Influence of different components of data error on the result of solving identification and approximation problems

Abstract

Abstract The dependences of approximation and identification errors for solving problems of approximation of functions associated with the presence of various components of data errors in the numerical solution are investigated. Using the study of problems that have an exact solution, we found patterns of increasing errors associated with rounding numbers and conditionality of the matrix of the equations system. It is shown that the error of identification (finding the coefficients of representation of the desired function as the linear combination of the basic ones) can be significantly greater than the error of approximation, despite the fact that the second one was estimated using crude values of the coefficients. The found dependencies limit the accuracy of the solution for approximation of other functions. Artificial distortion of the original data by random error is applied. It is shown that the identification error increases rapidly with increasing degree of approximation polynomial and significantly exceeds the amount of distortion. It is also shown that an increase in the number of specified points leads to a decrease of identification and approximation errors caused by random perturbation, inversely proportional to the number of points. Using examples of approximation of various functions, it is shown that the result of approximation, and especially identification, significantly depends on the convergence of the Taylor series (or its absence) on the segment under consideration.