Iterative Methods and Their Dynamics with Applications

Expanding the Applicability of Four Iterative Methods for Solving Least Squares Problems

“…The advantages (A) are obtained under the same computational cost as in earlier studies [2,3,10,11], since the computation of the Lipschitz constants in the old studies require the computation of the new Lipschitz constants as special cases. Hence, the applicability of these four methods to solve (1.1) is expanded.…”

“…Suppose x * satisfies (1.1), F has a continuous derivative in U (x * , ρ), F (x * ) has full rank and F satisfies the center Lipschitz condition on Ω with constant L 0 . Let ρ > 0 satisfy (see [2][3][4] for example), then we obtain a larger radius of convergence, tighter error bounds on the distances x n − x * and at least as precise information on the location of the solution x * , since…”

“…Special choices of Q n are considered in this study: Secant method [2,3,10]: Q n = [x n , x n−1 ; F ], where [·, ·; F ] : Ω 2 → L(R i , R j ) is a divided difference of order one. The convergence order of the method is 1+ √ 5 2 = 1.618 .…”

“…The GaussNewton method has convergence order two and has been studied in [2][3][4][5][6][7][8][9][10][11][12][13] under various Lipschitz-type conditions. In the present paper, we study the local convergence of these methods under more precise Lipschitz-type conditions to obtain the following advantages (say A):…”

Development of Optimal Eighth Order Derivative-Free Methods for Multiple Roots of Nonlinear Equations

“…Approximating a root (say, α) of a function is a very challenging task. It is also very important in many diverse areas such as Mathematical Biology, Physics, Chemistry, Economics and also Engineering to mention a few [1][2][3][4]. This is the case since problems from these areas are reduced to finding α.…”