Development of linear functional arithmetic and its application to solving problems of interval analysis

TL;DR

This paper introduces functional intervals and linear functional arithmetic to enhance interval analysis, demonstrating improved convergence and speed in solving minimization and zero-finding problems without derivative information.

Contribution

The work develops a new type of functional intervals and linear functional arithmetic, improving algorithm speed and convergence in interval analysis tasks.

Findings

High order of convergence observed in numerical experiments.

Significant speedup in algorithms using functional intervals.

Improved minimization algorithms for multivariable functions.

Abstract

The work is devoted to the construction of a new type of intervals -- functional intervals. These intervals are built on the idea of expanding boundaries from numbers to functions. Functional intervals have shown themselves to be promising for further study and use, since they have more rich algebraic properties compared to classical intervals lamy. In the work, linear functional arithmetic was constructed from one variable. This arithmetic was applied to solve such problems of interval analysis, as minimization of a function on an interval and finding zeros of a function on an interval. Results of numerical experiments for linear functional arithmetic showed a high order of convergence and a higher speed the growth of algorithms when using intervals of a new type, despite the fact that the calculations did not use information about derivative function. Also in the work, a modification of the minimization algorithms functions of several variables, based on the use of the function rational intervals of several variables. As a result, it was Improved speedup of algorithms, but only up to a certain number of unknowns.