Interval estimation of trigonometrical splines

I.G. Burova,N. Mastorakis,A. Bulucea,E.G. Ivanova,V. Mladenov,V.A. Kostin

doi:10.1051/matecconf/201929203001

Abstract

Quite often, it is necessary to quickly determine variation range of the function. If the function values are known at some points, then it is easy to construct the local spline approximation of this function and use the interval analysis rules. As a result, we get the area within which the approximation of this function changes. It is necessary to take into account the approximation error when studying the obtained area of change of function approximation. Thus, we get the range of changing the function with the approximation error. This paper discusses the features of using polynomial and trigonometrical splines of the third order approximation to determine the upper and lower boundaries of the area (domain) in which the values of the approximation are contained. Theorems of approximation by these local trigonometric and polynomial splines are formulated. The values of the constants in the estimates of the errors of approximation by the trigonometrical and polynomial splines are given. It is shown that these constants cannot be reduced. An algorithm for constructing the variation domain of the approximation of the function is described. The results of the numerical experiments are given.

Highlights

It is useful to determine the lower and upper bounds of the values of functions, eigenvalues of operators, solutions of systems of linear and nonlinear equations without without calculating a detailed numerical solution of the corresponding problems
This paper continues the series of papers on approximation by local polynomial and non-polynomial splines and interval estimation see [5], [6],[7]
The results of the application the trigonometrical splines for the approximation of functions are given in table 1

Summary

Introduction

It is useful to determine the lower and upper bounds of the values of functions, eigenvalues of operators, solutions of systems of linear and nonlinear equations without without calculating a detailed numerical solution of the corresponding problems. The solution of such problems is considered in many papers published recently. This paper continues the series of papers on approximation by local polynomial and non-polynomial splines and interval estimation see [5], [6],[7] For constructing this interval extension, we use techniques from interval analysis

Trigonometric splines

Comparison with polynomial splines

Approximation with right splines

Interval extention

Conclusion