Parametric Iterative Method for Addressing an Embedded-Steel Constitutive Model with Multiple Roots

Abstract

In this paper, an iterative procedure to find the solution of a nonlinear constitutive model for embedded steel reinforcement is introduced. The model presents different multiplicities, where parameters are randomly selected within a solvability region. To achieve this, a class of multipoint fixed-point iterative schemes for single roots is modified to find multiple roots, achieving the fourth order of convergence. Complex discrete dynamics techniques are employed to select the members with the most stable performance. The mechanical problem referred to earlier, as well as some academic problems involving multiple roots, are solved numerically to verify the theoretical analysis, robustness, and applicability of the proposed scheme.

Stability Analysis of a New Fourth-Order Optimal Iterative Scheme for Nonlinear Equations

“…It is widely known that fixed-point iterative methods play a fundamental role in scientific disciplines such as Celestial Mechanics (see for example [1,2]), Electrical Power Systems [3], Chemistry [4], Hydraulic Engineering [5], and Civil Engineering [6]. These algorithms provide approximate solutions when exact solutions are challenging to obtain or when problems are ill-conditioned, while offering computational efficiency in terms of both time and computational resources.…”

“…However, the coefficients of the rational operator are not simple. On the other hand, the family associated with K (−2,−1,0,1,γ) has simple coefficients and a radius of 36, which is greater than the radius of the family studied by Hueso et al (6). Therefore, we focus our analysis on this family.…”