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2014 | Book

Intervals Read chapter Read first chapter

Modal Interval Analysis

New Tools for Numerical Information

Authors: Miguel A. Sainz, Joaquim Armengol, Remei Calm, Pau Herrero, Lambert Jorba, Josep Vehi

Publisher: Springer International Publishing

Book Series : Lecture Notes in Mathematics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This book presents an innovative new approach to interval analysis. Modal Interval Analysis (MIA) is an attempt to go beyond the limitations of classic intervals in terms of their structural, algebraic and logical features. The starting point of MIA is quite simple: It consists in defining a modal interval that attaches a quantifier to a classical interval and in introducing the basic relation of inclusion between modal intervals through the inclusion of the sets of predicates they accept. This modal approach introduces interval extensions of the real continuous functions, identifies equivalences between logical formulas and interval inclusions, and provides the semantic theorems that justify these equivalences, along with guidelines for arriving at these inclusions. Applications of these equivalences in different areas illustrate the obtained results. The book also presents a new interval object: marks, which aspire to be a new form of numerical treatment of errors in measurements and computations.

Table of Contents

Frontmatter
Chapter 1. Intervals
Abstract
Classical, or set-theoretical intervals [1, 60–62] are a conceptual tool of computation with a sufficiently mature theoretical background to make the development of its techniques of application a major center of interest [33, 44].
Miguel A. Sainz, Joaquim Armengol, Remei Calm, Pau Herrero, Lambert Jorba, Josep Vehi
Chapter 2. Modal Intervals
Abstract
The semantical lack of the classical system of intervals cannot be resolved by remaining bound to the idea that identifies each interval [a, b] with the set of numerical values x for which the condition a ≤ x ≤ b holds. The way out must be found, therefore, through a restatement of the problem.
Miguel A. Sainz, Joaquim Armengol, Remei Calm, Pau Herrero, Lambert Jorba, Josep Vehi
Chapter 3. Modal Interval Extensions
Abstract
The problem discussed in this chapter is that of obtaining a class of interval functions \(F: {I}^{{\ast}}({\mathbb{R}}^{k}) \rightarrow {I}^{{\ast}}(\mathbb{R})\), consistently referring to the continuous functions f from \({\mathbb{R}}^{k}\) to \(\mathbb{R}\).
Miguel A. Sainz, Joaquim Armengol, Remei Calm, Pau Herrero, Lambert Jorba, Josep Vehi
Chapter 4. Interpretability and Optimality
Abstract
The Semantic Theorems show that \({f}^{{\ast}}(\boldsymbol{X})\) and \({f}^{{\ast}{\ast}}(\boldsymbol{X})\) are optimal from a semantic point of view, and clarify which ⊆ -sense of rounding is the right one when *-semantic or **-semantic are to be applied. They provide, therefore, a general norm that computational functions F from \({I}^{{\ast}}({\mathbb{R}}^{k})\) to \({I}^{{\ast}}(\mathbb{R})\) must satisfy to conform to the f ∗ or the f ∗∗-semantic, but this is still not a general procedure by which these functions may be effectively computed. These procedures will be provided by the modal syntactic extension of continuous real functions, as far as they satisfy certain suitability conditions.
Miguel A. Sainz, Joaquim Armengol, Remei Calm, Pau Herrero, Lambert Jorba, Josep Vehi
Chapter 5. Interval Arithmetic
Abstract
In this chapter the semantic interval extensions of the simplest elementary functions: the arithmetic operators (addition, multiplication, division), logarithm, exponential functions (including the hyperbolic functions), power function, and the trigonometric functions and their inverses, are considered, together with their most important properties and their arithmetic implementations.
Miguel A. Sainz, Joaquim Armengol, Remei Calm, Pau Herrero, Lambert Jorba, Josep Vehi
Chapter 6. Equations and Systems
Abstract
Similarly to the case of one interval equation A ∗ X = B, it is possible to treat the general problem of finding solutions for a system of linear interval equations A ∗X = B and to obtain a semantics for them, compatible with the necessary rounding.
Miguel A. Sainz, Joaquim Armengol, Remei Calm, Pau Herrero, Lambert Jorba, Josep Vehi
Chapter 7. Twins and f ∗Algorithm
Abstract
This chapter deals with the construction of an algorithm to obtain inner and outer approximations of the f ∗ extension of a continuous function f, in the case of non-monotony of f in the studied domain. One convenient approach, but not the only one, is to simultaneously work with both inner and outer approximations. This kind of interval representation, referred to as twins, have already been studied in the field of classical intervals [55, 64]. First of all, twins with modal intervals will be presented.
Miguel A. Sainz, Joaquim Armengol, Remei Calm, Pau Herrero, Lambert Jorba, Josep Vehi
Chapter 8. Marks
Abstract
Working on any digital scale, either a computation scale or a reading/writing measurement scale, digital values must be considered as intrinsically inexact. For example, consider an electrical circuit where a voltage measured with a voltmeter is 11. 3 V and a resistance of \(50\,\Omega \) is measured with an ohmmeter. These values are obviously associated to their measurement devices, which have their corresponding errors. A priori, one can think that these measurements and errors could be represented by intervals, but these values need to be represented in a digital scale and they could be considered valid or not in accordance to a certain tolerance.
Miguel A. Sainz, Joaquim Armengol, Remei Calm, Pau Herrero, Lambert Jorba, Josep Vehi
Chapter 9. Intervals of Marks
Abstract
Intervals, whether classical or modal, pretend to represent numerical information in a coherent way and, for that, one of the main problems is rounding. Indeed, using a digital scale with a finite number of digits, computations will have to be rounded in a convenient way. Working with non-interval numeric values, the best rounding is that which guarantees that the obtained value is “the closest” to the theoretical solution. Working with modal intervals the rule of rounding cannot be the same. Traditionally, the rounding process has been always a nuisance inherent in interval computation, but necessary to keep the semantic interpretations that these computations provide.
Miguel A. Sainz, Joaquim Armengol, Remei Calm, Pau Herrero, Lambert Jorba, Josep Vehi
Chapter 10. Some Related Problems
Abstract
This chapter presents some applications of modal intervals to practical problems in different fields. First, the minimax problem, tackled from the definitions of the modal *- and **-semantic extensions of a continuous function. Many real life problems of practical importance can be modelled as continuous minimax optimization problems.
Miguel A. Sainz, Joaquim Armengol, Remei Calm, Pau Herrero, Lambert Jorba, Josep Vehi
Backmatter
Metadata
Title
Modal Interval Analysis
Authors
Miguel A. Sainz
Joaquim Armengol
Remei Calm
Pau Herrero
Lambert Jorba
Josep Vehi
Copyright Year
2014
Electronic ISBN
978-3-319-01721-1
Print ISBN
978-3-319-01720-4
DOI
https://doi.org/10.1007/978-3-319-01721-1

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