An optimal scheme for multiple roots of nonlinear equations with eighth-order convergence

In this manuscript, we present a new general family of optimal iterative methods for finding multiple roots of nonlinear equations with known multiplicity using weight functions. An extensive convergence analysis is presented to verify the optimal eighth order convergence of the new family. Some special cases of the family are also presented which require only three functions and one derivative evaluation at each iteration to reach optimal eighth order convergence. A variety of numerical test functions along with some real-world problems such as beam designing model and Van der Waals’ equation of state are presented to ensure that the newly developed family efficiently competes with the other existing methods. The dynamical analysis of the proposed methods is also presented to validate the theoretical results by using graphical tools, termed as the basins of attraction.

Recently, some optimal fourth-order iterative methods for multiple roots of nonlinear equations are presented when the multiplicity m of the root is known. Different from these optimal iterative methods known already, this paper presents a new family of iterative methods using the modified Newton’s method as its first step. The new family, requiring one evaluation of the function and two evaluations of its first derivative, is of optimal order. Numerical examples are given to suggest that the new family can be competitive with other fourth-order methods and the modified Newton’s method for multiple roots.

There are a few optimal eighth order methods in literature for computing multiple zeros of a nonlinear function. Therefore, in this work our main focus is on developing a new family of optimal eighth order iterative methods for multiple zeros. The applicability of proposed methods is demonstrated on some real life and academic problems that illustrate the efficient convergence behavior. It is shown that the newly developed schemes are able to compete with other methods in terms of numerical error, convergence and computational time. Stability is also demonstrated by means of a pictorial tool, namely, basins of attraction that have the fractal-like shapes along the borders through which basins are symmetric.

In this paper we deal with iterative methods for approximating multiple roots of nonlinear equations, it is a special case where some particular aspects must be taken into account. We are interested in local convergence studies, that is in obtaining an open ball B(α, r) where the sequence xn generated by an iterative method, starting from any initial point in it generates a sequence {xn} that remains in the ball and converges to the solution of the problem, α. Then, it is interesting to obtain the largest possible value of r, but obviously, this depends on the conditions that the nonlinear function verifies.We define a new technique to simplify the process of obtaining the functions that bound the error equation just by using a characterization of the function using Taylor’s development around the root under the assumption of a bounding condition for the (m + 1)th derivative of the function f (x). This technique simplifies the usual intricate properties of divided differences that are used in already published papers about local convergence for multiple roots.Moreover, we complete the study in cases where the multiplicity m is unknown, so we estimate this factor by different strategies comparing the behavior of the corresponding estimations and how this fact affect to the original method.

The use of complex dynamics tools in order to deepen the knowledge of qualitative behaviour of iterative methods for solving non-linear equations is a growing area of research in the last few years with fruitful results. Most of the studies dealt with the analysis of iterative schemes for solving non-linear equations with simple roots; however, the case involving multiple roots remains almost unexplored. The main objective of this paper was to discuss the dynamical analysis of the rational map associated with an existing class of iterative procedures for multiple roots. This study was performed for cases of double and triple multiplicities, giving as a conjecture that the wideness of the convergence regions of the multiple roots increases when the multiplicity is higher and also that this family of parametric methods includes some specially fast and stable elements with global convergence.

In this paper, using the variant of Frontini-Sormani method, some higher order methods for finding the roots (simple and multiple) of nonlinear equations are proposed. In particular, we have constructed an optimal fourth order method and a family of sixth order method for finding a simple root. Further, an optimal fourth order method for finding a multiple root of a nonlinear equation is also proposed. We have used different weight functions to a cubically convergent For ntini-Sormani method for the construction of these methods. The proposed methods are tested on numerical examples and compare the results with some existing methods. Further, we have presented the basins of attraction of these methods to understand their dynamics visually.

Historically, work on multiple roots for the internal rate of return (PW = 0) equation focuses on tests for the root's uniqueness and interpreting the meaningfulness of computed roots. Instead, this article starts with problems that consist of P, A, and F cash flows. Then for all possible values of these cash flows, this article develops the boundary conditions for no, single, and multiple real roots of either positive or negative sign. The results of characterizing these roots are applied to two examples as a foundation for conclusions about those multiple root problems and hypotheses about other problems with multiple roots. The first example is an important class of resource development projects that may require significant expenditures for environmental remediation at the conclusion of operation. The second example of buying versus leasing a home is often used to demonstrate the value of economic analysis to students.

In this paper, an iterative method for multiple roots of nonlinear equations is presented. Its convergence order is analyzed and proved. It is shown that the proposed method has third-order convergence. To assess the validity and performance of the proposed method, some nonlinear equations are solved and the results are compared with the results from six other methods.

In numerical computation, locating multiple roots of nonlinear equations (NESs) in a single run is a challenging work. In order to solve the problem of population grouping and parameters settings during the evolutionary, a clustering-based adaptive speciation differential evolution, referred to as CASDE, is presented to deal with NESs. CASDE offers three advantages: 1) the clustering with dynamic clustering sizes is used to set clustering sizes for different problems; 2) adaptive parameter control at the niche level is proposed to enhance the search ability and efficiency; 3) re-initialization mechanism motivates the algorithm to search new roots and saves computing resources. To evaluate the performance of CASDE, we select 30 problems with different features as test suite. Experimental results indicate that the speciation clustering with dynamic clustering sizes, niche adaptive parameter control, and re-initialization mechanism when combined together in a synergistic manner can improve the ability to find multiple roots in a single run. Additionally, our method is also compared with other state-of-the-art methods, which is capable of obtaining better results in terms of peak ratio and success rate. Finally, two practical mechanical problems are used to verify the performance of CASDE, and it also demonstrates superior results.

We develop a family of fourth-order iterative methods using the weighted harmonic mean of two derivative functions to compute approximate multiple roots of nonlinear equations. They are proved to be optimally convergent in the sense of Kung-Traub’s optimal order. Numerical experiments for various test equations confirm well the validity of convergence and asymptotic error constants for the developed methods.

Asymptotic distributions of the latent roots of the covariance matrix with multiple population roots

Spectral break theory, as presented in this article, offers an algebraic-geometric framework unifying the concepts of renormalization and decoherence in the context of complex polynomials with multiplicity at least 2. The study focuses on transforming an unstable complex polynomial P , possessing at least one multiple root, into a simple polynomial Q , devoid of multiple or complex roots, through the composite operator . The renormalization operator R reduces root multiplicities to 1, while the decoherence operator D projects non-real roots onto their real parts. A spectral metric is defined to quantify the distance between the roots of P and Q, proving that ( d(P,Q) > 0 ) for any non-trivial P , establishing qualitative spectral non-proximity. Local continuity of the operators R and D is demonstrated on strata with fixed multiplicities in a non-compact space, leveraging the Weierstrass preparation theorem. Two indicators are introduced: e, measuring the impact of decoherence, and t, combining the spectral distance and e to assess the net effect of the spectral break. Examples illustrate that t reflects the dominance of either R or D . The study concludes with perspectives for extending the theory to non-linear polynomials and enhancing numerical algorithms for practical applications

On an application of symbolic computation and computer graphics to root-finders: The case of multiple roots of unknown multiplicity

Prevalence and morphology of multiple roots, root canals and C-shaped canals in mandibular premolars from cone-beam computed tomography images in a Thai population

RANK ADDITIVITY AND MATRIX POLYNOMIALS