f_{,i}$ is uniformly
continuous. In particular, there exists a $\delta_i>0$ such that if
$d(x,\tilde x)\le \delta_i$, then 
$|f_{,i}(x)-f_{,i}(\tilde x)|\le 2^{-(k+2)}$. 
\item According to \cite{Center}, the centered method has an accuracy
$\le O(\max(\Delta_1,...,\Delta_m))$. This means that for every $i$
from $1$ to $m$,
there exists a constant $C_i$ such that if all input intervals are
$\delta-$small, then the difference between the endpoints of the
actual range $[a^-_i,a^+_i]=f_{,i}({\bf x}_1,...,{\bf x}_m)$ 
and the corresponding endpoints of the estimated range 
${\bf e}_i=[e^-_i,e^+_i]$
does not exceed $C_i\cdot\delta$:
$$|e^-_i-a^-_i|\le C_i\cdot\delta,\ |e^+_i-a^+_i|\le C_i\cdot\delta.$$
\end{itemize}
Now, let us take $$\delta=\min(\delta_0,\delta_1,...,\delta_m,
{2^{-(k+2)}\over C_1}, ...,{2^{-(k+2)}\over C_1}).$$
Let us show that for this $\delta$, if the input intervals are
$\delta-$small, and $\tilde x\not\in A$, then none of the intervals 
${\bf e}_i$ contain zero and therefore, for these algorithms, the
above-described algorithm returns the exact interval range. 

Indeed, 
since $\tilde x\not\in A$, by definition of the set $A$, we have 
$|f_{,i}(\tilde x)|> 2^{-k}$ 
for all $i$. This means that either $f_{,i}(\tilde x)> 2^{-k}$, or
$f_{,i}(\tilde x)<-2^{-k}$. 
\begin{itemize}
\item Let us first consider the first case.
In this case, if $$x_j\in {\bf x}_j=[\tilde
x_j-\Delta_j,\tilde x_j+\Delta_j],$$ then (since the interval ${\bf
x}_j$ is $\delta-$small), we have $$|x_j-\tilde
x_j|\le\Delta_j\le\delta.$$ Hence, $$d(x,\tilde x)=\max |x_j-\tilde
x_j|\le \delta.$$ Due to our choice of $\delta$ as $\min
(...,\delta_i,...)$, we have $\delta\le\delta_i$ and therefore,
$d(x,\tilde x)\le \delta_i$. Due to our choice of $\delta_i$ this
inequality, in turn, leads to the inequality $|f_{,i}(x)-f_{,i}(\tilde
x)|\le 2^{-(k+2)}.$ In particular, we have $f_{,i}(x)\ge f_{,i}(\tilde
x)-2^{-(k+2)}$. Since $f_{,i}(\tilde x)>2^{-k}$, we can conclude
that $$f_{,i}(x)>2^{-k}-2^{-(k+2)}=3\cdot 2^{-(k+2)}.$$
So, if $x_1\in
{\bf x}_1$, ..., and 
$x_m\in {\bf x}_m$, then $f_{,i}(x)\ge 3\cdot 2^{-(k+2)}$. 
Therefore, the lower bound $a^-_i$ of the actual range $f_{,i}({\bf
x}_1,...,{\bf x}_m)$ satisfies the inequality $$a^-_i\ge 3\cdot 2^{-(k+2)}.$$ 

\item[] Now, because of our choice of $\delta$ as $$\delta=\min (...,
2^{-(k+2)}/C_i,...),$$ 
we have $\delta\le 2^{-(k+2)}/C_i$. Hence, $$C_i\cdot \delta\le
2^{-(k+2)}.$$ By the choice of $C_i$, this means that $|e^-_i-a^-_i|\le
2^{-(k+2)}$. 
Hence, $e^-_i\ge a^-_i-2^{-(k+2)}$. We already know that $a^-_i\ge
3\cdot 2^{-(k+2)}$. Hence, $$e^-_i\ge 3\cdot
2^{-(k+2)}-2^{-(k+2)}=2\cdot 2^{-(k+2)}>0.$$ The lower bound of the
interval ${\bf e}_i$ is positive, hence, this interval can only
contain positive numbers and cannot contain 0.

\item If $f_{,i}(\tilde x)<-2^{-k}$, then we can similarly conclude that the
upper bound $e^+_i$ of the interval ${\bf e}_i$ is negative; hence,
this interval contains only negative numbers and cannot contain 0.
\end{itemize} 
So, for these intervals, the above algorithm gives the exact range. Q.E.D.

\subsection{Proof of Theorem 2}
Theorem 2 follows from Theorem 1 if we take into consideration the
fact that if $A$ is invertible, then the solution $x=A^{-1}b$ 
of the system of linear equations $Ax=b$ is a rational
function of the coefficients $A_{ij}$ and $b_i$. 

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\end{document}

