Patrick Kahl

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

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.

Solving Differential Equations

Yet another problem where interval computations axe used is a problem of solving differential equations. In this chapter, we show that in general, this problem requires at least exponential time, and briefly describe the heuristics that are used to solve important particular classes of differential equations in polynomial time.

The Notions of Feasibility and NP-Hardness: Brief Introduction

The main goal of this book is to analyze computational complexity and feasibility of data processing and interval computations. In Chapter 1, we defined the basic problems of data processing and interval computations. In this chapter, we give a brief introduction to the notions related to feasibility and computational complexity.

What if Quantities are Discrete

The basic problem of interval computations that we analyze in this book is to compute the accuracy of the results of data processing. To be more precise, we know an algorithm f(x1,...,x n) that is used in data processing and the intervals xi of possible values of the input quantities x 1,..., x n and we want to describe the set of all possible values of f(x 1,..., x n) when x i ∈ x i.

Non-Interval Uncertainty II: Multi-Intervals and Their Generalizations

The basic problem of interval computations is to find all possible values of y = f(x1,..., xn) when we only have partial information about xi.

Properties of Interval Matrices II: Proofs and Auxiliary Results

In this chapter, we prove the results about computational complexity and feasibility of properties of interval matrices that were formulated in the previous chapter. Along the way, we also describe some important auxiliary results.

Engineering Corollary: Signal Processing is NP-Hard

In this chapter, we present another example of a practical problem which is computationally intractable in the presence of interval uncertainty: signal processing.

In the General Case, the Basic Problem of Interval Computations is Intractable

In this chapter, we describe the first negative result: that even for polynomials f(x1,..., xn), the basic problem of interval computations — the problem of computing the range f(x1,..., xn) for given intervals xi — is computationally intractable (NP-hard).

Solving Systems of Equations

In this chapter, we analyze the computational complexity and feasibility of yet another computational problem in which interval methods are often used: solving systems of equations. It turns out that already for systems of quadratic equations, solving these systems is NP-hard.

If Input Intervals are Narrow Enough, Then Interval Computations are Almost Always Easy

In the previous chapters, we have shown that in general, interval computations are NP-hard. This means, crudely speaking, that every algorithm that solves the interval computation problems requires, in some instances, unrealistic exponential time. Thus, the worst-case computational complexity of the problem is large. A natural question is: is this problem easy “ on average” (i.e., are complex instances rare), or is this problem difficult “ on average” too?