Numerical Solution for the Extrapolation Problem of Analytic Functions

Nikolaos P Bakas

doi:10.34133/2019/3903187

Abstract

In this work, a numerical solution for the extrapolation problem of a discrete set of n values of an unknown analytic function is developed. The proposed method is based on a novel numerical scheme for the rapid calculation of higher order derivatives, exhibiting high accuracy, with error magnitude of O(10−100) or less. A variety of integrated radial basis functions are utilized for the solution, as well as variable precision arithmetic for the calculations. Multiple alterations in the function's direction, with no curvature or periodicity information specified, are efficiently foreseen. Interestingly, the proposed procedure can be extended in multiple dimensions. The attained extrapolation spans are greater than two times the given domain length. The significance of the approximation errors is comprehensively analyzed and reported, for 5832 test cases.

Highlights

Prediction always differs from observation [1], and extrapolation remains an open challenge [2] or even a hopelessly ill-conditioned [3] problem, in a vast variety of scientific disciplines
The extrapolation span is based on the accuracy of the interpolation method ((9), (10), (12), (13), (14a), (14b), (14c), (14d), (14e), (14f)) as well as the numerical differentiation ((1), (5), (6), (6a), (6b), (6c), (6d), (6e), (6f), (8))
The numerical investigation of the extrapolation errors suggests that only utilizing high accuracy and a precise approximation scheme for a function as well as its derivatives, the extrapolation is attainable

Summary

Introduction

Prediction always differs from observation [1], and extrapolation remains an open challenge [2] or even a hopelessly ill-conditioned [3] problem, in a vast variety of scientific disciplines. It relies on numerical methods which attempt to predict the future, unknown values of a studied phenomenon, given a limited set of observations. Several methods have been proposed for time-series forecasting [2] and competitions regarding accuracy have been conducted, utilizing statistical [11, 17] and machine learning procedures [2, 10, 15]; the prediction horizon regards only a small percentage (∼ 20-30%) of the given data extension. Prediction procedures are essential for a vast variety of scientific fields, always based on numerical interpolation methods

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Discussion

Conclusion