Resumen de Efficient numerical methods for solving nonlinear problems
Marlon Ernesto Moscoso Martínez
The resolution of non-linear equations and systems is fundamental in various scientific and engineering fields, including physics, chemistry, biology, economics, and computer science. Numerical methods are crucial for solving these equations due to their complexity, which often results in multiple or no solutions, rendering traditional analytical methods inadequate. This research focuses on developing and analyzing new iterative schemes for solving non-linear equations and systems, emphasizing convergence, stability, and computational efficiency. Three key papers were published as part of this research. The first paper introduces a novel family of two-step iterative methods derived from a damped Newton scheme, which includes an additional Newton step with a weight function and a "frozen" derivative. This family, initially a four-parameter class with first-order convergence, becomes a single-parameter family with third-order convergence, which also exhibits exceptional stability and efficiency, validated through numerical tests. The second paper presents a new three-step iterative method, initially a three-parameter fourth-order family, which accelerates to a single-parameter sixth-order family. This method's convergence, complex dynamics, and numerical behavior are thoroughly studied, identifying stable members suitable for practical problems. The third paper extends the sixth-order family to systems of non-linear equations, creating a highly efficient single-parameter family. Dynamic and numerical analyses confirm the convergence, stability, and applicability of this extended family for large-scale problems. The research aims to overcome the limitations of some existing methods, offering robust and efficient solutions for non-linear equations and systems. The document is structured to cover the development, analysis, and validation of these methods, providing specific recommendations for their practical application in various scientific and engineering domains.
