Uniparametric Family Research Articles

Achieving Optimal Order in a Novel Family of Numerical Methods: Insights from Convergence and Dynamical Analysis Results

In this manuscript, we introduce a novel parametric family of multistep iterative methods designed to solve nonlinear equations. This family is derived from a damped Newton’s scheme but includes an additional Newton step with a weight function and a “frozen” derivative, that is, the same derivative than in the previous step. Initially, we develop a quad-parametric class with a first-order convergence rate. Subsequently, by restricting one of its parameters, we accelerate the convergence to achieve a third-order uni-parametric family. We thoroughly investigate the convergence properties of this final class of iterative methods, assess its stability through dynamical tools, and evaluate its performance on a set of test problems. We conclude that there exists one optimal fourth-order member of this class, in the sense of Kung–Traub’s conjecture. Our analysis includes stability surfaces and dynamical planes, revealing the intricate nature of this family. Notably, our exploration of stability surfaces enables the identification of specific family members suitable for scalar functions with a challenging convergence behavior, as they may exhibit periodical orbits and fixed points with attracting behavior in their corresponding dynamical planes. Furthermore, our dynamical study finds members of the family of iterative methods with exceptional stability. This property allows us to converge to the solution of practical problem-solving applications even from initial estimations very far from the solution. We confirm our findings with various numerical tests, demonstrating the efficiency and reliability of the presented family of iterative methods.

A FIXED-POINT TYPE RESULT FOR SOME NON-DIFFERENTIABLE FREDHOLM INTEGRAL EQUATIONS

In this paper, we present a new fixed-point result to draw conclusions about the existence and uniqueness of the solution for a nonlinear Fredholm integral equation of the second kind with non-differentiable Nemytskii operator. To do this, we will transform the problem of locating a fixed point for an integral operator into the problem of locating a solution of an integral equation. Thus, assuming conditions on the Nemytskii operator, we will obtain a global convergence domain for the solution of the considered integral equation, taking for this a uniparametric family of derivativefree iterative processes with quadratic convergence. This result provides us a new fixed-point result for the integral operator considered.

Fractal Complexity of a New Biparametric Family of Fourth Optimal Order Based on the Ermakov–Kalitkin Scheme

In this paper, we generalize the scheme proposed by Ermakov and Kalitkin and present a class of two-parameter fourth-order optimal methods, which we call Ermakov’s Hyperfamily. It is a substantial improvement of the classical Newton’s method because it optimizes one that extends the regions of convergence and is very stable. Another novelty is that it is a class containing as particular cases some classical methods, such as King’s family. From this class, we generate a new uniparametric family, which we call the KLAM, containing the classical Ostrowski and Chun, whose efficiency, stability, and optimality has been proven but also new methods that in many cases outperform these mentioned, as we prove. We demonstrate that it is of a fourth order of convergence, as well as being computationally efficienct. A dynamical study is performed allowing us to choose methods with good stability properties and to avoid chaotic behavior, implicit in the fractal structure defined by the Julia set in the related dynamic planes. Some numerical tests are presented to confirm the theoretical results and to compare the proposed methods with other known methods.

Performance of a New Sixth-Order Class of Iterative Schemes for Solving Non-Linear Systems of Equations

This manuscript is focused on a new parametric class of multi-step iterative procedures to find the solutions of systems of nonlinear equations. Starting from Ostrowski’s scheme, the class is constructed by adding a Newton step with a Jacobian matrix taken from the previous step and employing a divided difference operator, resulting in a triparametric scheme with a convergence order of four. The convergence order of the family can be accelerated to six by setting two parameters, resulting in a uniparametric family. We performed dynamic and numerical development to analyze the stability of the sixth-order family. Previous studies for scalar functions allow us to isolate those elements of the family with stable performance for solving practical problems. In this regard, we present dynamical planes showing the complexity of the family. In addition, the numerical properties of the class are analyzed with several test problems.

A note on the reduction of the slicing problem to centrally symmetric convex bodies

In this paper, we obtain the best possible value of the absolute constant C such that for every isotropic convex body K⊆Rn the following inequality (which was proved by Klartag and reduces the hyperplane conjecture to centrally symmetric convex bodies) is satisfied:LK≤CLKn+2(gK). Here LK denotes the isotropic constant of K, gK its covariogram function, which is log-concave, and, for any log-concave function g, Kn+2(g) is a convex body associated to the log-concave function g, which belongs to a uniparametric family introduced by Ball. In order to obtain this inequality, sharp inclusion results between the convex bodies in this family are obtained whenever g satisfies a better type of concavity than the log-concavity, as gK is, indeed 1n-concave.

A significant improvement of a family of secant-type methods

Secant-type iterative methods are usually used to solve nonlinear systems of equations where the operator involved is nondifferentiable or its derivative is costly. From a uniparametric family of secant-type methods, that is reduced to the secant method and Newton’s method for some particular values of the parameter involved, with operational cost similar to that of the secant and Newton methods and superlinear convergence, we construct a biparametric family of iterative methods with quadratic convergence, study their local convergence, their efficiency and, based on the dynamics of the methods, their accessibility.

