Updating to Optimal Parametric Values by Memory-Dependent Methods: Iterative Schemes of Fractional Type for Solving Nonlinear Equations
In the paper, two nonlinear variants of the Newton method are developed for solving nonlinear equations. The derivative-free nonlinear fractional type of the one-step iterative scheme of a fourth-order convergence contains three parameters, whose optimal values are obtained by a memory-dependent upd...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-03-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/12/7/1032 |
| _version_ | 1827286757799886848 |
|---|---|
| author | Chein-Shan Liu Chih-Wen Chang |
| author_facet | Chein-Shan Liu Chih-Wen Chang |
| author_sort | Chein-Shan Liu |
| collection | DOAJ |
| description | In the paper, two nonlinear variants of the Newton method are developed for solving nonlinear equations. The derivative-free nonlinear fractional type of the one-step iterative scheme of a fourth-order convergence contains three parameters, whose optimal values are obtained by a memory-dependent updating method. Then, as the extensions of a one-step linear fractional type method, we explore the fractional types of two- and three-step iterative schemes, which possess sixth- and twelfth-order convergences when the parameters’ values are optimal; the efficiency indexes are <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msqrt><mn>6</mn></msqrt></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mroot><mn>12</mn><mn>3</mn></mroot></semantics></math></inline-formula>, respectively. An extra variable is supplemented into the second-degree Newton polynomial for the data interpolation of the two-step iterative scheme of fractional type, and a relaxation factor is accelerated by the memory-dependent method. Three memory-dependent updating methods are developed in the three-step iterative schemes of linear fractional type, whose performances are greatly strengthened. In the three-step iterative scheme, when the first step involves using the nonlinear fractional type model, the order of convergence is raised to sixteen. The efficiency index also increases to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mroot><mn>16</mn><mn>3</mn></mroot></semantics></math></inline-formula>, and a third-degree Newton polynomial is taken to update the values of optimal parameters. |
| first_indexed | 2024-04-24T10:39:59Z |
| format | Article |
| id | doaj.art-4678cf04d5cf43049df49d4f730598a7 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-04-24T10:39:59Z |
| publishDate | 2024-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-4678cf04d5cf43049df49d4f730598a72024-04-12T13:22:40ZengMDPI AGMathematics2227-73902024-03-01127103210.3390/math12071032Updating to Optimal Parametric Values by Memory-Dependent Methods: Iterative Schemes of Fractional Type for Solving Nonlinear EquationsChein-Shan Liu0Chih-Wen Chang1Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 202301, TaiwanDepartment of Mechanical Engineering, National United University, Miaoli 360302, TaiwanIn the paper, two nonlinear variants of the Newton method are developed for solving nonlinear equations. The derivative-free nonlinear fractional type of the one-step iterative scheme of a fourth-order convergence contains three parameters, whose optimal values are obtained by a memory-dependent updating method. Then, as the extensions of a one-step linear fractional type method, we explore the fractional types of two- and three-step iterative schemes, which possess sixth- and twelfth-order convergences when the parameters’ values are optimal; the efficiency indexes are <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msqrt><mn>6</mn></msqrt></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mroot><mn>12</mn><mn>3</mn></mroot></semantics></math></inline-formula>, respectively. An extra variable is supplemented into the second-degree Newton polynomial for the data interpolation of the two-step iterative scheme of fractional type, and a relaxation factor is accelerated by the memory-dependent method. Three memory-dependent updating methods are developed in the three-step iterative schemes of linear fractional type, whose performances are greatly strengthened. In the three-step iterative scheme, when the first step involves using the nonlinear fractional type model, the order of convergence is raised to sixteen. The efficiency index also increases to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mroot><mn>16</mn><mn>3</mn></mroot></semantics></math></inline-formula>, and a third-degree Newton polynomial is taken to update the values of optimal parameters.https://www.mdpi.com/2227-7390/12/7/1032nonlinear equationnonlinear perturbation of Newton methodfractional type iterative schemesmulti-step iterative schemememory-dependent method |
| spellingShingle | Chein-Shan Liu Chih-Wen Chang Updating to Optimal Parametric Values by Memory-Dependent Methods: Iterative Schemes of Fractional Type for Solving Nonlinear Equations Mathematics nonlinear equation nonlinear perturbation of Newton method fractional type iterative schemes multi-step iterative scheme memory-dependent method |
| title | Updating to Optimal Parametric Values by Memory-Dependent Methods: Iterative Schemes of Fractional Type for Solving Nonlinear Equations |
| title_full | Updating to Optimal Parametric Values by Memory-Dependent Methods: Iterative Schemes of Fractional Type for Solving Nonlinear Equations |
| title_fullStr | Updating to Optimal Parametric Values by Memory-Dependent Methods: Iterative Schemes of Fractional Type for Solving Nonlinear Equations |
| title_full_unstemmed | Updating to Optimal Parametric Values by Memory-Dependent Methods: Iterative Schemes of Fractional Type for Solving Nonlinear Equations |
| title_short | Updating to Optimal Parametric Values by Memory-Dependent Methods: Iterative Schemes of Fractional Type for Solving Nonlinear Equations |
| title_sort | updating to optimal parametric values by memory dependent methods iterative schemes of fractional type for solving nonlinear equations |
| topic | nonlinear equation nonlinear perturbation of Newton method fractional type iterative schemes multi-step iterative scheme memory-dependent method |
| url | https://www.mdpi.com/2227-7390/12/7/1032 |
| work_keys_str_mv | AT cheinshanliu updatingtooptimalparametricvaluesbymemorydependentmethodsiterativeschemesoffractionaltypeforsolvingnonlinearequations AT chihwenchang updatingtooptimalparametricvaluesbymemorydependentmethodsiterativeschemesoffractionaltypeforsolvingnonlinearequations |