{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:51:47Z","timestamp":1760143907530,"version":"build-2065373602"},"reference-count":38,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,3,7]],"date-time":"2024-03-07T00:00:00Z","timestamp":1709769600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000780","name":"European Union\u2019s Horizon 2020 research and innovation program","doi-asserted-by":"publisher","award":["810660"],"award-info":[{"award-number":["810660"]}],"id":[{"id":"10.13039\/501100000780","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>Function approximation is a fundamental process in a variety of problems in computational mechanics, structural engineering, as well as other domains that require the precise approximation of a phenomenon with an analytic function. This work demonstrates a unified approach to these techniques, utilizing partial sums of the Taylor series in a high arithmetic precision. In particular, the proposed approach is capable of interpolation, extrapolation, numerical differentiation, numerical integration, solution of ordinary and partial differential equations, and system identification. The method employs Taylor polynomials and hundreds of digits in the computations to obtain precise results. Interestingly, some well-known problems are found to arise in the calculation accuracy and not methodological inefficiencies, as would be expected. In particular, the approximation errors are precisely predictable, the Runge phenomenon is eliminated, and the extrapolation extent may a priory be anticipated. The attained polynomials offer a precise representation of the unknown system as well as its radius of convergence, which provides a rigorous estimation of the prediction ability. The approximation errors are comprehensively analyzed for a variety of calculation digits and test problems and can be reproduced by the provided computer code.<\/jats:p>","DOI":"10.3390\/computation12030053","type":"journal-article","created":{"date-parts":[[2024,3,7]],"date-time":"2024-03-07T08:59:37Z","timestamp":1709801977000},"page":"53","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Taylor Polynomials in a High Arithmetic Precision as Universal Approximators"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7297-0693","authenticated-orcid":false,"given":"Nikolaos","family":"Bakas","sequence":"first","affiliation":[{"name":"Computation-Based Science and Technology Research Center, The Cyprus Institute, 20 Konstantinou Kavafi Str., 2121 Nicosia, Cyprus"},{"name":"Department of Information Technology & AI Lab, American College of Greece, Deree. 6 Gravias Str., Aghia Paraskevi, 15342 Athens, Greece"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3903187","DOI":"10.34133\/2019\/3903187","article-title":"Numerical Solution for the Extrapolation Problem of Analytic Functions","volume":"2019","author":"Bakas","year":"2019","journal-title":"Research"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1080\/10586458.2005.10128931","article-title":"A comparison of three high-precision quadrature schemes","volume":"14","author":"Bailey","year":"2005","journal-title":"Exp. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1016\/j.enganabound.2011.07.008","article-title":"Multiquadric and its shape parameter\u2014A numerical investigation of error estimate, condition number, and round-off error by arbitrary precision computation","volume":"36","author":"Cheng","year":"2012","journal-title":"Eng. Anal. Bound. Elem."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"614","DOI":"10.1016\/j.enganabound.2006.11.011","article-title":"Error estimate, optimal shape factor, and high precision computation of multiquadric collocation method","volume":"31","author":"Huang","year":"2007","journal-title":"Eng. Anal. Bound. Elem."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"102801","DOI":"10.1016\/j.sysarc.2022.102801","article-title":"CLARINET: A quire-enabled RISC-V-based framework for posit arithmetic empiricism","volume":"135","author":"Sharma","year":"2023","journal-title":"J. Syst. Archit."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1007\/s42514-023-00150-2","article-title":"ddRingAllreduce: A high-precision RingAllreduce algorithm","volume":"5","author":"Lei","year":"2023","journal-title":"CCF Trans. High Perform. Comput."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1016\/j.apm.2024.01.012","article-title":"Mathematical modelling for high precision ray tracing in optical design","volume":"128","author":"Wu","year":"2024","journal-title":"Appl. Math. Model."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Friebel, K.F.A., Bi, J., and Castrillon, J. (2023, January 14\u201316). Base2: An IR for Binary Numeral Types. Proceedings of the 13th International Symposium on Highly Efficient Accelerators and Reconfigurable Technologies, Kusatsu, Japan.","DOI":"10.1145\/3597031.3597048"},{"key":"ref_9","unstructured":"Granlund, T. (2024, January 13). The GNU Multiple Precision Arithmetic Library. Available online: https:\/\/gmplib.org\/."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"101428","DOI":"10.1016\/j.softx.2023.101428","article-title":"JGMP: Java bindings and wrappers for the GMP library","volume":"23","author":"Amato","year":"2023","journal-title":"SoftwareX"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"162","DOI":"10.1016\/j.cam.2005.08.015","article-title":"Multivariate approximation by a combination of modified Taylor polynomials","volume":"196","author":"Guessab","year":"2006","journal-title":"J. Comput. Appl. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1016\/S0377-0427(99)00360-X","article-title":"Generalization of Taylor\u2019s theorem and Newton\u2019s method via a new family of determinantal interpolation formulas and its applications","volume":"126","author":"Kalantari","year":"2000","journal-title":"J. Comput. Appl. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1023\/A:1024467732637","article-title":"Verified Integration of ODEs and Flows Using Differential Algebraic Methods on High-Order Taylor Models","volume":"10","author":"Berz","year":"1998","journal-title":"Reliab. Comput."},{"key":"ref_14","first-page":"291","article-title":"The approximate solution of high-order linear Volterra-Fredholm integro-differential equations in terms of Taylor polynomials","volume":"112","author":"Sezer","year":"2000","journal-title":"Appl. Math. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1007\/s12190-020-01397-6","article-title":"A novel approach for the numerical approximation to the solution of singularly perturbed differential-difference equations with small shifts","volume":"65","author":"Ranjan","year":"2021","journal-title":"J. Appl. Math. Comput."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"308","DOI":"10.1137\/090774707","article-title":"Impossibility of Fast Stable Approximation of Analytic Functions from Equispaced Samples","volume":"53","author":"Platte","year":"2011","journal-title":"SIAM Rev."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/0893-9659(92)90014-Z","article-title":"Defeating the Runge phenomenon for equispaced polynomial interpolation via Tikhonov regularization","volume":"5","author":"Boyd","year":"1992","journal-title":"Appl. Math. Lett."},{"key":"ref_18","first-page":"1948","article-title":"Study of regional geomagnetic model of Fujian and adjacent areas based on 3D Taylor Polynomial model","volume":"59","author":"Zhang","year":"2016","journal-title":"Acta Geophys. Sin."},{"key":"ref_19","first-page":"158","article-title":"Divergence (Runge Phenomenon) for least-squares polynomial approximation on an equispaced grid and Mock-Chebyshev subset interpolation","volume":"210","author":"Boyd","year":"2009","journal-title":"Appl. Math. Comput."},{"key":"ref_20","first-page":"484","article-title":"Exponentially-convergent strategies for defeating the runge phenomenon for the approximation of non-periodic functions, part I: Single-interval schemes","volume":"5","author":"Boyd","year":"2009","journal-title":"Commun. Comput. Phys."},{"key":"ref_21","unstructured":"Knaplock, R. (1715). Principles of Linear Perspective, British Library."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1175","DOI":"10.1007\/s00419-014-0962-7","article-title":"Optimum design of thin plates via frequency optimization using BEM","volume":"85","author":"Babouskos","year":"2015","journal-title":"Arch. Appl. Mech."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1545","DOI":"10.1007\/s00419-014-0944-9","article-title":"Buckling of cylindrical shell panels: A MAEM solution","volume":"85","author":"Yiotis","year":"2015","journal-title":"Arch. Appl. Mech."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1137\/141000671","article-title":"Julia: A fresh approach to numerical computing","volume":"59","author":"Bezanson","year":"2017","journal-title":"SIAM Rev."},{"key":"ref_25","unstructured":"Apostol, T.M. (1967). Calculus, John Wiley & Sons."},{"key":"ref_26","unstructured":"Browder, A. (2012). Mathematical Analysis: An Introduction, Springer Science & Business Media."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"715","DOI":"10.5802\/aif.1184","article-title":"Partial sums of Taylor series on a circle","volume":"39","author":"Katsoprinakis","year":"2011","journal-title":"Ann. L\u2019Institut Fourier"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1293","DOI":"10.5802\/aif.1549","article-title":"Universal Taylor series","volume":"46","author":"Nestoridis","year":"2011","journal-title":"Ann. de L\u2019Institut Fourier"},{"key":"ref_29","unstructured":"Press, W.H., and Teukolsky, S.A. (2007). VWT, and FBP, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Horn, R.A., and Johnson, C.R. (1991). Topics in Matrix Analysis, Cambridge University Press.","DOI":"10.1017\/CBO9780511840371"},{"key":"ref_31","unstructured":"Ycart, B. (2012). A case of mathematical eponymy: The Vandermonde determinant. arXiv."},{"key":"ref_32","unstructured":"Turner, L.R. (1966). Inverse of the Vandermonde Matrix with Applications."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1007\/s10208-018-9384-1","article-title":"Stable extrapolation of analytic functions","volume":"19","author":"Demanet","year":"2019","journal-title":"Found. Comput. Math."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"256","DOI":"10.1115\/1.3269509","article-title":"Advanced Mechanics of Materials (4th Ed.)","volume":"110","author":"Boresi","year":"1988","journal-title":"J. Vib. Acoust. Stress Reliab. Des."},{"key":"ref_35","first-page":"12","article-title":"System identification by the analog equation method","volume":"10","author":"Katsikadelis","year":"1995","journal-title":"WIT Trans. Model. Simul."},{"key":"ref_36","unstructured":"Katsikadelis, J.T. (2014). The Boundary Element Method for Plate Analysis, Elsevier."},{"key":"ref_37","unstructured":"Gregory, F.H. (2011). Arithmetic and Reality: A Development of Popper\u2019s Ideas. Philos. Math. Educ. J., 26."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1145\/1236463.1236468","article-title":"MPFR: A multiple-precision binary floating-point library with correct rounding","volume":"33","author":"Fousse","year":"2007","journal-title":"ACM Trans. Math. Softw."}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/12\/3\/53\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:10:05Z","timestamp":1760105405000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/12\/3\/53"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,3,7]]},"references-count":38,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2024,3]]}},"alternative-id":["computation12030053"],"URL":"https:\/\/doi.org\/10.3390\/computation12030053","relation":{},"ISSN":["2079-3197"],"issn-type":[{"type":"electronic","value":"2079-3197"}],"subject":[],"published":{"date-parts":[[2024,3,7]]}}}