Kurchatov-type methods for non-differentiable Hammerstein-type integral equations

We consider a generic type of nonlinear Hammerstein-type integral equations with the particularity of having non-differentiable kernel of Nemystkii type. So, in order to solve it we consider a uniparametric family of iterative processes derivative free, with the main advantage that for a special value of the involved parameter the iterative method obtained coincides with Newton’s method, that is due to the fact of evaluating the divided difference operator when the two values are the same. We perform a qualitative convergence study by choosing an auxiliary point, that allow us to obtain the existence and separation of solutions of the given equation, that is, local and semilocal convergence balls can be obtained.

On the relationship between Lozi maps and max-type difference equations

In the present work we revise a transformation that links generalized Lozi maps with max-type difference equations. In this view, according to the technique of topological conjugation, we relate the dynamics of a concrete Lozi map with a complete uniparametric family of max-equations, and we apply this fact to investigate the dynamics of two particular families. Moreover, we present some numerical simulations related to the topic and, finally, we propose some open problems that look into the relationship established between generalized Lozi maps and max-equations.

Light ring images of double photon spheres in black hole and wormhole spacetimes

The silhouette of a black hole having a critical curve (an unstable bound photon orbit) when illuminated by an optically thin accretion disk whose emission is confined to the equatorial plane shows a distinctive central brightness depression (the shadow) whose outer edge consists of a series of strongly lensed, self-similar rings superimposed with the disk's direct emission. While the size and shape of the critical curve depend only on the background geometry, the pattern of bright and dark regions (including the size and depth of the shadow itself) in the image is strongly influenced by the (astro)physics of the accretion disk. This aspect makes it difficult to extract clean and clear observational discriminators between the Kerr black hole and other compact objects. In the presence of a second critical curve, however, observational differences become apparent. In this work we shall consider some spherically symmetric black hole and wormhole geometries characterized by the presence of a second critical curve, via a uniparametric family of extensions of the Schwarzschild metric. By assuming three toy models of geometrically thin accretion disks, we show the presence of additional light rings in the intermediate region between the two critical curves. The observation of such rings could represent a compelling evidence for the existence of black hole mimickers having multiple critical curves.

Shadows and optical appearance of black bounces illuminated by a thin accretion disk

We study the light rings and shadows of an uniparametric family of spherically symmetric geometries interpolating between the Schwarzschild solution, a regular black hole, and a traversable wormhole, and dubbed as black bounces, all of them sharing the same critical impact parameter. We consider the ray-tracing method in order to study the impact parameter regions corresponding to the direct, lensed, and photon ring emissions, finding a broadening of all these regions for black bounce solutions as compared to the Schwarzschild one. Using this, we determine the optical appearance of black bounces when illuminated by three standard toy models of optically and geometrically thin accretion disks viewed in face-on orientation.

Chaos and Stability in a New Iterative Family for Solving Nonlinear Equations

In this paper, we present a new parametric family of three-step iterative for solving nonlinear equations. First, we design a fourth-order triparametric family that, by holding only one of its parameters, we get to accelerate its convergence and finally obtain a sixth-order uniparametric family. With this last family, we study its convergence, its complex dynamics (stability), and its numerical behavior. The parameter spaces and dynamical planes are presented showing the complexity of the family. From the parameter spaces, we have been able to determine different members of the family that have bad convergence properties, as attracting periodic orbits and attracting strange fixed points appear in their dynamical planes. Moreover, this same study has allowed us to detect family members with especially stable behavior and suitable for solving practical problems. Several numerical tests are performed to illustrate the efficiency and stability of the presented family.

A New Class of Fifth and Sixth Order Root-Finding Methods with Its Dynamics and Applications

In this paper, we deal with the construction, analysis and applications of a modified uniparametric family of methods to solve nonlinear equations in R. We study the convergence of new methods which shows the order of convergence is at least five and for a particular value 3/2 of the parameter γ, the method is sixth order convergent. We discuss several applications such as Max Planck’s conservative law, chemical equilibrium and multi-factor effect to demonstrate the productiveness and capability of the suggested method (for γ = 3/2 ). At every iteration our method is compared with Maroju et al. method[1] and Parhi and Gupta method[2] in terms of the values |f (xn)| and |xn − xn−1|. From the numerical experiments, advantages of our method is observed. Furthermore, we study the complex dynamics to determine the stability and dynamical properties of the methods.

Local Convergence and Complex Dynamics of a Uni-parametric Family of Iterative Schemes

Using the Lipschitz continuous derivative, we study the local convergence analysis of a third and fourth-order convergent uni-parametric class of iterative schemes. This technique avoids the usual practice of Taylor expansion in convergence analysis and extends the applicability of the family by using the assumption based on the first-order derivative only. Our study provides the radii of balls of convergence and computable error bounds along with the uniqueness of the solution. Also, complex dynamical properties of the family are discussed to select good schemes in the view of numerical stability. Numerical illustrations show that our analysis is useful in solving such problems where previous studies fail to solve.

