Enhancing the Convergence Order from <i>p</i> to <i>p</i> + 3 in Iterative Methods for Solving Nonlinear Systems of Equations without the Use of Jacobian Matrices
In this paper, we present an innovative technique that improves the convergence order of iterative schemes that do not require the evaluation of Jacobian matrices. As far as we know, this is the first technique that allows us the achievement of an increase, from <i>p</i> to <inline-fo...
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MDPI AG
2023-10-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/20/4238 |
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| author | Alicia Cordero Miguel A. Leonardo-Sepúlveda Juan R. Torregrosa María P. Vassileva |
| author_facet | Alicia Cordero Miguel A. Leonardo-Sepúlveda Juan R. Torregrosa María P. Vassileva |
| author_sort | Alicia Cordero |
| collection | DOAJ |
| description | In this paper, we present an innovative technique that improves the convergence order of iterative schemes that do not require the evaluation of Jacobian matrices. As far as we know, this is the first technique that allows us the achievement of an increase, from <i>p</i> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>+</mo><mn>3</mn></mrow></semantics></math></inline-formula> units, in the order of convergence. This is constructed from any Jacobian-free scheme of order <i>p</i>. We conduct comprehensive numerical tests first in academical examples to validate the theoretical results, showing the efficiency and effectiveness of the new Jacobian-free schemes. Then, we apply them on the non-differentiable partial differential equations that models the nutrient diffusion in a biological substrate. |
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| language | English |
| last_indexed | 2024-03-10T21:04:26Z |
| publishDate | 2023-10-01 |
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| spelling | doaj.art-0c5fca1cde9643dcb323b93fa83204e42023-11-19T17:13:08ZengMDPI AGMathematics2227-73902023-10-011120423810.3390/math11204238Enhancing the Convergence Order from <i>p</i> to <i>p</i> + 3 in Iterative Methods for Solving Nonlinear Systems of Equations without the Use of Jacobian MatricesAlicia Cordero0Miguel A. Leonardo-Sepúlveda1Juan R. Torregrosa2María P. Vassileva3Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, Camino de Vera, s/n, 46022 Valencia, SpainÁrea de Ciencia Básica y Ambiental, Instituto Tecnológico de Santo Domingo (INTEC), Av. Los Próceres, Gala, Santo Domingo 10602, Dominican RepublicInstituto de Matemática Multidisciplinar, Universitat Politècnica de València, Camino de Vera, s/n, 46022 Valencia, SpainÁrea de Ciencia Básica y Ambiental, Instituto Tecnológico de Santo Domingo (INTEC), Av. Los Próceres, Gala, Santo Domingo 10602, Dominican RepublicIn this paper, we present an innovative technique that improves the convergence order of iterative schemes that do not require the evaluation of Jacobian matrices. As far as we know, this is the first technique that allows us the achievement of an increase, from <i>p</i> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>+</mo><mn>3</mn></mrow></semantics></math></inline-formula> units, in the order of convergence. This is constructed from any Jacobian-free scheme of order <i>p</i>. We conduct comprehensive numerical tests first in academical examples to validate the theoretical results, showing the efficiency and effectiveness of the new Jacobian-free schemes. Then, we apply them on the non-differentiable partial differential equations that models the nutrient diffusion in a biological substrate.https://www.mdpi.com/2227-7390/11/20/4238iterative methodsnonlinear systemslocal convergenceJacobian-free scheme |
| spellingShingle | Alicia Cordero Miguel A. Leonardo-Sepúlveda Juan R. Torregrosa María P. Vassileva Enhancing the Convergence Order from <i>p</i> to <i>p</i> + 3 in Iterative Methods for Solving Nonlinear Systems of Equations without the Use of Jacobian Matrices Mathematics iterative methods nonlinear systems local convergence Jacobian-free scheme |
| title | Enhancing the Convergence Order from <i>p</i> to <i>p</i> + 3 in Iterative Methods for Solving Nonlinear Systems of Equations without the Use of Jacobian Matrices |
| title_full | Enhancing the Convergence Order from <i>p</i> to <i>p</i> + 3 in Iterative Methods for Solving Nonlinear Systems of Equations without the Use of Jacobian Matrices |
| title_fullStr | Enhancing the Convergence Order from <i>p</i> to <i>p</i> + 3 in Iterative Methods for Solving Nonlinear Systems of Equations without the Use of Jacobian Matrices |
| title_full_unstemmed | Enhancing the Convergence Order from <i>p</i> to <i>p</i> + 3 in Iterative Methods for Solving Nonlinear Systems of Equations without the Use of Jacobian Matrices |
| title_short | Enhancing the Convergence Order from <i>p</i> to <i>p</i> + 3 in Iterative Methods for Solving Nonlinear Systems of Equations without the Use of Jacobian Matrices |
| title_sort | enhancing the convergence order from i p i to i p i 3 in iterative methods for solving nonlinear systems of equations without the use of jacobian matrices |
| topic | iterative methods nonlinear systems local convergence Jacobian-free scheme |
| url | https://www.mdpi.com/2227-7390/11/20/4238 |
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