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In this article, we construct an efficient family of simultaneous methods for finding all the distinct as well as multiple roots of polynomial equations. Convergence analysis proves that the order of convergence of newly constructed family of simultaneous methods is seventeen. Fractal-based simultaneous iterative algorithms are thoroughly examined. Using self-similar features, fractal-based simultaneous schemes can converge to solutions faster, saving computational time and resources necessary for solving nonlinear equations. Fractals analysis illustrates the newly developed method's global convergence behavior when compared to single root-finding procedures for solving fractional order polynomials that arise in complex engineering applications. Some real problems from various branches of engineering along with some higher degree polynomials are considered as test examples to show the global convergence property of simultaneous methods, performance and efficiency of the proposed family of methods. Further computational efficiencies, CPU time and residual graphs are also drawn to validate the efficiency, robustness of the newly introduced family of methods as compared to the existing methods in the literature. [ABSTRACT FROM AUTHOR]

In this paper, a Newton-type iterative scheme for solving nonlinear systems is designed. In the process of proving the convergence order, we use the higher derivatives of the function and show that the convergence order of this iterative method is six. In order to avoid the influence of the existence of higher derivatives on the proof of convergence, we mainly discuss the convergence of this iterative method under weak conditions. In Banach space, the local convergence of the iterative scheme is established by using the ω -continuity condition of the first-order Fréchet derivative, and the application range of the iterative method is extended. In addition, we also give the radius of a convergence sphere and the uniqueness of its solution. Finally, the superiority of the new iterative method is illustrated by drawing attractive basins and comparing them with the average iterative times of other same-order iterative methods. Additionally, we utilize this iterative method to solve both nonlinear systems and nonlinear matrix sign functions. The applicability of this study is demonstrated by solving practical chemical problems. [ABSTRACT FROM AUTHOR]

In this paper, we present an innovative technique that improves the convergence order of iterative schemes that do not require the evaluation of Jacobian matrices. As far as we know, this is the first technique that allows us the achievement of an increase, from p to p + 3 units, in the order of convergence. This is constructed from any Jacobian-free scheme of order p. We conduct comprehensive numerical tests first in academical examples to validate the theoretical results, showing the efficiency and effectiveness of the new Jacobian-free schemes. Then, we apply them on the non-differentiable partial differential equations that models the nutrient diffusion in a biological substrate. [ABSTRACT FROM AUTHOR]

We introduce a twice differentiable augmented Lagrangian for nonlinear optimization with general inequality constraints and show that a strict local minimizer of the original problem is an approximate strict local solution of the augmented Lagrangian. A novel augmented Lagrangian method of multipliers (ALM) is then presented. Our method is originated from a generalization of the Hestenes-Powell augmented Lagrangian, and is a combination of the augmented Lagrangian and the interior-point technique. It shares a similar algorithmic framework with existing ALMs for optimization with inequality constraints, but it can use the second derivatives and does not depend on projections on the set of inequality constraints. In each iteration, our method solves a twice continuously differentiable unconstrained optimization subproblem on primal variables. The dual iterates, penalty and smoothing parameters are updated adaptively. The global and local convergence are analyzed. Without assuming any constraint qualification, it is proved that the proposed method has strong global convergence. The method may converge to either a Karush-Kuhn-Tucker (KKT) point or a singular stationary point when the converging point is a minimizer. It may also converge to an infeasible stationary point of nonlinear program when the problem is infeasible. Furthermore, our method is capable of rapidly detecting the possible infeasibility of the solved problem. Under suitable conditions, it is locally linearly convergent to the KKT point, which is consistent with ALMs for optimization with equality constraints. The preliminary numerical experiments on some small benchmark test problems demonstrate our theoretical results. [ABSTRACT FROM AUTHOR]

A novel local and semi-local convergence theorem for the four-step nonlinear scheme is presented. Earlier studies on local convergence were conducted without particular assumption on Lipschitz constant. In first part, the main local convergence theorems with a weak ϰ -average (assuming it as a positively integrable function and dropping the essential property of ND) are obtained. In comparison to previous research, in another part, we employ majorizing sequences that are more accurate in their precision along with the certain form of ϰ average Lipschitz criteria. A finer local and semi-local convergence criteria, boosting its utility, by relaxing the assumptions is derived. Applications in engineering to a variety of specific cases, such as object motion governed by a system of differential equations, are illustrated. [ABSTRACT FROM AUTHOR]

