Inverse Family of Numerical Methods for Approximating All Simple and Roots with Multiplicity of Nonlinear Polynomial Equations with Engineering Applications

A new inverse family of the iterative method is interrogated in the present article for simultaneously estimating all distinct and multiple roots of nonlinear polynomial equations. Convergence analysis proves that the order of convergence of the newly constructed family of methods is two. The computer algebra systems CAS-Mathematica is used to determine the lower bound of convergence order, which justifies the local convergence of the newly developed method. Some nonlinear models from physics, chemistry, and engineering sciences are considered to demonstrate the performance and efficiency of the newly constructed family of inverse simultaneous methods in comparison to classical methods in the literature. The computational time in seconds and residual error graph of the inverse simultaneous methods are also presented to elaborate their convergence behavior.

Left behind and united by populism? The multiple roots of populism in feelings of lacking societal recognition

A prominent but underspecified explanation for the rise of populism points to individuals’ feelings of being “left behind” by the development of society. At its core lies the claim that support for populism is driven by the feeling of being denied the societal recognition one deserves. Our contribution builds on the insight that individuals can feel to lack recognition in different ways and for different reasons. We argue that—due to this multifaceted character—the common perception of being neglected societal recognition unites otherwise heterogeneous segments of the population in their support for populism. Relying on data from the GLES Pre-Election Cross-Section 2021, we will investigate the multiple roots of populist attitudes in feelings of lacking societal recognition in two steps. First, we will test the hypothesis that distinct feelings of lacking recognition dominate in different social segments. Second, we will test the hypothesis that each of these distinct feelings of lacking recognition are associated with populist attitudes.

Efficient iterative methods for finding simultaneously all the multiple roots of polynomial equation

Two new iterative methods for the simultaneous determination of all multiple as well as distinct roots of nonlinear polynomial equation are established, using two suitable corrections to achieve a very high computational efficiency as compared to the existing methods in the literature. Convergence analysis shows that the orders of convergence of the newly constructed simultaneous methods are 10 and 12. At the end, numerical test examples are given to check the efficiency and numerical performance of these simultaneous methods.

Memorizing Schröder’s Method as an Efficient Strategy for Estimating Roots of Unknown Multiplicity

In this paper, we propose, to the best of our knowledge, the first iterative scheme with memory for finding roots whose multiplicity is unknown existing in the literature. It improves the efficiency of a similar procedure without memory due to Schröder and can be considered as a seed to generate higher order methods with similar characteristics. Once its order of convergence is studied, its stability is analyzed showing its good properties, and it is compared numerically in terms of their basins of attraction with similar schemes without memory for finding multiple roots.

Radiological aspects of idiopathic lumbosacral plexitis

Idiopathic Lumbosacral Plexitis is a disease that do not have definitive causes but even starts to be studied by Sander Evans et al. in 1981 hypotheses were put forward related to viral infections, vaccines and heterologous serum. Although more recently studies have showed that the disease can develop without immunological history. Multiple roots and nerves are affected and it can be bilateral. The diferential diagnosis of other pathologies must involves research for tumours, traumas and diabetes. The findings of the disease include: autoimmune demyelination and vasculitis with axonal lesion. The clinicalconditionare characterized by intensive pain, paresis, hypoesthesia in lower members, and a limping gait. The prognosis is positive, however do not have a completely remission without squeals. The treatment has not been defined but is common to uses corticosteroids and immunoglobulin.

A Graphic Method for Detecting Multiple Roots Based on Self-Maps of the Hopf Fibration and Nullity Tolerances

The aim of this paper is to study, from a topological and geometrical point of view, the iteration map obtained by the application of iterative methods (Newton or relaxed Newton’s method) to a polynomial equation. In fact, we present a collection of algorithms that avoid the problem of overflows caused by denominators close to zero and the problem of indetermination which appears when simultaneously the numerator and denominator are equal to zero. This is solved by working with homogeneous coordinates and the iteration of self-maps of the Hopf fibration. As an application, our algorithms can be used to check the existence of multiple roots for polynomial equations as well as to give a graphical representation of the union of the basins of attraction of simple roots and the union of the basins of multiple roots. Finally, we would like to highlight that all the algorithms developed in this work have been implemented in Julia, a programming language with increasing use in the mathematical community.

Derivative-Free King’s Scheme for Multiple Zeros of Nonlinear Functions

There is no doubt that the fourth-order King’s family is one of the important ones among its counterparts. However, it has two major problems: the first one is the calculation of the first-order derivative; secondly, it has a linear order of convergence in the case of multiple roots. In order to improve these complications, we suggested a new King’s family of iterative methods. The main features of our scheme are the optimal convergence order, being free from derivatives, and working for multiple roots (m≥2). In addition, we proposed a main theorem that illustrated the fourth order of convergence. It also satisfied the optimal Kung–Traub conjecture of iterative methods without memory. We compared our scheme with the latest iterative methods of the same order of convergence on several real-life problems. In accordance with the computational results, we concluded that our method showed superior behavior compared to the existing methods.

Experimental and Numerical Simulation Study on the Shear Strength Characteristics of Magnolia multiflora Root-Soil Composites

The shear strength of the soil refers to the ultimate strength of the soil against shear failure, which is one of the important indicators used to measure slope stability. This paper presents a simulation of direct shear tests on root-soil composites with different root embedding angles under different stress conditions. By comparing and analyzing the simulation results of ABAQUS software and the laboratory test results, the enhancement effect of plant roots on soil shear strength was explored. Conclusions can be drawn as follows: the excellent agreement between numerical models and laboratory shear tests suggested that the developed model can quickly and conveniently predict the shear strength of the root-soil composites. The shear strength was related to the rooting arrangement. For a single root system, when the inclination angle of the root was about 64° to the shear direction, the shear resistance of soil was much improved, while the root reinforcement had less effect when the inclination angle was greater than 90°. In the case of multiple roots, the hybrid rooting method can more effectively improve the shear resistance of the root-soil composite. Therefore, in the practical application of using the root to strengthen the soil, the angle of a single root and arrangement of multiple roots should be comprehensively considered.