# Informal Introduction: Data Processing, Interval Computations, and Computational Complexity

• Anatoly Lakeyev
• Jiří Rohn
• Patrick Kahl
Chapter
Part of the Applied Optimization book series (APOP, volume 10)

## Abstract

This introduction starts with material aimed mainly at those readers who axe not well familiar with interval computations and/or with the computational complexity aspects of data processing and interval computations. It provides the motivation for the basic mathematical and computational problems that we will be analyzing in this book. Readers who are well familiar with these problems can skip the bulk of this chapter and go straight to the last section that briefly outlines the structure of the book.

In brief, this chapter’s analysis starts with the the following problem (that is considered one of the basic problems of interval computations): given a function f (x1,..., xn) of n real variables, and n intervals xi, = [x i, x̄i], compute the range
$$y = f\left( {{x_1},...{x_n}} \right) = \left\{ {f\left( {{x_1},...{x_n}} \right)/{x_1} \in {x_1},...{x_n} \in {x_n}} \right\}$$

A typical application of this problem is: from the measurements, we know the approximate values x̃i of physical quantities xi, and we know the guaranteed accuracy Δi of each measurement. As a result, we know that xi belongs to the interval xi = [x̃i - Δi, x̃i + Δi]. We also know the algorithm f(x1,..., xn) that transforms the values xi, into the value of the desired quantity y. We want to know the set of possible values of y. For a continuous function f (x1,..., xn), this set is an interval (we will denote it by y = [y, ȳ]). So, the question is: can we compute the endpoints y and y of this interval ȳ in reasonable time?

## Keywords

Arithmetic Operation Interval Computation Guarantee Estimate Exact Range Metic Operation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Authors and Affiliations

• 1
• Anatoly Lakeyev
• 2
• Jiří Rohn
• 3
• Patrick Kahl
• 4
1. 1.University of Texas at El PasoUSA
2. 2.Computing CenterRussian Academy of SciencesIrkutskRussia
3. 3.Charles University and Academy of SciencesPragueCzech Republic
4. 4.IBMTucsonUSA