| Инд. авторы: | Шарая И.А. |
| Заглавие: | Метод граничных интервалов для визуализации полиэдральных множеств решений |
| Библ. ссылка: | Шарая И.А. Метод граничных интервалов для визуализации полиэдральных множеств решений // Вычислительные технологии. - 2015. - Т.20. - № 1. - С.75-103. - ISSN 1560-7534. - EISSN 2313-691X. |
| Внешние системы: | РИНЦ: 23273855; |
| Реферат: | eng: A new method, called “boundary intervals method”, is proposed for investigation and visualization of the sets determined by systems of linear algebraic inequalities or represented as the union of solution sets to a finite number of systems of linear inequalities. Software that allows one “to visually see” various solution sets is helpful in decision making, in education, as well as for investigation of the solution sets and debugging algorithms for their estimation. At present, there exist several visualization packages for the solution sets to linear systems of relations (inequalities and equations) that can only handle systems with no more than three relations and work unsatisfactorily with unbounded and thin solution sets. All these approaches rest upon computation and further use of vertices of the polyhedral sets. The new boundary intervals method we develop is based on using so-called boundary intervals instead of vertices, which enables us to overcome the limitations of the previous approaches. The proposed method is useful, in particular, for visualization of AE-solution sets to interval linear relations systems that can consist of equations, inequalities or both. Such a possibility is based on the fact that the intersection of any AE-solution set with separate orthants of the entire space is a polyhedral set for which the determining systems of linear inequalities can be easily written out from the initial interval system of relations. The paper describes basics of the boundary intervals method for systems with two and three unknown variables. Also, we present software packages lineq and IntLinIncXX implementing the boundary intervals method and designed for visualization of the solution sets to systems of linear relations, both interval and usual noninterval. rus: Предложен метод граничных интервалов для исследования и визуализации множества, которое описано системой линейных неравенств или представлено как объединение множеств решений конечного набора таких систем. Изложены основы метода для систем с двумя и тремя неизвестными. |
| Ключевые слова: | Polyhedral set; Polyhedron; Polytope; system of linear inequalities; boundary interval; визуализация; полиэдральное множество; полиэдр; многогранник; политоп; система линейных неравенств; граничный интервал; visualization; |
| Издано: | 2015 |
| Физ. характеристика: | с.75-103 |
| Цитирование: | 1. Шарая И.А. lineq - пакет программ для визуализации множеств решений систем линейных неравенств. Версия для Matlab: Релиз 25.04.2012. Адрес доступа: http://www. nsc.ru/interval/sharaya/irash.html 2. Sharaya, I.A. Boundary intervals and visualization of AE-solution sets for interval system of linear equations. 15th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetics and Verified Numerics - SCAN’2012, September 23-29, 2012, Novosibirsk, Russia. Book of Abstracts. Institute of Computational Technologies, Novosibirsk, 2012. P. 166-167. Available at: http://conf.nsc.ru/files/conferences/scan2012/140000/ scan2012Abstracts.pdf 3. Sharaya, I.A. Boundary intervals and visualization of AE-solution sets for interval system of linear equations [online presentation] // 15th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetics and Verified Numerics - SCAN’2012, Novosibirsk, Russia, September 23-29, 2012. Available at: http://conf.nsc.ru/files/ conferences/scan2012/142985/Sharaya-scan2012.pdf 4. Shary, S.P. A new technique in systems analysis under interval uncertainty and ambiguity // Reliable Computing. 2002. Vol. 8, No. 5. P. 321-419. Available at: http://www.ict.nsc.ru/ shary/Papers/ANewTech.pdf 5. INTLAB - INTerval LABoratory, the Matlab toolbox for reliable computing. Available at: http://www.ti3.tu-harburg.de/rump/intlab/ 6. Popova, E., Kramer, W. Visualization of parametric solution sets. Preprint BUWWRSWT 2006/10, Bergische Universit¨at Wuppertal, 2006. Available at: http://www2.math. uni-wuppertal.de/wrswt/preprints/prep_06_10-2.pdf 7. Kramer, W. intpakX - an interval arithmetic package for Maple. Scientific Computing, Computer Arithmetic and Validated Numerics, 2006. Proceedings of SCAN 2006, 12th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics, Duisburg, Germany, September 26-29, 2006. IEEE Computer Society Press, 2007, page 27. DOI: 10.1109/SCAN.2006.29 8. Kramer, W. Computing and visualizing solution sets of interval linear systems // Serdica J. of Comput. 2007. Vol. 1, No. 4. P. 455-468. Available at: http://serdica-comp.math.bas. bg/index.php/serdicajcomputing/article/download/33/30 9. Popova, E., Kramer, W. Visualizing parametric solution sets // BIT Numerical Mathematics. 2008. Vol. 48, iss. 1. P. 95-115. DOI: 10.1007/s10543-007-0159-3 10. AE-solset.ps - программа на языке PostScript для визуализации множеств AE-решений интервальных 2×2-систем линейных алгебраических уравнений. Адрес доступа: http://www.nsc.ru/interval/Programing/AE-solset.ps AEsolset.ps - a PostScript program for visualization of AE-solution sets to interval linear 2 × 2-systems of equations. Available at: http://www.nsc.ru/interval/Programing/AEsolset.ps 11. Емеличев В.А., Ковалёв М.М., Кравцов М.К. Многогранники, грани, оптимизация. M.: Наука, 1981. 12. Келли Дж.Л. Общая топология. Изд. 2-е. Пер. с англ. М.: Наука, 1981. 13. Matlab - The language of technical computing. Available at: http://www.mathworks.com/ products/matlab 14. Scilab - free and open source software for numerical computation. Available at: http://www. scilab.org 15. Шарая И.А. IntLinIncR2 - пакет программ для визуализации множеств решений интервальных линейных систем отношений с двумя неизвестными. Версия для Matlab: Релиз 01.09.2014. Адрес доступа: http://www.nsc.ru/interval/Programing, http://interval. ict.nsc.ru/sharaya/irash.html 16. Шарая И.А. IntLinIncR3 - пакет программ для визуализации множеств решений интервальных линейных систем отношений с тремя неизвестными. Версия для Matlab: Релиз 01.09.2014. Адрес доступа: http://www.nsc.ru/interval/Programing, http://interval. ict.nsc.ru/sharaya/irash.html |