This paper tests for convergence in labour productivity at the local level in 10 sub‐Saharan countries, disaggregated into 1136 administrative entities. This work combines nighttime lights data and a unique set of population censuses to produce local measures of growth, employment and sectoral shares. We find evidence of unconditional convergence across sectors in the range of 2%. However, convergence is heterogeneous and conditional on both manufacturing and services employment shares. Convergence is also associated with proximity to the main city, moderate population density, low land suitability and relatively moderate temperature. Finally, the within effect dominates the between effect. [ABSTRACT FROM AUTHOR]

In this paper, we study the nonlinear matrix equation (NME) X + ∑ i = 1 m A i * X − 1 A i = Q . We transform this equation into an equivalent zero-point equation, then we use Newton's iteration method to solve the equivalent equation. Under some mild conditions, we obtain the domain of approximation solutions and prove that the sequence of approximation solutions generated by Newton's iteration method converges to the unique solution of this equation. In addition, the error estimation of the approximation solution is given. Finally, the comparison of two well-known approaches with Newton's iteration method by some numerical examples demonstrates the superiority of Newton's iteration method in the convergence speed. [ABSTRACT FROM AUTHOR]

The secant method is the most used iterative method to solve an operator equation where the operator involved is nondifferentiable. A known problem that arises when applying this method is its accessibility. Then, we try to improve it by using the technique of decomposition of the operator involved, so that the operator is in turn the sum of two operators, one differentiable and the other continuous but not differentiable. For this, we use a family of Newton‐secant‐type iterative methods that arises from Newton's method and from a well‐known family of secant‐type iterative methods. We study the accessibility of the new methods in two different ways: From the convergence balls of the methods, obtained from a local study of the convergence, and from a dynamic study of the methods. Some examples related to chemistry are also presented to prove the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we provide a detailed local convergence analysis of a one-parameter family of iteration methods for the simultaneous approximation of polynomial zeros due to Ivanov (Numer. Algor. 75(4): 1193–1204, 2017). Thus, we obtain two local convergence theorems that provide sufficient conditions to guarantee the Q-cubic convergence of all members of the family. Among the other contributions, our results unify the latest such kind of results of the well known Dochev–Byrnev and Ehrlich methods. Several practical applications are further given to emphasize the advantages of the studied family of methods and to show the applicability of the theoretical results. [ABSTRACT FROM AUTHOR]

The present work describes the asymptotic local shape of a graph drawn uniformly at random from all connected simple planar graphs with n labelled vertices. We establish a novel uniform infinite planar graph (UIPG) as quenched limit in the local topology as n → ∞. We also establish such limits for random 2-connected planar graphs and maps as their number of edges tends to infinity. Our approach encompasses a new probabilistic view on the Tutte decomposition. This allows us to follow the path along the decomposition of connectivity from planar maps to planar graphs in a uniform way, basing each step on condensation phenomena for random walks under subexponentiality and Gibbs partitions. Using large deviation results, we recover the asymptotic formula by Giménez and Noy (2009) for the number of planar graphs. [ABSTRACT FROM AUTHOR]

The local convergence analysis of the multi-step seventh order method to solve nonlinear equations is presented in this paper. The point of this paper is that our proposed study requires a weak hypothesis where the Fréchet derivative of the nonlinear operator satisfies the ψ -continuity condition, which thereby extends the applicability of the method when both Lipschitz and Hölder conditions fail. The convergence in this study is considered under the hypotheses on the first-order derivative without involving derivatives of the higher-order. To find a subset of the original convergence domain, a strategy is devised here. As a result, the new Lipschitz constants are at least as tight as the old ones, allowing for a more precise convergence analysis in the local convergence case. Some concrete numerical examples showing the performance of the method over some existing schemes are presented in this article. [ABSTRACT FROM AUTHOR]

