Lambert Jorba

Formal Solution to Systems of Interval Linear or Non-Linear Equations

This is the first of two papers which present the Modal Interval Analysis as a framework where the search and interpretation of formal solutions for a set of simultaneous interval linear or non-linear equations is started on, together with the interval estimations for sets of solutions of real-valued systems in which coefficients and right-hand sides belong to certain intervals. The main purpose of this first paper is to show that the modal intervals are a suitable tool to approach problems where logical reference appear. Thus, it is possible to give a logical meaning to general solutions of the system of equations, obtained, in the linear case, by means of an algorithm whose convergence conditions are studied.

Interval Estimations of Solution Sets to Real-Valued Systems of Linear or Non-Linear Equations

This is a second paper devoted to present the Modal Interval Analysis as a framework where the search of formal solutions for a set of simultaneous interval linear or non-linear equations is started on, together with the interval estimations for sets of solutions of real-valued systems in which coefficients and right-hand sides belong to certain intervals. The main purpose of this second paper is to show that the modal intervals are a suitable tool to approach problems where logical references appear, for example, to find interval estimates of a special class of generalized sets of solutions of real-valued linear and non-linear systems, the UE-solution sets.

Some Related Problems

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.

Twins and f ∗ Algorithm

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.

Modal Interval Extensions

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 \(\mathbb{R}\).

Interpretability and Optimality

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.

Equations and Systems

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.

Intervals of Marks

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.