Publication

Abstract : The Potra-Pták-Chebyshev technique family, which focuses on an approximate unique solution of a nonlinear system of equations, is investigated in this chapter for its local convergence. The hypothesis of this investigation shows convergence only for the first derivative. Our methodology avoids the usual Taylor expansions that require higher-order derivatives by merely using generalized Lipschitz-type requirements on the derivative initially. Additionally, our innovative approach provides error bounds on the involved distances, a calculable radius of convergence, and estimates on the uniqueness of the solution based on certain functions that emerge in these generalized conditions. Such estimates are not produced by methods employing Taylor series with higher derivatives, which may not exist, be exceedingly expensive to compute, or be impossible to compute. The convergence order is derived using computational orders of convergence, which don’t need higher derivatives. This methodology can be used for any iterative method that makes use of high-order derivatives and Taylor expansions. The study of local convergence based on Lipschitz constants is important since it shows the degree of difficulty in choosing initial guesses.