Generalized Convergence for Multi-Step Schemes under Weak Conditions
We have developed a local convergence analysis for a general scheme of high-order convergence, aiming to solve equations in Banach spaces. A priori estimates are developed based on the error distances. This way, we know in advance the number of iterations required to reach a predetermined error tole...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2024-01-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/12/2/220 |
| _version_ | 1797343041977057280 |
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| author | Ramandeep Behl Ioannis K. Argyros Hashim Alshehri Samundra Regmi |
| author_facet | Ramandeep Behl Ioannis K. Argyros Hashim Alshehri Samundra Regmi |
| author_sort | Ramandeep Behl |
| collection | DOAJ |
| description | We have developed a local convergence analysis for a general scheme of high-order convergence, aiming to solve equations in Banach spaces. A priori estimates are developed based on the error distances. This way, we know in advance the number of iterations required to reach a predetermined error tolerance. Moreover, a radius of convergence is determined, allowing for a selection of initial points assuring the convergence of the scheme. Furthermore, a neighborhood that contains only one solution to the equation is specified. Notably, we present the generalized convergence of these schemes under weak conditions. Our findings are based on generalized continuity requirements and contain a new semi-local convergence analysis (with a majorizing sequence) not seen in earlier studies based on Taylor series and derivatives which are not present in the scheme. We conclude with a good collection of numerical results derived from applied science problems. |
| first_indexed | 2024-03-08T10:41:57Z |
| format | Article |
| id | doaj.art-10c2fc1eb11e41208e06ad81ec16695e |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-08T10:41:57Z |
| publishDate | 2024-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-10c2fc1eb11e41208e06ad81ec16695e2024-01-26T17:31:24ZengMDPI AGMathematics2227-73902024-01-0112222010.3390/math12020220Generalized Convergence for Multi-Step Schemes under Weak ConditionsRamandeep Behl0Ioannis K. Argyros1Hashim Alshehri2Samundra Regmi3Mathematical Modelling and Applied Computation Research Group (MMAC), Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi ArabiaDepartment of Computing and Mathematical Sciences, Cameron University, Lawton, OK 73505, USAMathematical Modelling and Applied Computation Research Group (MMAC), Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi ArabiaDepartment of Mathematics, University of Houston, Houston, TX 77205, USAWe have developed a local convergence analysis for a general scheme of high-order convergence, aiming to solve equations in Banach spaces. A priori estimates are developed based on the error distances. This way, we know in advance the number of iterations required to reach a predetermined error tolerance. Moreover, a radius of convergence is determined, allowing for a selection of initial points assuring the convergence of the scheme. Furthermore, a neighborhood that contains only one solution to the equation is specified. Notably, we present the generalized convergence of these schemes under weak conditions. Our findings are based on generalized continuity requirements and contain a new semi-local convergence analysis (with a majorizing sequence) not seen in earlier studies based on Taylor series and derivatives which are not present in the scheme. We conclude with a good collection of numerical results derived from applied science problems.https://www.mdpi.com/2227-7390/12/2/220multi-step schemeball convergencecomplete normed spacenonlinear systems |
| spellingShingle | Ramandeep Behl Ioannis K. Argyros Hashim Alshehri Samundra Regmi Generalized Convergence for Multi-Step Schemes under Weak Conditions Mathematics multi-step scheme ball convergence complete normed space nonlinear systems |
| title | Generalized Convergence for Multi-Step Schemes under Weak Conditions |
| title_full | Generalized Convergence for Multi-Step Schemes under Weak Conditions |
| title_fullStr | Generalized Convergence for Multi-Step Schemes under Weak Conditions |
| title_full_unstemmed | Generalized Convergence for Multi-Step Schemes under Weak Conditions |
| title_short | Generalized Convergence for Multi-Step Schemes under Weak Conditions |
| title_sort | generalized convergence for multi step schemes under weak conditions |
| topic | multi-step scheme ball convergence complete normed space nonlinear systems |
| url | https://www.mdpi.com/2227-7390/12/2/220 |
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