secant method
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This study discusses the analysis of the modification of Newton-Secant method and solving nonlinear equations having a multiplicity of  by using a modified Newton-Secant method. A nonlinear equation that has a multiplicity   is an equation that has more than one root. The first step is to analyze the modification of the Newton-Secant method, namely to construct a mathematical model of the Newton-Secant method using the concept of the Newton method and the concept of the Secant method. The second step is to construct a modified mathematical model of the Newton-Secant method by adding the parameter . After obtaining the modified formula for the Newton-Secant method, then applying the method to solve a nonlinear equations that have a multiplicity . In this case, it is applied to the nonlinear equation which has a multiplicity of . The solution is done by selecting two different initial values, namely  and . Furthermore, to determine the effectivity of this method, the researcher compared the result with the Newton-Raphson method, the Secant method, and the Newton-Secant method that has not been modified. The obtained results from the analysis of modification of Newton-Secant method is an iteration formula of the modified Newton-Secant method. And for the result of  using a modified Newton-Secant method with two different initial values, the root of  is obtained approximately, namely  with less than iterations. whereas when using the Newton-Raphson method, the Secant method, and the Newton-Secant method, the root  is also approximated, namely  with more than  iterations. Based on the problem to find the root of the nonlinear equation  it can be concluded that the modified Newton-Secant method is more effective than the Newton-Raphson method, the Secant method, and the Newton-Secant method that has not been modified

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AbstractWe study the discrete dynamical system defined on a subset of $$R^2$$ R 2 given by the iterates of the secant method applied to a real polynomial p. Each simple real root $$\alpha $$ α of p has associated its basin of attraction $${\mathcal {A}}(\alpha )$$ A ( α ) formed by the set of points converging towards the fixed point $$(\alpha ,\alpha )$$ ( α , α ) of S. We denote by $${\mathcal {A}}^*(\alpha )$$ A ∗ ( α ) its immediate basin of attraction, that is, the connected component of $${\mathcal {A}}(\alpha )$$ A ( α ) which contains $$(\alpha ,\alpha )$$ ( α , α ) . We focus on some topological properties of $${\mathcal {A}}^*(\alpha )$$ A ∗ ( α ) , when $$\alpha $$ α is an internal real root of p. More precisely, we show the existence of a 4-cycle in $$\partial {\mathcal {A}}^*(\alpha )$$ ∂ A ∗ ( α ) and we give conditions on p to guarantee the simple connectivity of $${\mathcal {A}}^*(\alpha )$$ A ∗ ( α ) .

The secant method is a very effective numerical procedure used for solving nonlinear equations of the form f(x)=0. In a recent work (A. Sidi, Generalization of the secant method for nonlinear equations. Appl. Math. E-Notes, 8:115–123, 2008), we presented a generalization of the secant method that uses only one evaluation of f(x) per iteration, and we provided a local convergence theory for it that concerns real roots. For each integer k, this method generates a sequence {xn} of approximations to a real root of f(x), where, for n≥k, xn+1=xn−f(xn)/pn,k′(xn), pn,k(x) being the polynomial of degree k that interpolates f(x) at xn,xn−1,…,xn−k, the order sk of this method satisfying 1<sk<2. Clearly, when k=1, this method reduces to the secant method with s1=(1+5)/2. In addition, s1<s2<s3<⋯, such that limk→∞sk=2. In this note, we study the application of this method to simple complex roots of a function f(z). We show that the local convergence theory developed for real roots can be extended almost as is to complex roots, provided suitable assumptions and justifications are made. We illustrate the theory with two numerical examples.

The main purpose of this paper is to obtain the real roots of an expression using the Numerical method, bisection method, Newton's method and secant method. Root analysis is calculated using specific, precise starting points and numerical methods and is represented by Maple. In this research, we used Maple software to analyze the roots of nonlinear equations by special methods, and by showing geometric diagrams, we examined the relevant examples. In this process, the Newton-Raphson method, the algorithm for root access, is fully illustrated by Maple. Also, the secant method and the bisection method were demonstrated by Maple by solving examples and drawing graphs related to each method.

This study is aimed to pinpoint the cases or situations that makes the performance of secant method superior than the performance of New- ton’s method in solving certain selected nonlinear equations. Despite the convergence order of Newton’s method that is higher, that does not necessarily mean it performs faster than secant method for all nonlinear problems. In addition, both methods are almost identical in their approach and secant method has advantages that can make its performances superior or equivalent to Newton’s method for some nonlinear equations. Four numerical experiments are given to clarify the situations in which the performance of secant method are proven to be superior over Newton’s method.

Cable tension monitoring is vital for the health monitoring of cable-stayed bridges. During the service of bridges, cable tension fluctuates rather than monotonously changes. However, existing research works pay little attention to the influence of tension history. In this article, the influence of the tension history on the monitoring of cable tension was studied. To guide the experiment, the magnetization theory of ferromagnetic materials and the electromagnetic induction principle were combined to analyze the theory of the magnetoelastic inductance method. The magnetoelastic inductance method characterized cable tension by sensor inductance. Based on the theoretical analysis, tension monitoring experiments were carried out to figure out the influences of design tension and tension variation. Experimental results showed the design tension and the tension variation influenced the relationship between the inductance and the tension. To monitor the fluctuating tension, a secant method was proposed. When the tension changed less than 30% of the design tension, the tension can be ascertained by the secant method. The experimental results demonstrated that the influence of the tension history should be considered when the design tension was different or the tension variation was large. Besides, the influence of the tension history analyzed in this article is suitable for other tension monitoring methods based on the magnetic properties of ferromagnetic materials.