An Efficient Optimal Derivative-Free Fourth-Order Method and Its Memory Variant for Non-Linear Models and Their Dynamics
We propose a new optimal iterative scheme without memory free from derivatives for solving non-linear equations. There are many iterative schemes existing in the literature which either diverge or fail to work when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" d...
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MDPI AG
2023-03-01
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| Series: | Mathematical and Computational Applications |
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| Online Access: | https://www.mdpi.com/2297-8747/28/2/48 |
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| author | Himani Sharma Munish Kansal Ramandeep Behl |
| author_facet | Himani Sharma Munish Kansal Ramandeep Behl |
| author_sort | Himani Sharma |
| collection | DOAJ |
| description | We propose a new optimal iterative scheme without memory free from derivatives for solving non-linear equations. There are many iterative schemes existing in the literature which either diverge or fail to work when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>f</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>. However, our proposed scheme works even in these cases. In addition, we extended the same idea for iterative methods with memory with the help of self-accelerating parameters estimated from the current and previous approximations. As a result, the order of convergence increased from four to seven without the addition of any further functional evaluation. To confirm the theoretical results, numerical examples and comparisons with some of the existing methods are included which reveal that our scheme is more efficient than the existing schemes. Furthermore, basins of attraction are also included to describe a clear picture of the convergence of the proposed method as well as some of the existing methods. |
| first_indexed | 2024-03-11T04:46:39Z |
| format | Article |
| id | doaj.art-3f323edea3234fdeb0b01c452eee23e4 |
| institution | Directory Open Access Journal |
| issn | 1300-686X 2297-8747 |
| language | English |
| last_indexed | 2024-03-11T04:46:39Z |
| publishDate | 2023-03-01 |
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| series | Mathematical and Computational Applications |
| spelling | doaj.art-3f323edea3234fdeb0b01c452eee23e42023-11-17T20:19:34ZengMDPI AGMathematical and Computational Applications1300-686X2297-87472023-03-012824810.3390/mca28020048An Efficient Optimal Derivative-Free Fourth-Order Method and Its Memory Variant for Non-Linear Models and Their DynamicsHimani Sharma0Munish Kansal1Ramandeep Behl2School of Mathematics, Thapar Institute of Engineering and Technology, Patiala 147004, IndiaSchool of Mathematics, Thapar Institute of Engineering and Technology, Patiala 147004, IndiaMathematical Modelling and Applied Computation Research Group (MMAC), Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi ArabiaWe propose a new optimal iterative scheme without memory free from derivatives for solving non-linear equations. There are many iterative schemes existing in the literature which either diverge or fail to work when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>f</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>. However, our proposed scheme works even in these cases. In addition, we extended the same idea for iterative methods with memory with the help of self-accelerating parameters estimated from the current and previous approximations. As a result, the order of convergence increased from four to seven without the addition of any further functional evaluation. To confirm the theoretical results, numerical examples and comparisons with some of the existing methods are included which reveal that our scheme is more efficient than the existing schemes. Furthermore, basins of attraction are also included to describe a clear picture of the convergence of the proposed method as well as some of the existing methods.https://www.mdpi.com/2297-8747/28/2/48non-linear equationiterative method with memory<i>R</i>-order of convergencebasin of attraction |
| spellingShingle | Himani Sharma Munish Kansal Ramandeep Behl An Efficient Optimal Derivative-Free Fourth-Order Method and Its Memory Variant for Non-Linear Models and Their Dynamics Mathematical and Computational Applications non-linear equation iterative method with memory <i>R</i>-order of convergence basin of attraction |
| title | An Efficient Optimal Derivative-Free Fourth-Order Method and Its Memory Variant for Non-Linear Models and Their Dynamics |
| title_full | An Efficient Optimal Derivative-Free Fourth-Order Method and Its Memory Variant for Non-Linear Models and Their Dynamics |
| title_fullStr | An Efficient Optimal Derivative-Free Fourth-Order Method and Its Memory Variant for Non-Linear Models and Their Dynamics |
| title_full_unstemmed | An Efficient Optimal Derivative-Free Fourth-Order Method and Its Memory Variant for Non-Linear Models and Their Dynamics |
| title_short | An Efficient Optimal Derivative-Free Fourth-Order Method and Its Memory Variant for Non-Linear Models and Their Dynamics |
| title_sort | efficient optimal derivative free fourth order method and its memory variant for non linear models and their dynamics |
| topic | non-linear equation iterative method with memory <i>R</i>-order of convergence basin of attraction |
| url | https://www.mdpi.com/2297-8747/28/2/48 |
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