A Comparison Study of the Classical and Modern Results of Semi-Local Convergence of Newton–Kantorovich Iterations-II
This article is an independently written continuation of an earlier study with the same title [Mathematics, 2022, 10, 1225] on the Newton Process (NP). This process is applied to solve nonlinear equations. The complementing features are: the smallness of the initial approximation is expressed explic...
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2022-05-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/10/11/1839 |
| _version_ | 1797492649893036032 |
|---|---|
| author | Samundra Regmi Ioannis K. Argyros Santhosh George Michael I. Argyros |
| author_facet | Samundra Regmi Ioannis K. Argyros Santhosh George Michael I. Argyros |
| author_sort | Samundra Regmi |
| collection | DOAJ |
| description | This article is an independently written continuation of an earlier study with the same title [Mathematics, 2022, 10, 1225] on the Newton Process (NP). This process is applied to solve nonlinear equations. The complementing features are: the smallness of the initial approximation is expressed explicitly in turns of the Lipschitz or Hölder constants and the convergence order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>+</mo><mi>p</mi></mrow></semantics></math></inline-formula> is shown for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo><mo>.</mo></mrow></semantics></math></inline-formula> The first feature becomes attainable by further simplifying proofs of convergence criteria. The second feature is possible by choosing a bit larger upper bound on the smallness of the initial approximation. This way, the completed convergence analysis is finer and can replace the classical one by Kantorovich and others. A two-point boundary value problem (TPBVP) is solved to complement this article. |
| first_indexed | 2024-03-10T01:06:45Z |
| format | Article |
| id | doaj.art-bd1b545d8d3843048979efaf1bc217c2 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T01:06:45Z |
| publishDate | 2022-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-bd1b545d8d3843048979efaf1bc217c22023-11-23T14:25:26ZengMDPI AGMathematics2227-73902022-05-011011183910.3390/math10111839A Comparison Study of the Classical and Modern Results of Semi-Local Convergence of Newton–Kantorovich Iterations-IISamundra Regmi0Ioannis K. Argyros1Santhosh George2Michael I. Argyros3Department of Mathematics, University of Houston, Houston, TX 77204, USADepartment of Mathematical Sciences, Cameron University, Lawton, OK 73505, USADepartment of Mathematical and Computational Sciences, National Institute of Technology Karnataka, Karnataka 575 025, IndiaDepartment of Computer Science, University of Oklahoma, Norman, OK 73019, USAThis article is an independently written continuation of an earlier study with the same title [Mathematics, 2022, 10, 1225] on the Newton Process (NP). This process is applied to solve nonlinear equations. The complementing features are: the smallness of the initial approximation is expressed explicitly in turns of the Lipschitz or Hölder constants and the convergence order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>+</mo><mi>p</mi></mrow></semantics></math></inline-formula> is shown for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo><mo>.</mo></mrow></semantics></math></inline-formula> The first feature becomes attainable by further simplifying proofs of convergence criteria. The second feature is possible by choosing a bit larger upper bound on the smallness of the initial approximation. This way, the completed convergence analysis is finer and can replace the classical one by Kantorovich and others. A two-point boundary value problem (TPBVP) is solved to complement this article.https://www.mdpi.com/2227-7390/10/11/1839iterative processesBanach spacesemi-local convergence |
| spellingShingle | Samundra Regmi Ioannis K. Argyros Santhosh George Michael I. Argyros A Comparison Study of the Classical and Modern Results of Semi-Local Convergence of Newton–Kantorovich Iterations-II Mathematics iterative processes Banach space semi-local convergence |
| title | A Comparison Study of the Classical and Modern Results of Semi-Local Convergence of Newton–Kantorovich Iterations-II |
| title_full | A Comparison Study of the Classical and Modern Results of Semi-Local Convergence of Newton–Kantorovich Iterations-II |
| title_fullStr | A Comparison Study of the Classical and Modern Results of Semi-Local Convergence of Newton–Kantorovich Iterations-II |
| title_full_unstemmed | A Comparison Study of the Classical and Modern Results of Semi-Local Convergence of Newton–Kantorovich Iterations-II |
| title_short | A Comparison Study of the Classical and Modern Results of Semi-Local Convergence of Newton–Kantorovich Iterations-II |
| title_sort | comparison study of the classical and modern results of semi local convergence of newton kantorovich iterations ii |
| topic | iterative processes Banach space semi-local convergence |
| url | https://www.mdpi.com/2227-7390/10/11/1839 |
| work_keys_str_mv | AT samundraregmi acomparisonstudyoftheclassicalandmodernresultsofsemilocalconvergenceofnewtonkantorovichiterationsii AT ioanniskargyros acomparisonstudyoftheclassicalandmodernresultsofsemilocalconvergenceofnewtonkantorovichiterationsii AT santhoshgeorge acomparisonstudyoftheclassicalandmodernresultsofsemilocalconvergenceofnewtonkantorovichiterationsii AT michaeliargyros acomparisonstudyoftheclassicalandmodernresultsofsemilocalconvergenceofnewtonkantorovichiterationsii AT samundraregmi comparisonstudyoftheclassicalandmodernresultsofsemilocalconvergenceofnewtonkantorovichiterationsii AT ioanniskargyros comparisonstudyoftheclassicalandmodernresultsofsemilocalconvergenceofnewtonkantorovichiterationsii AT santhoshgeorge comparisonstudyoftheclassicalandmodernresultsofsemilocalconvergenceofnewtonkantorovichiterationsii AT michaeliargyros comparisonstudyoftheclassicalandmodernresultsofsemilocalconvergenceofnewtonkantorovichiterationsii |