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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| Main Authors: | , , , , |
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| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2025-02-01
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| Series: | Open Mathematics |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/math-2024-0115 |
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| Summary: | 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. |
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| ISSN: | 2391-5455 |