In this paper, we study the simplest quadratic matrix equation: Q(X)=X2+BX+C=0$$ \mathcal{Q}(X)={X}^2+ BX+C=0 $$. We transform this equation into an equivalent fixed‐point equation, and based on it, we construct the Krasnoselskij method. From this transformation, we can obtain iterative schemes more accurate than the successive approximation method. Moreover, under suitable conditions, we establish different results for the existence and localization of a solution for this equation with the Krasnoselskij method. Finally, we see numerically that the predictor–corrector iterative scheme, with the Krasnoselskij method as a predictor and the Newton method as corrector method, can improve the numerical application of the Newton method when approximating a solution of the quadratic matrix equation. [ABSTRACT FROM AUTHOR]

The local convergence analysis of multi-step, high-order Jarratt-like schemes is extended for solving Banach space valued systems of equations using the derivative instead of up to the ninth derivative as in previous works. Our idea expands the usage of the scheme in cases not considered earlier and can also be utilized in other schemes, too. Experiments test the theoretical results. [ABSTRACT FROM AUTHOR]

We study the local convergence of classical quasi-Newton methods for nonlinear optimization. Although it was well established a long time ago that asymptotically these methods converge superlinearly, the corresponding rates of convergence still remain unknown. In this paper, we address this problem. We obtain first explicit non-asymptotic rates of superlinear convergence for the standard quasi-Newton methods, which are based on the updating formulas from the convex Broyden class. In particular, for the well-known DFP and BFGS methods, we obtain the rates of the form (n L 2 μ 2 k) k / 2 and (nL μ k) k / 2 respectively, where k is the iteration counter, n is the dimension of the problem, μ is the strong convexity parameter, and L is the Lipschitz constant of the gradient. [ABSTRACT FROM AUTHOR]

In this paper, a new version of a secant-type method for solving constrained mixed generalized equations is addressed. The method is a combination of the secant method applied to generalized equations with the conditional gradient method. We use the contraction mapping principle to establish the convergence results. Moreover, by assuming the Lipschitz condition on the gradient and the metric regularity property, we show that the sequence generated by the proposed algorithm is well-defined and locally convergent for a solution with linear or superlinear rate. [ABSTRACT FROM AUTHOR]

In this paper, we study local convergence of high-order Tensor Methods for solving convex optimization problems with composite objective. We justify local superlinear convergence under the assumption of uniform convexity of the smooth component, having Lipschitz-continuous high-order derivative. The convergence both in function value and in the norm of minimal subgradient is established. Global complexity bounds for the Composite Tensor Method in convex and uniformly convex cases are also discussed. Lastly, we show how local convergence of the methods can be globalized using the inexact proximal iterations. [ABSTRACT FROM AUTHOR]

In this work, a time-fractional biharmonic equation with a Caputo derivative of fractional order α ∈ (0 , 1) is considered, whose solutions exhibit a weak singularity at initial time t = 0. For this problem, a system of two second-order differential equations is derived by introducing a intermediate variable p = − Δ u , then discretised the system using the standard finite element method in space together with the L1 discretisation of Caputo derivative on graded mesh in time. The H 1 -norm stability result of the method is established, and then a sharp H 1 -norm local convergent result is presented. Finally, numerical experiments are provided to further verify our theoretical analysis for each fixed value of α. [ABSTRACT FROM AUTHOR]

Optimization is important in machine learning problems, and quasi-Newton methods have a reputation as the most efficient numerical methods for smooth unconstrained optimization. In this paper, we study the explicit superlinear convergence rates of quasi-Newton methods and address two open problems mentioned by Rodomanov and Nesterov (2021b). First, we extend Rodomanov and Nesterov (2021b)'s results to random quasi-Newton methods, which include common DFP, BFGS, SR1 methods. Such random methods employ a random direction for updating the approximate Hessian matrix in each iteration. Second, we focus on the specific quasi-Newton methods: SR1 and BFGS methods. We provide improved versions of greedy and random methods with provable better explicit (local) superlinear convergence rates. Our analysis is closely related to the approximation of a given Hessian matrix, unconstrained quadratic objective, as well as the general strongly convex, smooth, and strongly self-concordant functions. [ABSTRACT FROM AUTHOR]

