A Family of Higher Order Scheme for Multiple Roots
We have developed a two-point iterative scheme for multiple roots that achieves fifth order convergence by using two function evaluations and two derivative evaluations each iteration. Weight function approach is utilized to frame the scheme. The weight function named as <inline-formula><ma...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-01-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/15/1/228 |
| _version_ | 1797436847102623744 |
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| author | Tajinder Singh Himani Arora Lorentz Jäntschi |
| author_facet | Tajinder Singh Himani Arora Lorentz Jäntschi |
| author_sort | Tajinder Singh |
| collection | DOAJ |
| description | We have developed a two-point iterative scheme for multiple roots that achieves fifth order convergence by using two function evaluations and two derivative evaluations each iteration. Weight function approach is utilized to frame the scheme. The weight function named as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mo>(</mo><msub><mi>υ</mi><mi>t</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> is used, which is a function of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>υ</mi><mi>t</mi></msub></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>υ</mi><mi>t</mi></msub></semantics></math></inline-formula> is a function of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ω</mi><mi>t</mi></msub></semantics></math></inline-formula>, i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>υ</mi><mi>t</mi></msub><mo>=</mo><mfrac><msub><mi>ω</mi><mi>t</mi></msub><mrow><mn>1</mn><mo>+</mo><mi>a</mi><msub><mi>ω</mi><mi>t</mi></msub></mrow></mfrac><mo>,</mo></mrow></semantics></math></inline-formula> where <i>a</i> is a real number and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ω</mi><mi>t</mi></msub><mo>=</mo><msup><mfenced separators="" open="(" close=")"><mfrac><mrow><mi>g</mi><mo>(</mo><msub><mi mathvariant="monospace">y</mi><mi>t</mi></msub><mo>)</mo></mrow><mrow><mi>g</mi><mo>(</mo><msub><mi mathvariant="monospace">x</mi><mi>t</mi></msub><mo>)</mo></mrow></mfrac></mfenced><mfrac><mn>1</mn><mover accent="true"><mi>m</mi><mo stretchy="false">˜</mo></mover></mfrac></msup></mrow></semantics></math></inline-formula> is a multi-valued function. The consistency of the newly generated methods is ensured numerically and through the basins of attraction. Four complex functions are considered to compare the new methods with existing schemes via basins of attraction, and all provided basins of attraction possess reflection symmetry. Further, five numerical examples are used to verify the theoretical results and to contrast the presented schemes with some recognized schemes of fifth order. The results obtained have proved that the new schemes are better than the existing schemes of the same nature. |
| first_indexed | 2024-03-09T11:07:32Z |
| format | Article |
| id | doaj.art-cfaedee9e6b747b5ae55c261f3bac5ae |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-09T11:07:32Z |
| publishDate | 2023-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-cfaedee9e6b747b5ae55c261f3bac5ae2023-12-01T00:53:48ZengMDPI AGSymmetry2073-89942023-01-0115122810.3390/sym15010228A Family of Higher Order Scheme for Multiple RootsTajinder Singh0Himani Arora1Lorentz Jäntschi2Department of Mathematics, Guru Nanak Dev University, Amritsar 143005, Punjab, IndiaDepartment of Mathematics, Guru Nanak Dev University, Amritsar 143005, Punjab, IndiaInstitute of Doctoral Studies, Babeş-Bolyai University, 400084 Cluj-Napoca, RomaniaWe have developed a two-point iterative scheme for multiple roots that achieves fifth order convergence by using two function evaluations and two derivative evaluations each iteration. Weight function approach is utilized to frame the scheme. The weight function named as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mo>(</mo><msub><mi>υ</mi><mi>t</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> is used, which is a function of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>υ</mi><mi>t</mi></msub></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>υ</mi><mi>t</mi></msub></semantics></math></inline-formula> is a function of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ω</mi><mi>t</mi></msub></semantics></math></inline-formula>, i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>υ</mi><mi>t</mi></msub><mo>=</mo><mfrac><msub><mi>ω</mi><mi>t</mi></msub><mrow><mn>1</mn><mo>+</mo><mi>a</mi><msub><mi>ω</mi><mi>t</mi></msub></mrow></mfrac><mo>,</mo></mrow></semantics></math></inline-formula> where <i>a</i> is a real number and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ω</mi><mi>t</mi></msub><mo>=</mo><msup><mfenced separators="" open="(" close=")"><mfrac><mrow><mi>g</mi><mo>(</mo><msub><mi mathvariant="monospace">y</mi><mi>t</mi></msub><mo>)</mo></mrow><mrow><mi>g</mi><mo>(</mo><msub><mi mathvariant="monospace">x</mi><mi>t</mi></msub><mo>)</mo></mrow></mfrac></mfenced><mfrac><mn>1</mn><mover accent="true"><mi>m</mi><mo stretchy="false">˜</mo></mover></mfrac></msup></mrow></semantics></math></inline-formula> is a multi-valued function. The consistency of the newly generated methods is ensured numerically and through the basins of attraction. Four complex functions are considered to compare the new methods with existing schemes via basins of attraction, and all provided basins of attraction possess reflection symmetry. Further, five numerical examples are used to verify the theoretical results and to contrast the presented schemes with some recognized schemes of fifth order. The results obtained have proved that the new schemes are better than the existing schemes of the same nature.https://www.mdpi.com/2073-8994/15/1/228nonlinear equationsmultiple rootsNewton–Raphson’s methoditerative methods |
| spellingShingle | Tajinder Singh Himani Arora Lorentz Jäntschi A Family of Higher Order Scheme for Multiple Roots Symmetry nonlinear equations multiple roots Newton–Raphson’s method iterative methods |
| title | A Family of Higher Order Scheme for Multiple Roots |
| title_full | A Family of Higher Order Scheme for Multiple Roots |
| title_fullStr | A Family of Higher Order Scheme for Multiple Roots |
| title_full_unstemmed | A Family of Higher Order Scheme for Multiple Roots |
| title_short | A Family of Higher Order Scheme for Multiple Roots |
| title_sort | family of higher order scheme for multiple roots |
| topic | nonlinear equations multiple roots Newton–Raphson’s method iterative methods |
| url | https://www.mdpi.com/2073-8994/15/1/228 |
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