Extension of King’s Iterative Scheme by Means of Memory for Nonlinear Equations
We developed a new family of optimal eighth-order derivative-free iterative methods for finding simple roots of nonlinear equations based on King’s scheme and Lagrange interpolation. By incorporating four self-accelerating parameters and a weight function in a single variable, we extend the proposed...
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MDPI AG
2023-05-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/15/5/1116 |
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| author | Saima Akram Maira Khalid Moin-ud-Din Junjua Shazia Altaf Sunil Kumar |
| author_facet | Saima Akram Maira Khalid Moin-ud-Din Junjua Shazia Altaf Sunil Kumar |
| author_sort | Saima Akram |
| collection | DOAJ |
| description | We developed a new family of optimal eighth-order derivative-free iterative methods for finding simple roots of nonlinear equations based on King’s scheme and Lagrange interpolation. By incorporating four self-accelerating parameters and a weight function in a single variable, we extend the proposed family to an efficient iterative scheme with memory. Without performing additional functional evaluations, the order of convergence is boosted from 8 to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>15.51560</mn></mrow></semantics></math></inline-formula>, and the efficiency index is raised from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1.6817</mn></mrow></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1.9847</mn></mrow></semantics></math></inline-formula>. To compare the performance of the proposed and existing schemes, some real-world problems are selected, such as the eigenvalue problem, continuous stirred-tank reactor problem, and energy distribution for Planck’s radiation. The stability and regions of convergence of the proposed iterative schemes are investigated through graphical tools, such as 2D symmetric basins of attractions for the case of memory-based schemes and 3D stereographic projections in the case of schemes without memory. The stability analysis demonstrates that our newly developed schemes have wider symmetric regions of convergence than the existing schemes in their respective domains. |
| first_indexed | 2024-03-11T03:16:59Z |
| format | Article |
| id | doaj.art-eab616f97b1a443d9e78120abaab4845 |
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| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-11T03:16:59Z |
| publishDate | 2023-05-01 |
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| series | Symmetry |
| spelling | doaj.art-eab616f97b1a443d9e78120abaab48452023-11-18T03:31:10ZengMDPI AGSymmetry2073-89942023-05-01155111610.3390/sym15051116Extension of King’s Iterative Scheme by Means of Memory for Nonlinear EquationsSaima Akram0Maira Khalid1Moin-ud-Din Junjua2Shazia Altaf3Sunil Kumar4Department of Mathematics, Government College Women University Faisalabad, Faisalabad 38000, PakistanCentre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya, Multan 60000, PakistanDepartment of Mathematics, Ghazi University, Dera Ghazi Khan 32200, PakistanDepartment of Mathematics and Statistics, Institute of Southern Punjab, Multan 60800, PakistanDepartment of Mathematics, University Centre for Research and Development, Chandigarh University, Mohali 140413, IndiaWe developed a new family of optimal eighth-order derivative-free iterative methods for finding simple roots of nonlinear equations based on King’s scheme and Lagrange interpolation. By incorporating four self-accelerating parameters and a weight function in a single variable, we extend the proposed family to an efficient iterative scheme with memory. Without performing additional functional evaluations, the order of convergence is boosted from 8 to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>15.51560</mn></mrow></semantics></math></inline-formula>, and the efficiency index is raised from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1.6817</mn></mrow></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1.9847</mn></mrow></semantics></math></inline-formula>. To compare the performance of the proposed and existing schemes, some real-world problems are selected, such as the eigenvalue problem, continuous stirred-tank reactor problem, and energy distribution for Planck’s radiation. The stability and regions of convergence of the proposed iterative schemes are investigated through graphical tools, such as 2D symmetric basins of attractions for the case of memory-based schemes and 3D stereographic projections in the case of schemes without memory. The stability analysis demonstrates that our newly developed schemes have wider symmetric regions of convergence than the existing schemes in their respective domains.https://www.mdpi.com/2073-8994/15/5/1116nonlinear equationmultipoint iterative methodsconvergence orderwith-memory methodefficiency indexpolynomiography |
| spellingShingle | Saima Akram Maira Khalid Moin-ud-Din Junjua Shazia Altaf Sunil Kumar Extension of King’s Iterative Scheme by Means of Memory for Nonlinear Equations Symmetry nonlinear equation multipoint iterative methods convergence order with-memory method efficiency index polynomiography |
| title | Extension of King’s Iterative Scheme by Means of Memory for Nonlinear Equations |
| title_full | Extension of King’s Iterative Scheme by Means of Memory for Nonlinear Equations |
| title_fullStr | Extension of King’s Iterative Scheme by Means of Memory for Nonlinear Equations |
| title_full_unstemmed | Extension of King’s Iterative Scheme by Means of Memory for Nonlinear Equations |
| title_short | Extension of King’s Iterative Scheme by Means of Memory for Nonlinear Equations |
| title_sort | extension of king s iterative scheme by means of memory for nonlinear equations |
| topic | nonlinear equation multipoint iterative methods convergence order with-memory method efficiency index polynomiography |
| url | https://www.mdpi.com/2073-8994/15/5/1116 |
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