• A novel asymptotic optimization strategy is proposed to adjust the operational conditions iteratively based on the Bayesian optimization method. • The specified quality target is incorporated into the GP framework and EI acquisition function to improve the convergence property. • The improved EI is further combined with BOBYQA to accelerate the convergence, and an efficient warm start rule is developed when switching from BO to BOBYQA. • The comparison results in case studies show that the proposed quality control method can efficiently achieve the specified quality target. How to take as few experiments as possible to achieve the desired quality target is a challenging and essential topic of process quality control, especially for complex and expensive manufacturing processes with high-dimensional quality outputs. This paper proposes a Bayesian optimization (BO)-based strategy utilizing quality information with adaptive local convergence. First, a series of pretreatment setting rules of the Gaussian process model is proposed to reduce the uncertainty of BO. Then, the quality target is adopted into the Gaussian process model, and the acquisition function is improved. Finally, to balance global exploration and local exploitation and reduce the experimental cost, an adaptive criterion is designed to switch global BO search to local search. The operating conditions can be efficiently updated until the quality target is reached. Two applications with high-dimensional quality outputs are presented to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Firstly, we define an (r , r , ⋯ , r) -slice tensor, and then solve an (r , r , ⋯ , r) -slice tensor equation by direct method. Secondly, we propose a new splitting of the coefficient tensor to develop a slice tensor splitting method, which is different from the tensor splitting methods in [1,2]. At each iteration of this method, an (r , r , ⋯ , r) -slice tensor equation is solved. Finally, we prove the local convergence of the proposed method under appropriate conditions, and give numerical experiments to illustrate the effectiveness of this method. [ABSTRACT FROM AUTHOR]

Levenberg–Marquardt methods (LMMs) are the most typical algorithms for solving nonlinear equations F (x) = 0 , where F : R n → R m is a continuously differentiable function. They sequentially solve subproblems represented as squared residual of the Newton equations with the L 2 regularization to determine the search direction. However, since the subproblems of the LMMs are usually reduced to linear equations with n variables, it takes much time to solve them when m ≪ n. In this paper, we propose a new LMM which generalizes the L 2 regularization of the subproblems of the ordinary LMMs. By virtue of the generalization, we can choose a suitable regularization term for each given problem. Moreover, we show that a sequence generated by the proposed method converges globally and quadratically under some reasonable assumptions. Finally, we conduct numerical experiments to confirm that the proposed method performs better than the existing LMMs for some problems that satisfy m ≪ n. [ABSTRACT FROM AUTHOR]

Attitude motion parameters estimation of the space non-cooperative tumbling target is the premise of the on-orbit target capture task. However, existing methods cannot balance the estimation accuracy and efficiency. In this paper, an adaptive hybrid Kalman filter is presented to estimate the attitude, angular velocity and inertia motion parameters of the space non-cooperative tumbling target. First, the dynamic model of the non-cooperative tumbling target is derived, and the nonlinear discrete state-space equation is obtained. Then, a hybrid Kalman filter is designed by combining the extended Kalman filter and the unscented Kalman filter. The adaptive technique is introduced to track time-varying uncertain measurement noise caused by complex space disturbances. After that, based on the first-order linearization technique, the sufficient condition for the local convergence of the proposed method is proved. Finally, the numerical simulation experiment was conducted to compare the proposed method with the adaptive extended Kalman filter and the adaptive unscented Kalman filter. The experimental results show that the estimation accuracy of the proposed method is much higher than that of the adaptive extended Kalman filter and slightly lower than that of the adaptive unscented Kalman filter, and the iteration-average Central-Processing-Unit time is 64.2% smaller than that of the adaptive unscented Kalman filter and slightly longer than that of the adaptive extended Kalman filter. The proposed method achieves a much higher accuracy with a slight loss of iteration-average Central-Processing-Unit time. [ABSTRACT FROM AUTHOR]