Bifurcations along the Boundary Curves of Red Fixed Components in the Parameter Space for Uniparametric, Jarratt-Type Simple-Root Finders

Bifurcations have been studied with an extensive analysis of boundary curves of red, fixed components in the parametric space for a uniparametric family of simple-root finders under the Möbius conjugacy map applied to a quadratic polynomial. An elementary approach from the perspective of a plane curve theory properly describes the geometric figures resembling a circle or cardioid to characterize the underlying boundary curves that are parametrically expressed. Moreover, exact bifurcation points for satellite components on the boundaries have been found, according to the fact that the tangent line at a bifurcation point simultaneously touches the red fixed component and the satellite component. Computational experiments implemented with examples well reflect the significance of the theoretical backgrounds pursued in this paper.

The dynamical analysis of a uniparametric family of three-point optimal eighth-order multiple-root finders under the Möbius conjugacy map on the Riemann sphere

In this paper, we present dynamical viewpoints under the Mobius conjugacy map on the Riemann sphere for a uniparametric family of optimal eighth-order multiple-root finders. Various conjugacy properties are investigated including the invariance of the fixed point and its multiplier, which enables the family of iterative methods to trace along the consistent orbits in position. The parameter spaces and dynamical planes are studied and illustrated to visualize the periodic components and their geometric properties. Both theoretical and computational analyses along with a numerical algorithm are carried out regarding the bifurcation points of satellite and primitive components.

Can the cosmological dark sector be modeled by a single scalar field?

In a previous paper it was shown that a minimally coupled scalar field of mass $M \sim H_0$ can describe both components of the dark sector in a unified way. In the solution found, the dark energy component decays linearly with the Hubble parameter, with a homogeneous creation of dark matter. In the present note we show that a $\Lambda$CDM dark sector can also be modeled by such a single field. More generally, we show that the system of Klein-Gordon and Einstein equations admits a uniparametric family of solutions that is equivalent to a non-adiabatic (with zero sound speed) generalised Chaplygin gas.

Dynamical Analysis via Möbius Conjugacy Map on a Uniparametric Family of Optimal Fourth-Order Multiple-Zero Solvers with Rational Weight Functions

A triparametric family of fourth-order multiple-zero solvers have been proposed. In this paper, we select among them a uniparametric family of optimal fourth-order multiple-zero solvers with rational weight functions and pursue their dynamics by exploring the relevant parameter spaces and dynamical planes, by means of Möbius conjugacy map applied to a prototype polynomial of the form (z-A)m(z-B)m. The resulting dynamics is best illustrated through various stability surfaces and parameter spaces as well as dynamical planes.

One-dimensional families of Riemann surfaces of genus g with $$4\text {g}+4$$ 4 g + 4 automorphims

We prove that the maximal number $$ag+b$$ of automorphisms of equisymmetric and complex-uniparametric families of Riemann surfaces appearing in all genera is $$4g+4$$ . For each integer $$g\ge 2$$ we find an equisymmetric complex-uniparametric family $${\mathcal {A}}_{g}$$ of Riemann surfaces of genus g having automorphism group of order $$4g+4$$ . For $$g\equiv -1\,{\mathrm {mod}}\,4$$ we present another uniparametric family $${\mathcal {K}}_{g}$$ with automorphism group of order $$4g+4$$ . The family $${\mathcal {A}}_{g}$$ contains the Accola–Maclachlan surface and the family $${\mathcal {K}}_{g}$$ contains the Kulkarni surface.

Existence and multiplicity of entire radial spacelike graphs with prescribed mean curvature function in certain Friedmann–Lemaître–Robertson–Walker spacetimes

We provide sufficient conditions for the existence of a uniparametric family of entire spacelike graphs with prescribed mean curvature in a Friedmann–Lemaître–Robertson–Walker spacetime with flat fiber. The proof is based on the analysis of the associated homogeneous Dirichlet problem on a Euclidean ball together with suitable bounds for the gradient which permit the prolongability of the solution to the whole space.

A family of second derivative free fourth order continuation method for solving nonlinear equations

In this paper, we present a parameter based iterative method free from the second derivative for solving nonlinear equations of type f(x)=0. Ezquerro and Hernández (1999) discussed the convergence analysis of a uniparametric family of an iterative method in Banach space. Based on this idea, we propose a uniparametric family of second derivative free iterative method in R. We discuss the convergence analysis and observe that it possesses a fourth-order convergence for solving nonlinear equations in α∈R. Several numerical examples are worked out with our proposed methods for parameter α=1, α=2 and the existing fourth-order iterative method proposed in King (1973), Chun (2007), Soleymani et al. (2012) and Maheshwari (2009). Finally, from the comparison, we observe that our method is more efficient than existing methods. Finally, we compare basins of attraction of our methods with the second derivative free fourth-order iterative method proposed in King (1973), Chun (2007), Soleymani et al. (2012) and Maheshwari (2009) observe that the proposed scheme is more efficient.