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In this work, a numerical solution for the extrapolation problem of a discrete set of^{−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.

Prediction always differs from observation [

More specifically, in the case of analytic functions, where the given data follow an unknown rigor mathematical law, any extrapolation results are too weak to be interesting [

The purpose of this study was to provide a generic numerical solution for the extrapolation problem. It was attained for extended extrapolation horizons of even greater than 200% the given domain length if the set of data are derived from an unknown analytic function and their precision is high. The rationale of the proposed method adheres to the following three stages: (a) interpolation of the set of values

Let

Given values of

Around the endpoint

Radial Basis Functions (RBFs) networks are universal approximators while Integrated RBFs are capable of attaining precise approximation for a function and its derivatives [

Let _{j} and _{j} (Figure

The definitions of the radial basis functions_{i} and_{j}.

The Gaussian RBF is defined by

Respectively, the Shifted Logarithmic RBF is defined by

Exploiting the computed derivatives with high accuracy at the end of the domain ^{-1}. Hence, the interpolation of the function within the end increment

In Algorithm

1: Formulate vectors

2: Formulate matrix

3: Formulate vector

3:

4:

5:

6:

7:

9:

The efficiency of the procedure was verified through numerical examples for a variety of highly nonlinear functions and their extrapolation spans. The extrapolation span is based on the accuracy of the interpolation method ((_{j} are selected near the

The features which have an effect on the calculations, as well as their values (in parentheses) utilized in the parametric investigation, are the number of Taylor terms utilized, indicated as number of derivatives, (25, 50, 75); the number of computer digits used for the arbitrary precision calculations (500, 1000, 2000); the span of the given domain

The ^{−10}) and for 25 and 75 derivatives (MD = 54.2206, p-value = ^{−10} for 500 to 1000 digits and MD = 33.9628, p-value = 9.5606 ^{−10} for 500 to 2000 digits). The number of divisions exhibits a clear difference as well, in Figure ^{−10} for 50 to 100 divisions and MD = 49.6978, p-value = 9.5606 ^{−10} for 50 to 200 divisions). The univariate linear correlation of the condition number of ^{2} of 0.2559 (p-value = 3.1608 ^{−149}). For the condition number of^{323} for the condition of^{−323} for the inversion errors.

In order to further investigate more complex associations among the studied parameters and the extrapolation error ^{2} for the predicted versus actual ^{2} for the test set was 0.9658 and the highest predictive features were the number of divisions, the number of integrations, and number of digits (Figure

In Figure

Extrapolation of analytic functions

In Figure

Extrapolation of function

Numerical differentiation is highly sensitive to noise [^{−100}) or less (Figures

The approximation scheme ((_{i}, but for any intermediate point between two given points and, hence, for points_{j} (Figure ^{−100}) or even exact zero (supplementary database). The arbitrary precision has also been used for the robust computation of ^{−20}-10^{−30}).

The association of the condition number of ^{2} (0.25595) with slightly negative slope (Figure ^{323}, that is, the maximum real number considered by the software [

In brief, a method for the extrapolation of analytic functions is developed, accompanied by a systematic investigation of the involved parameters. The proposed scheme for the numerical differentiation exhibited low enough errors to permit adequate extrapolation spans. The constituted database of the numerical experiments highlights the fact that, for the same problem formulation (^{−20}), the predictions are limited to short length. Only if the measurements contain some hundreds of significant digits, the proposed solution is efficient. As this is difficult to be accomplished by laboratory instruments, this work’s findings provide strong evidence that we are far from lengthy predictions in physics, biology, and engineering and also even more far from phenomena studied in health and social sciences. However, the parametric investigation suggests that the precision in calculations and the utilized methods are vastly significant, as the extrapolation horizons achieved by the proposed numerical scheme are about an order of magnitude higher than those in the existing literature, highlighting the potentiality of predictions.

All the software operations ran on an Intel i7-6700 CPU @3.40GHz with 32GB memory and SSD hard disk, and the execution time was accordingly tracked. The errors of the derivatives in the relevant literature are reported in the Supplementary Data File

The author declares no conflicts of interest.

This study would not have been possible without professor’s John T. Katsikadelis helpful discussions, valuable suggestions, and clarifications regarding the integrated radial basis functions as well as his confidence that the problem is solvable. Special thanks are due to Dr. Natia R. Anastasi for reviewing the original manuscript. I would also like to thank Dr. Savvas Chatzichristofis, Director of Informatics at Neapolis University Pafos, Nikitas Giannakopoulos, Ass. Professor of Medicine, University of Würzburg, Germany, and Angeliki Sivitanidou and Thomas Dimopoulos, Lecturers at Neapolis University, for suggestions regarding language.

The supplementary materials are divided into two parts: (A) The supplementary figures and tables for the statistical analyses of parameters affecting extrapolation accuracy, and the bibliometric review figures. (B) The supplementary database: Data File S1. experiments.xlsx (attached): database for the parametric investigation of extrapolation errors. Figure S1: bibliometric map of papers with keyword extrapolation. Utilizing a bibliometric procedure [^{−10}, from Table S7). Figure S9: variance of ^{−7} - Table S8). Figure S10: variance of

^{2}-continuous