Finding a numerical solution of linear algebraic equations is known to present an ill-posed in the sense that small perturbation in the right hand side may lead to large errors in the solution. It is important to verify the accuracy of an approximate solution by taking into account all possible errors in the elements of the matrix, and of the vector at the right hand side as well as roundoff errors. There may be computational difficulties with ill-posed systems as well. If to apply standard methods such as the method of Gauss elimination to such systems it may be not possible to obtain the correct solution though discrepancy can be less accuracy of data errors. Besides, a small discrepancy will not always guarantee proximity to a correct solution. Actually there is no need for preliminary assessment whether a given system of linear algebraic equations is inherently ill-conditioned or well-conditioned. In this paper we consider a new approach to the solution of algebraic systems, which is based on statistical effect in matrices of big order. It will be shown that the conditionality of the systems of equation may change with a high probability, if the matrix distorted by random noise. After applying some standard methods, we may introduce the received "chaotic" solution is used as a source of a priori information a more general variational problem.
| DOI | 10.11648/j.acm.20150403.25 |
| Published in | Applied and Computational Mathematics (Volume 4, Issue 3, June 2015) |
| Page(s) | 220-224 |
| 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. |
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Copyright © The Author(s), 2024. Published by Science Publishing Group |
Ill-Posed Problems, Condition Numbers, Random Matrix
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APA Style
Vladimir V. Ternovski, Mikhail M. Khapaev, Alexander S. Grushicin. (2015). Ill-Posed Algebraic Systems with Noise Data. Applied and Computational Mathematics, 4(3), 220-224. https://doi.org/10.11648/j.acm.20150403.25
ACS Style
Vladimir V. Ternovski; Mikhail M. Khapaev; Alexander S. Grushicin. Ill-Posed Algebraic Systems with Noise Data. Appl. Comput. Math. 2015, 4(3), 220-224. doi: 10.11648/j.acm.20150403.25
AMA Style
Vladimir V. Ternovski, Mikhail M. Khapaev, Alexander S. Grushicin. Ill-Posed Algebraic Systems with Noise Data. Appl Comput Math. 2015;4(3):220-224. doi: 10.11648/j.acm.20150403.25
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Download@article{10.11648/j.acm.20150403.25,
author = {Vladimir V. Ternovski and Mikhail M. Khapaev and Alexander S. Grushicin},
title = {Ill-Posed Algebraic Systems with Noise Data},
journal = {Applied and Computational Mathematics},
volume = {4},
number = {3},
pages = {220-224},
doi = {10.11648/j.acm.20150403.25},
url = {https://doi.org/10.11648/j.acm.20150403.25},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20150403.25},
abstract = {Finding a numerical solution of linear algebraic equations is known to present an ill-posed in the sense that small perturbation in the right hand side may lead to large errors in the solution. It is important to verify the accuracy of an approximate solution by taking into account all possible errors in the elements of the matrix, and of the vector at the right hand side as well as roundoff errors. There may be computational difficulties with ill-posed systems as well. If to apply standard methods such as the method of Gauss elimination to such systems it may be not possible to obtain the correct solution though discrepancy can be less accuracy of data errors. Besides, a small discrepancy will not always guarantee proximity to a correct solution. Actually there is no need for preliminary assessment whether a given system of linear algebraic equations is inherently ill-conditioned or well-conditioned. In this paper we consider a new approach to the solution of algebraic systems, which is based on statistical effect in matrices of big order. It will be shown that the conditionality of the systems of equation may change with a high probability, if the matrix distorted by random noise. After applying some standard methods, we may introduce the received "chaotic" solution is used as a source of a priori information a more general variational problem.},
year = {2015}
}
TY - JOUR T1 - Ill-Posed Algebraic Systems with Noise Data AU - Vladimir V. Ternovski AU - Mikhail M. Khapaev AU - Alexander S. Grushicin Y1 - 2015/06/19 PY - 2015 N1 - https://doi.org/10.11648/j.acm.20150403.25 DO - 10.11648/j.acm.20150403.25 T2 - Applied and Computational Mathematics JF - Applied and Computational Mathematics JO - Applied and Computational Mathematics SP - 220 EP - 224 PB - Science Publishing Group SN - 2328-5613 UR - https://doi.org/10.11648/j.acm.20150403.25 AB - Finding a numerical solution of linear algebraic equations is known to present an ill-posed in the sense that small perturbation in the right hand side may lead to large errors in the solution. It is important to verify the accuracy of an approximate solution by taking into account all possible errors in the elements of the matrix, and of the vector at the right hand side as well as roundoff errors. There may be computational difficulties with ill-posed systems as well. If to apply standard methods such as the method of Gauss elimination to such systems it may be not possible to obtain the correct solution though discrepancy can be less accuracy of data errors. Besides, a small discrepancy will not always guarantee proximity to a correct solution. Actually there is no need for preliminary assessment whether a given system of linear algebraic equations is inherently ill-conditioned or well-conditioned. In this paper we consider a new approach to the solution of algebraic systems, which is based on statistical effect in matrices of big order. It will be shown that the conditionality of the systems of equation may change with a high probability, if the matrix distorted by random noise. After applying some standard methods, we may introduce the received "chaotic" solution is used as a source of a priori information a more general variational problem. VL - 4 IS - 3 ER -
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