Formation and examination of a few new iterative methods for nonlinear equation numerical solutions
DOI:
https://doi.org/10.29070/pn6z3d65Keywords:
Formation, Iterative Methods, Nonlinear Equation, Numerical SolutionsAbstract
The extensive use of numerical solutions to nonlinear equations in fields as diverse as physics, engineering, economics, and more makes this a vital topic in scientific computing and applied mathematics. Although efficient, classical methods such as Newton-Raphson and Secant methods have drawbacks include being sensitive to initial estimations, diverging in some circumstances, and requiring more computing power to evaluate derivatives. Our goal in this research was to find better iterative solutions to nonlinear equations of the type f(x)=0 that are more resilient and have a faster convergence rate. We came up with a few new approaches and tested them out. By adding weight functions and higher-order correction terms to conventional schemes, the newly created approaches are able to adapt and expand upon them. Proofs of convergence order, error estimates, and stability criteria are all part of the extensive theoretical analysis that is given. These techniques improve computing efficiency by achieving higher-order convergence (up to fourth and sixth order) without increasing the number of function or derivative evaluations each iteration. The performance of the suggested approaches is compared to current techniques through a series of numerical tests performed on benchmark nonlinear equations. These novel iterative approaches show good promise as instruments for addressing complicated nonlinear problems in real-world applications, thanks to their enhanced convergence behaviour.
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