The Enhanced Fixed Point Method: An Extremely Simple Procedure to Accelerate the Convergence of the Fixed Point Method to Solve Nonlinear Algebraic Equations
This work proposes the Enhanced Fixed Point Method (EFPM) as a straightforward modification to the problem of finding an exact or approximate solution for a linear or nonlinear algebraic equation. The proposal consists of providing a versatile method that is easy to employ and systematic. Therefore,...
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| Format: | Article |
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MDPI AG
2022-10-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/20/3797 |
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| author | Uriel Filobello-Nino Hector Vazquez-Leal Jesús Huerta-Chua Jaime Martínez-Castillo Agustín L. Herrera-May Mario Alberto Sandoval-Hernandez Victor Manuel Jimenez-Fernandez |
| author_facet | Uriel Filobello-Nino Hector Vazquez-Leal Jesús Huerta-Chua Jaime Martínez-Castillo Agustín L. Herrera-May Mario Alberto Sandoval-Hernandez Victor Manuel Jimenez-Fernandez |
| author_sort | Uriel Filobello-Nino |
| collection | DOAJ |
| description | This work proposes the Enhanced Fixed Point Method (EFPM) as a straightforward modification to the problem of finding an exact or approximate solution for a linear or nonlinear algebraic equation. The proposal consists of providing a versatile method that is easy to employ and systematic. Therefore, it is expected that this work contributes to breaking the paradigm that an effective modification for a known method has to be necessarily long and complicated. As a matter of fact, the method expresses an algebraic equation in terms of the same equation but multiplied for an adequate factor, which most of the times is just a simple numeric factor. The main idea is modifying the original equation, slightly changing it for others in such a way that both have the same solution. Next, the modified equation is expressed as a fixed point problem and the proposed parameters are employed to accelerate the convergence of the fixed point problem for the original equation. Since the Newton method results from a possible fixed point problem of an algebraic equation, we will see that it is relatively easy to get modified versions of the Newton method with orders of convergence major than two. We will see in this work the convenience of this procedure. |
| first_indexed | 2024-03-09T19:52:07Z |
| format | Article |
| id | doaj.art-55fdd944a6324734bc0af7fc94def822 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T19:52:07Z |
| publishDate | 2022-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-55fdd944a6324734bc0af7fc94def8222023-11-24T01:07:09ZengMDPI AGMathematics2227-73902022-10-011020379710.3390/math10203797The Enhanced Fixed Point Method: An Extremely Simple Procedure to Accelerate the Convergence of the Fixed Point Method to Solve Nonlinear Algebraic EquationsUriel Filobello-Nino0Hector Vazquez-Leal1Jesús Huerta-Chua2Jaime Martínez-Castillo3Agustín L. Herrera-May4Mario Alberto Sandoval-Hernandez5Victor Manuel Jimenez-Fernandez6Facultad de Instrumentación Electrónica, Universidad Veracruzana, Cto. Gonzalo Aguirre Beltrán S/N, Xalapa 91000, Veracruz, MexicoFacultad de Instrumentación Electrónica, Universidad Veracruzana, Cto. Gonzalo Aguirre Beltrán S/N, Xalapa 91000, Veracruz, MexicoInstituto Tecnológico Superior de Poza Rica, Tecnológico Nacional de México, Luis Donaldo Colosio Murrieta S/N, Arroyo del Maíz, Poza Rica 93230, Veracruz, MexicoCentro de Investigación en Micro y Nanotecnología, Universidad Veracruzana, Boca del Río 94294, Veracruz, MexicoCentro de Investigación en Micro y Nanotecnología, Universidad Veracruzana, Boca del Río 94294, Veracruz, MexicoInstituto Tecnológico Superior de Poza Rica, Tecnológico Nacional de México, Luis Donaldo Colosio Murrieta S/N, Arroyo del Maíz, Poza Rica 93230, Veracruz, MexicoFacultad de Instrumentación Electrónica, Universidad Veracruzana, Cto. Gonzalo Aguirre Beltrán S/N, Xalapa 91000, Veracruz, MexicoThis work proposes the Enhanced Fixed Point Method (EFPM) as a straightforward modification to the problem of finding an exact or approximate solution for a linear or nonlinear algebraic equation. The proposal consists of providing a versatile method that is easy to employ and systematic. Therefore, it is expected that this work contributes to breaking the paradigm that an effective modification for a known method has to be necessarily long and complicated. As a matter of fact, the method expresses an algebraic equation in terms of the same equation but multiplied for an adequate factor, which most of the times is just a simple numeric factor. The main idea is modifying the original equation, slightly changing it for others in such a way that both have the same solution. Next, the modified equation is expressed as a fixed point problem and the proposed parameters are employed to accelerate the convergence of the fixed point problem for the original equation. Since the Newton method results from a possible fixed point problem of an algebraic equation, we will see that it is relatively easy to get modified versions of the Newton method with orders of convergence major than two. We will see in this work the convenience of this procedure.https://www.mdpi.com/2227-7390/10/20/3797nonlinear algebraic equationsexact solutionsapproximate solutionsfixed pointfixed point theoremfixed point iteration |
| spellingShingle | Uriel Filobello-Nino Hector Vazquez-Leal Jesús Huerta-Chua Jaime Martínez-Castillo Agustín L. Herrera-May Mario Alberto Sandoval-Hernandez Victor Manuel Jimenez-Fernandez The Enhanced Fixed Point Method: An Extremely Simple Procedure to Accelerate the Convergence of the Fixed Point Method to Solve Nonlinear Algebraic Equations Mathematics nonlinear algebraic equations exact solutions approximate solutions fixed point fixed point theorem fixed point iteration |
| title | The Enhanced Fixed Point Method: An Extremely Simple Procedure to Accelerate the Convergence of the Fixed Point Method to Solve Nonlinear Algebraic Equations |
| title_full | The Enhanced Fixed Point Method: An Extremely Simple Procedure to Accelerate the Convergence of the Fixed Point Method to Solve Nonlinear Algebraic Equations |
| title_fullStr | The Enhanced Fixed Point Method: An Extremely Simple Procedure to Accelerate the Convergence of the Fixed Point Method to Solve Nonlinear Algebraic Equations |
| title_full_unstemmed | The Enhanced Fixed Point Method: An Extremely Simple Procedure to Accelerate the Convergence of the Fixed Point Method to Solve Nonlinear Algebraic Equations |
| title_short | The Enhanced Fixed Point Method: An Extremely Simple Procedure to Accelerate the Convergence of the Fixed Point Method to Solve Nonlinear Algebraic Equations |
| title_sort | enhanced fixed point method an extremely simple procedure to accelerate the convergence of the fixed point method to solve nonlinear algebraic equations |
| topic | nonlinear algebraic equations exact solutions approximate solutions fixed point fixed point theorem fixed point iteration |
| url | https://www.mdpi.com/2227-7390/10/20/3797 |
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