Hybrid two-point methods using only one inverse for solving nonlinear equations
UDC 519.6, 517.98 The task of solving of nonlinear equations often involves the use of iterative methods that require multiple inversions of linear operators, which can be computationally costly, especially for large matrices or complex operators. We propose hybrid two-point methods that overcome th...
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| Date: | 2026 |
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| Main Authors: | , , , |
| Format: | Article |
| Language: | English |
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Institute of Mathematics, NAS of Ukraine
2026
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/9002 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1866663705637289984 |
|---|---|
| author | Argyros, I. K. Shakhno, S. Havdiak, M. Argyros, M. I. Argyros, I. K. Shakhno, S. Havdiak, M. Argyros, M. I. |
| author_facet | Argyros, I. K. Shakhno, S. Havdiak, M. Argyros, M. I. Argyros, I. K. Shakhno, S. Havdiak, M. Argyros, M. I. |
| author_institution_txt_mv | [
{
"author": "I. K. Argyros",
"institution": "Department of Computing and Mathematical Sciences, Cameron University, Lawton, OK, USA"
},
{
"author": "S. Shakhno",
"institution": "Department of Theory of Optimal Processes, Ivan Franko National University of Lviv, Lviv, Ukraine"
},
{
"author": "M. Havdiak",
"institution": "Department of Theory of Optimal Processes, Ivan Franko National University of Lviv, Lviv, Ukraine"
},
{
"author": "M. I. Argyros",
"institution": "College of Computing and Engineering, Nova Southeastern University, Fort Lauderdale, Florida, USA"
}
] |
| author_sort | Argyros, I. K. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2026-05-30T12:43:57Z |
| description | UDC 519.6, 517.98
The task of solving of nonlinear equations often involves the use of iterative methods that require multiple inversions of linear operators, which can be computationally costly, especially for large matrices or complex operators. We propose hybrid two-point methods that overcome the indicated limitation by requiring only a single inversion of a fixed linear operator throughout the entire process. This approach preserves the rapid convergence and avoids the repeated inversions of $Q'(y_n)$ in each step. We provide both semilocal and local convergence analyses, utilizing a majorant function to control the Fréchet derivative of the operator. Numerical experiments confirm that the performance of our methods is comparable with the classical approaches but is characterized by much lower computational costs. These findings highlight the potentials of this technique as a practical alternative to Newton's method and other iterative methods requiring operator inversions. |
| doi_str_mv | 10.3842/umzh.v78i5-6.9002 |
| first_indexed | 2026-05-30T01:00:36Z |
| format | Article |
| fulltext | |
| id | umjimathkievua-article-9002 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-05-31T01:00:33Z |
| publishDate | 2026 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | |
| spelling | umjimathkievua-article-90022026-05-30T12:43:57Z Hybrid two-point methods using only one inverse for solving nonlinear equations Hybrid two-point methods using only one inverse for solving nonlinear equations Argyros, I. K. Shakhno, S. Havdiak, M. Argyros, M. I. Argyros, I. K. Shakhno, S. Havdiak, M. Argyros, M. I. Two Point Methods; Banach space; Divided difference; Hybrid two Point Method; Convergence UDC 519.6, 517.98 The task of solving of nonlinear equations often involves the use of iterative methods that require multiple inversions of linear operators, which can be computationally costly, especially for large matrices or complex operators. We propose hybrid two-point methods that overcome the indicated limitation by requiring only a single inversion of a fixed linear operator throughout the entire process. This approach preserves the rapid convergence and avoids the repeated inversions of $Q'(y_n)$ in each step. We provide both semilocal and local convergence analyses, utilizing a majorant function to control the Fréchet derivative of the operator. Numerical experiments confirm that the performance of our methods is comparable with the classical approaches but is characterized by much lower computational costs. These findings highlight the potentials of this technique as a practical alternative to Newton's method and other iterative methods requiring operator inversions. УДК 519.6, 517.98 Гібридні двоточкові методи з використанням лише однієї інверсії для розв'язання нелінійних рівнянь Для розв'язання нелінійних рівнянь зазвичай застосовують ітераційні методи, які передбачають повторне обчислення обернених операторів. Це може призводити до великих обчислювальних затрат, особливо при роботі з великими матрицями або складними операторами. В нашій роботі запропоновано гібридні двокрокові методи, які вирішують цю проблему, потребуючи лише одне обернення для фіксованого лінійного оператора протягом усього процесу. Такий підхід зберігає швидку збіжність, уникаючи повторного обернення $Q'(y_n)$ на кожному кроці. Проведено аналіз локальної та напівлокальної збіжності з використанням мажорантної функції для контролю похідної Фреше оператора. Результати числових експериментів підтверджують, що запропоновані методи працюють на рівні класичних підходів. При цьому вони характеризуються значно меншими обчислювальними затратами. Отримані результати демонструють перспективність запропонованого підходу, як практичної альтернативи методу Ньютона та інших ітераційних методів, що потребують обчислення обернених операторів. Institute of Mathematics, NAS of Ukraine 2026-05-29 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/9002 10.3842/umzh.v78i5-6.9002 Ukrains’kyi Matematychnyi Zhurnal; Vol. 78 No. 5-6 (2026); 361–362 Український математичний журнал; Том 78 № 5-6 (2026); 361–362 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/9002/10653 Copyright (c) 2026 I. K. Argyros, S. Shakhno, M. Havdiak, M. I. Argyros |
| spellingShingle | Argyros, I. K. Shakhno, S. Havdiak, M. Argyros, M. I. Argyros, I. K. Shakhno, S. Havdiak, M. Argyros, M. I. Hybrid two-point methods using only one inverse for solving nonlinear equations |
| title | Hybrid two-point methods using only one inverse for solving nonlinear equations |
| title_alt | Hybrid two-point methods using only one inverse for solving nonlinear equations |
| title_full | Hybrid two-point methods using only one inverse for solving nonlinear equations |
| title_fullStr | Hybrid two-point methods using only one inverse for solving nonlinear equations |
| title_full_unstemmed | Hybrid two-point methods using only one inverse for solving nonlinear equations |
| title_short | Hybrid two-point methods using only one inverse for solving nonlinear equations |
| title_sort | hybrid two-point methods using only one inverse for solving nonlinear equations |
| topic_facet | Two Point Methods Banach space Divided difference Hybrid two Point Method Convergence |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/9002 |
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