On Some Improved Harmonic Mean Newton-Like Methods for Solving Systems of Nonlinear Equations
In this work, we have developed a fourth order Newton-like method based on harmonic mean and its multi-step version for solving system of nonlinear equations. The new fourth order method requires evaluation of one function and two first order Fréchet derivatives for each iteration. The multi-step ve...
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| Format: | Article |
| Language: | English |
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MDPI AG
2015-10-01
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| Series: | Algorithms |
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| Online Access: | http://www.mdpi.com/1999-4893/8/4/895 |
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| author | Diyashvir Kreetee Rajiv Babajee Kalyanasundaram Madhu Jayakumar Jayaraman |
| author_facet | Diyashvir Kreetee Rajiv Babajee Kalyanasundaram Madhu Jayakumar Jayaraman |
| author_sort | Diyashvir Kreetee Rajiv Babajee |
| collection | DOAJ |
| description | In this work, we have developed a fourth order Newton-like method based on harmonic mean and its multi-step version for solving system of nonlinear equations. The new fourth order method requires evaluation of one function and two first order Fréchet derivatives for each iteration. The multi-step version requires one more function evaluation for each iteration. The proposed new scheme does not require the evaluation of second or higher order Fréchet derivatives and still reaches fourth order convergence. The multi-step version converges with order 2r+4, where r is a positive integer and r ≥ 1. We have proved that the root α is a point of attraction for a general iterative function, whereas the proposed new schemes also satisfy this result. Numerical experiments including an application to 1-D Bratu problem are given to illustrate the efficiency of the new methods. Also, the new methods are compared with some existing methods. |
| first_indexed | 2024-12-22T13:21:37Z |
| format | Article |
| id | doaj.art-6eb57ea675dc448f8ed7a3871a0010bb |
| institution | Directory Open Access Journal |
| issn | 1999-4893 |
| language | English |
| last_indexed | 2024-12-22T13:21:37Z |
| publishDate | 2015-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Algorithms |
| spelling | doaj.art-6eb57ea675dc448f8ed7a3871a0010bb2022-12-21T18:24:28ZengMDPI AGAlgorithms1999-48932015-10-018489590910.3390/a8040895a8040895On Some Improved Harmonic Mean Newton-Like Methods for Solving Systems of Nonlinear EquationsDiyashvir Kreetee Rajiv Babajee0Kalyanasundaram Madhu1Jayakumar Jayaraman2African Network for Policy Research & Advocacy for Sustainability (ANPRAS), Midlands, Curepipe 52501, MauritiusDepartment of Mathematics, Pondicherry Engineering College, Pondicherry 605014, IndiaDepartment of Mathematics, Pondicherry Engineering College, Pondicherry 605014, IndiaIn this work, we have developed a fourth order Newton-like method based on harmonic mean and its multi-step version for solving system of nonlinear equations. The new fourth order method requires evaluation of one function and two first order Fréchet derivatives for each iteration. The multi-step version requires one more function evaluation for each iteration. The proposed new scheme does not require the evaluation of second or higher order Fréchet derivatives and still reaches fourth order convergence. The multi-step version converges with order 2r+4, where r is a positive integer and r ≥ 1. We have proved that the root α is a point of attraction for a general iterative function, whereas the proposed new schemes also satisfy this result. Numerical experiments including an application to 1-D Bratu problem are given to illustrate the efficiency of the new methods. Also, the new methods are compared with some existing methods.http://www.mdpi.com/1999-4893/8/4/895system of nonlinear equationNewton’s methodorder of convergencepoint of attraction |
| spellingShingle | Diyashvir Kreetee Rajiv Babajee Kalyanasundaram Madhu Jayakumar Jayaraman On Some Improved Harmonic Mean Newton-Like Methods for Solving Systems of Nonlinear Equations Algorithms system of nonlinear equation Newton’s method order of convergence point of attraction |
| title | On Some Improved Harmonic Mean Newton-Like Methods for Solving Systems of Nonlinear Equations |
| title_full | On Some Improved Harmonic Mean Newton-Like Methods for Solving Systems of Nonlinear Equations |
| title_fullStr | On Some Improved Harmonic Mean Newton-Like Methods for Solving Systems of Nonlinear Equations |
| title_full_unstemmed | On Some Improved Harmonic Mean Newton-Like Methods for Solving Systems of Nonlinear Equations |
| title_short | On Some Improved Harmonic Mean Newton-Like Methods for Solving Systems of Nonlinear Equations |
| title_sort | on some improved harmonic mean newton like methods for solving systems of nonlinear equations |
| topic | system of nonlinear equation Newton’s method order of convergence point of attraction |
| url | http://www.mdpi.com/1999-4893/8/4/895 |
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