Numerical solutions for nonlinear equations: Development and analysis of new iterative methods
DOI:
https://doi.org/10.29070/pqfty615Keywords:
Numerical Solutions, Nonlinear Equations, Development, Iterative MethodsAbstract
In scientific and technical computations, the numerical solution of nonlinear equations is essential, as analytical solutions are not always applicable or accessible. New iterative approaches to efficiently and accurately solve nonlinear equations of the type f(x)=0, f(x)=0 are the subject of this work, which also rigorously analyses existing methods. While Secant algorithms and Newton-Raphson techniques are frequently employed, they have some drawbacks when it comes to convergence speed, reliance on initial estimations, and sensitivity to the function's nature. In response to these difficulties, the study presents hybrid and modified iterative systems that, under relaxed settings, show better convergence characteristics, namely stability and speed. Without substantially raising computing cost, the suggested approaches are built utilising multipoint evaluations or higher-order derivatives. In order to determine the convergence order, error boundaries, and stability requirements, a thorough convergence study is conducted. By conducting thorough numerical tests on benchmark nonlinear equations and comparing the outcomes with preexisting classical procedures, we confirm the efficacy and resilience of the novel methods. Suitable for tackling complicated nonlinear problems encountered in real-world applications, the suggested iterative approaches offer a considerable increase in accuracy and efficiency, as demonstrated by the findings.
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References
1. Adomian, G. (2019). Nonlinear stochastic systems and applications to physics. Kluwer Academic Publishers.
2. Babajee, D. K. R. (2015). Several improvements of the 2-point third-order midpoint iterative method using weight function. Applied Mathematics and Computation, 218(15), 7958–7966.
3. Ezquerro, J. A., & Hernández, M. A. (2003). A uniparametric Halley-type iteration with free second derivative. International Journal of Pure and Applied Mathematics, 6(1), 99–110.
4. Ezquerro, J. A., & Hernández, M. A. (2004). On Halley-type iterations with free second derivative. Journal of Computational and Applied Mathematics, 170(2), 455–459.
5. Babajee, D. K. R., Dauhoo, M. Z., Darvishi, M. T., Karami, A., & Barati, A. (2014). Analysis of two Chebyshev-like third-order methods free from second derivatives for solving system of nonlinear equations. Journal of Computational and Applied Mathematics, 233, 2002–2015.
6. Chua, L. O., Desoer, C. A., & Kuh, E. S. (2007). Linear and nonlinear circuits. McGraw Hill.
7. Heydari, M., Horseini, S. M., & Loghmani, G. M. (2014). Convergence of family of third-order methods free from second-order derivatives for finding the multiple roots of nonlinear equations. World Applied Sciences Journal, 11(5), 507-512.
8. Homeier, H. H. H. (2003). A modified Newton method for root finding with cubic convergence. Journal of Computational and Applied Mathematics, 157(1), 227–230.
9. Chun, C. (2005). Iterative methods improving Newton’s method by decomposition. Computers and Mathematics with Applications, 50(10–12), 1559–1568.
10. Frontini, M., & Sormani, E. (2004). Third-order methods from quadrature formulae for solving systems of nonlinear equations. Applied Mathematics and Computation, 149(3), 771–782.
11. Frontini, M., & Sormani, F. (2003). Some variants of Newton’s method with third-order convergence. Journal of Computational and Applied Mathematics, 140, 419–426.
12. Adomian, G., & Rach, R. (2015). On the solution of algebraic equations by decomposition method. Mathematics Analysis Applications, 105, 141–166.
13. Geum, Y. H., & Kim, Y. I. (2015). A uniparametric family of three-step eighth-order multipoint iterative methods for simple roots. Applied Mathematics Letters, 24, 929–935.
14. Ghanbari, B. (2015). A new general fourth-order family of methods for finding simple roots of nonlinear equations. Journal of King Saud University-Science, 23(4), 395–398.
15. Kahya, E. (2007). A class of exponential quadratically convergent iterative formulae for unconstrained optimization. Applied Mathematics and Computation, 186(2), 1010–1017.
