Unified convergence domains of Newton-like methods for solving operator equations

On two high‐order families of frozen Newton‐type methods

“…We present three cases corresponding to three different situations in Table 1. Tables 3-5 show the results obtained for the k-step iterative method (1). In each table, we can read the necessary iteration number I, the computational cost  in terms of products, the CEI, and the computational time of MAPLE execution to get one correct decimal of the approximated solution .…”

“…In this case, 1 = 1∕3 and 0 = 12 1 . Notice that for m = 9 (see Table 5), the minimum value of is obtained by taking k = 3 in the k-step iterative method (1).…”

“…This paper is organized as follows. In Section 2, we study the semilocal convergence for the family (1). We also present the corresponding theorem for the family (2).…”

“…[1][2][3][4][5][6][7][8][9] The Newton method is second-order convergent under some regularity assumptions. The classical third-order methods use second-order Fréchet derivatives.…”

The Convergence Ball and Error Analysis of the Relaxed Secant Method

“…This method and Newton-like methods have been studied well by many authors (see [1][2][3][4][5][6][7][8][9][10][11][12]). Newton's method requires that is differentiable.…”