If experimental tables are numerically integrated using quadrature formulas, then the measurement errors of the physical instrument is not taken into account. The result of such numerical integration will be inaccurate because of the accumulation of errors due to the summation of random values, and the residual term of the quadrature formula cannot be calculated using solely classical concepts. The traditional approach consists of applying various smoothing algorithms. In this case, methods are used that are unrelated to the problem of integrating itself, which leads to excessive smoothing of the result. The authors propose a method for numerical integration of inaccurate numerical functions that minimizes the residual term of the quadrature formula for the set of unknown values based on the error confidence intervals by using ill-posed problem algorithms. The high level of effectiveness of this new method, for which it is sufficient to know the error level of the signal, is demonstrated through examples.
| Published in | Applied and Computational Mathematics (Volume 3, Issue 2) |
| DOI | 10.11648/j.acm.20140302.14 |
| Page(s) | 63-67 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2014. Published by Science Publishing Group |
Quadrature Formula, Ill-Posed Problem, Tikhonov Regularization
| [1] | A. Tikhonov and V. Arsenin, Solutions of ill-posed problems. Winston, Washington, DC(1977). |
| [2] | A. Bakushinsky and A. Goncharsky, Ill-posed problems: theory and applications. Springer Netherlands (1994). |
| [3] | T. Prvan , Integrating noisy data, Appl.Math.Lett.(1995) Vol. 8, No.6: 83-87. |
| [4] | V.V. Ternovskii and M.M. Khapaev, Reconstruction of periodic function from noisy input data , Doklady Mathematics(2009) Vol.79,No.1:81-82. |
APA Style
Vladimir V. Ternovski, Mikhail M. Khapaev. (2014). Method for Integrating Tabular Functions that Considers Errors. Applied and Computational Mathematics, 3(2), 63-67. https://doi.org/10.11648/j.acm.20140302.14
ACS Style
Vladimir V. Ternovski; Mikhail M. Khapaev. Method for Integrating Tabular Functions that Considers Errors. Appl. Comput. Math. 2014, 3(2), 63-67. doi: 10.11648/j.acm.20140302.14
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Download@article{10.11648/j.acm.20140302.14,
author = {Vladimir V. Ternovski and Mikhail M. Khapaev},
title = {Method for Integrating Tabular Functions that Considers Errors},
journal = {Applied and Computational Mathematics},
volume = {3},
number = {2},
pages = {63-67},
doi = {10.11648/j.acm.20140302.14},
url = {https://doi.org/10.11648/j.acm.20140302.14},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140302.14},
abstract = {If experimental tables are numerically integrated using quadrature formulas, then the measurement errors of the physical instrument is not taken into account. The result of such numerical integration will be inaccurate because of the accumulation of errors due to the summation of random values, and the residual term of the quadrature formula cannot be calculated using solely classical concepts. The traditional approach consists of applying various smoothing algorithms. In this case, methods are used that are unrelated to the problem of integrating itself, which leads to excessive smoothing of the result. The authors propose a method for numerical integration of inaccurate numerical functions that minimizes the residual term of the quadrature formula for the set of unknown values based on the error confidence intervals by using ill-posed problem algorithms. The high level of effectiveness of this new method, for which it is sufficient to know the error level of the signal, is demonstrated through examples.},
year = {2014}
}
TY - JOUR T1 - Method for Integrating Tabular Functions that Considers Errors AU - Vladimir V. Ternovski AU - Mikhail M. Khapaev Y1 - 2014/05/20 PY - 2014 N1 - https://doi.org/10.11648/j.acm.20140302.14 DO - 10.11648/j.acm.20140302.14 T2 - Applied and Computational Mathematics JF - Applied and Computational Mathematics JO - Applied and Computational Mathematics SP - 63 EP - 67 PB - Science Publishing Group SN - 2328-5613 UR - https://doi.org/10.11648/j.acm.20140302.14 AB - If experimental tables are numerically integrated using quadrature formulas, then the measurement errors of the physical instrument is not taken into account. The result of such numerical integration will be inaccurate because of the accumulation of errors due to the summation of random values, and the residual term of the quadrature formula cannot be calculated using solely classical concepts. The traditional approach consists of applying various smoothing algorithms. In this case, methods are used that are unrelated to the problem of integrating itself, which leads to excessive smoothing of the result. The authors propose a method for numerical integration of inaccurate numerical functions that minimizes the residual term of the quadrature formula for the set of unknown values based on the error confidence intervals by using ill-posed problem algorithms. The high level of effectiveness of this new method, for which it is sufficient to know the error level of the signal, is demonstrated through examples. VL - 3 IS - 2 ER -
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