Exploring homotopy with hyperspherical tracking to find complex roots with application to electrical circuits
In the field of applied sciences, systems are frequently modeled using mathematical frameworks that include systems of nonlinear algebraic equations. Identifying their roots, whether real or complex, is of critical importance. The widespread use of complex numbers in science and engineering highligh...
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| Format: | Article |
| Language: | English |
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De Gruyter
2025-02-01
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| Series: | Open Mathematics |
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| Online Access: | https://doi.org/10.1515/math-2024-0115 |
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| author | Sandoval-Hernandez Mario A. Jimenez-Islas Hugo Vazquez-Leal Hector Quemada-Villagómez Miriam L. Lopez-Gonzalez María de la Luz |
| author_facet | Sandoval-Hernandez Mario A. Jimenez-Islas Hugo Vazquez-Leal Hector Quemada-Villagómez Miriam L. Lopez-Gonzalez María de la Luz |
| author_sort | Sandoval-Hernandez Mario A. |
| collection | DOAJ |
| description | In the field of applied sciences, systems are frequently modeled using mathematical frameworks that include systems of nonlinear algebraic equations. Identifying their roots, whether real or complex, is of critical importance. The widespread use of complex numbers in science and engineering highlights the importance of accurately determining the complex roots of equations. This article presents a study in which the complex roots of a system of equations are identified through an approach that utilizes homotopy continuation, with the curve being traced using a hyperspherical path tracking technique. Furthermore, this article details five case studies on electrical networks where this method is applied to solve systems of equations containing imaginary coefficients to find mesh currents. The path tracking shows the behavior of system equation in each case study. Finally, an analysis of the precision of the solutions obtained in these case studies is provided, demonstrating an accuracy of up to 15 SDs in a single iteration during the refinement stage. |
| format | Article |
| id | doaj-art-565c0b4fd0e94296a7edd041563199da |
| institution | Kabale University |
| issn | 2391-5455 |
| language | English |
| publishDate | 2025-02-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Open Mathematics |
| spelling | doaj-art-565c0b4fd0e94296a7edd041563199da2025-02-10T13:24:36ZengDe GruyterOpen Mathematics2391-54552025-02-012311910.1515/math-2024-0115Exploring homotopy with hyperspherical tracking to find complex roots with application to electrical circuitsSandoval-Hernandez Mario A.0Jimenez-Islas Hugo1Vazquez-Leal Hector2Quemada-Villagómez Miriam L.3Lopez-Gonzalez María de la Luz4Centro de Bachillerato Tecnológico Industrial y de Servicios, No. 190. Av. 15 S/N esq. calle 11, col. Venustiano, Carranza, 94297 Boca del Río, Veracruz, MexicoTecnológico Nacional de México en Celaya, Antonio García Cubas 600, Fovissste, 38010 Celaya, Gto, MexicoFacultad de Instrumentación Electrónica, Universidad Veracruzana, Cto. Aguirre Beltrán S/N, Zona Universitaria, 91090 Xalapa, Veracruz, MexicoTecnológico Nacional de México en Celaya, Antonio García Cubas 600, Fovissste, 38010 Celaya, Gto, MexicoDepartamento de Enfermería y Obstetricia, Universidad de Guanajuato, Lascuráin de Retana, No. 5, Col. Centro, 36000. Guanajuato, Gto, MexicoIn the field of applied sciences, systems are frequently modeled using mathematical frameworks that include systems of nonlinear algebraic equations. Identifying their roots, whether real or complex, is of critical importance. The widespread use of complex numbers in science and engineering highlights the importance of accurately determining the complex roots of equations. This article presents a study in which the complex roots of a system of equations are identified through an approach that utilizes homotopy continuation, with the curve being traced using a hyperspherical path tracking technique. Furthermore, this article details five case studies on electrical networks where this method is applied to solve systems of equations containing imaginary coefficients to find mesh currents. The path tracking shows the behavior of system equation in each case study. Finally, an analysis of the precision of the solutions obtained in these case studies is provided, demonstrating an accuracy of up to 15 SDs in a single iteration during the refinement stage.https://doi.org/10.1515/math-2024-0115homotopy continuation methodspherical path tracking methodscomplex rootsmultiple operating pointselectrical circuits010165d2565h0465h1068w30 |
| spellingShingle | Sandoval-Hernandez Mario A. Jimenez-Islas Hugo Vazquez-Leal Hector Quemada-Villagómez Miriam L. Lopez-Gonzalez María de la Luz Exploring homotopy with hyperspherical tracking to find complex roots with application to electrical circuits Open Mathematics homotopy continuation method spherical path tracking methods complex roots multiple operating points electrical circuits 0101 65d25 65h04 65h10 68w30 |
| title | Exploring homotopy with hyperspherical tracking to find complex roots with application to electrical circuits |
| title_full | Exploring homotopy with hyperspherical tracking to find complex roots with application to electrical circuits |
| title_fullStr | Exploring homotopy with hyperspherical tracking to find complex roots with application to electrical circuits |
| title_full_unstemmed | Exploring homotopy with hyperspherical tracking to find complex roots with application to electrical circuits |
| title_short | Exploring homotopy with hyperspherical tracking to find complex roots with application to electrical circuits |
| title_sort | exploring homotopy with hyperspherical tracking to find complex roots with application to electrical circuits |
| topic | homotopy continuation method spherical path tracking methods complex roots multiple operating points electrical circuits 0101 65d25 65h04 65h10 68w30 |
| url | https://doi.org/10.1515/math-2024-0115 |
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