We present two theoretical results and two surprising conjectures concerning convergence properties of Broyden's method for smooth nonlinear systems of equations. First, we show that when Broyden's method is applied to a nonlinear mapping F : R n → R n with n - d affine component functions and the initial matrix B 0 is chosen suitably, then the generated sequence (u k , F (u k) , B k) k ≥ 1 can be identified with a lower-dimensional sequence that is also generated by Broyden's method. This property enables us to prove, second, that for such mixed linear–nonlinear systems of equations a proper choice of B 0 ensures 2d-step q-quadratic convergence, which improves upon the previously known 2n steps. Numerical experiments of high precision confirm the faster convergence and show that it is not available if B 0 deviates from the correct choice. In addition, the experiments suggest two surprising possibilities: It seems that Broyden's method is (2 d - 1) -step q-quadratically convergent for d > 1 and that it admits a q-order of convergence of 2 1 / (2 d) . These conjectures are new even for d = n . [ABSTRACT FROM AUTHOR]

Nonconvex distributed constrained optimization over multi-agent systems plays an increasingly important role in control community. However, the duality gap for a nonconvex distributed constrained optimization problem is not always zero. This brings some difficulties for the algorithm design and convergence analysis about such problems. In this paper, to eliminate the duality gap, the original distributed optimization problem is converted to an equivalent distributed one by using a partial power reformulation. Next, for solving this equivalence problem, a continuous-time algorithm is proposed over the multi-agent system where each agent exchanges local information with other agents under an undirected communication graph. It is proved that the proposed continuous-time algorithm locally converges to a strict local optimal solution of the considered original nonconvex distributed optimization problem. Numerical examples show the validness of the presented algorithm. Finally, the algorithm is applied for optimal placement. [ABSTRACT FROM AUTHOR]

A numerical algorithm for the solution of an inverse coefficient problem for nonstationary, nonlinear production-destruction type model is proposed and tested on an example of the Lorenz'63 system. With an ensemble of adjoint problem solutions, the inverse problem is transformed into a quasi-linear matrix problem and solved with Newton-type algorithm. Two different ways of the adjoint ensemble construction are compared. In the first one, a trigonometric basis is used. In the second one in situ measurements are taken into account. Local convergence properties of the algorithm are studied numerically to find out when the use of more data can lead to the degradation of the reconstruction results. [ABSTRACT FROM AUTHOR]

We study the local convergence analysis of two higher-order methods using Hölder continuity condition on the first Frechet derivative to solve nonlinear equations in Banach spaces. Hölder continuous first derivative is used to extend the applicability of the method on such problems for which Lipschitz condition fails. Also, this convergence analysis generalizes the local convergence analysis based on Lipschitz continuity condition. Our analysis provides the radius of convergence ball and error bounds along with the uniqueness of the solution. Numerical examples like Hammerstein integral equation and a system of nonlinear equations are solved to verify our theoretical results. [ABSTRACT FROM AUTHOR]

In this article, we suggest the local analysis of a uni-parametric third and fourth order class of iterative algorithms for addressing nonlinear equations in Banach spaces. The proposed local convergence is established using an w-continuity condition on the first Fréchet derivative. In this way, the utility of the discussed schemes is extended and the application of Taylor expansion in convergence analysis is removed. Furthermore, this study provides radii of convergence balls and the uniqueness of the solution along with the calculable error distances. The dynamical analysis of the discussed family is also presented. Finally, we provide numerical explanations that show the suggested analysis performs well in the situation where the earlier approach cannot be implemented. [ABSTRACT FROM AUTHOR]

In this paper, we present a sensitivity result for quadratic second-order cone programming under the weak form of second-order sufficient condition. Based on this result, we analyze the local convergence of an SQP-type method for nonlinear second-order cone programming. The subproblems of this method at each iteration are quadratic second-order cone programming problems. Compared with the local convergence analysis done before, we do not need the assumption that the Hessian matrix of the Lagrangian function is positive definite. Besides, the iteration sequence which is proved to be superlinearly convergent does not contain the Lagrangian multiplier. [ABSTRACT FROM AUTHOR]

In this paper we present the convergence factor of the successive quadratic approximation method for solving the nonlinear eigenvalue problems explicitly. To reduce the computational cost, we propose the inexact versions of the successive approximation methods and analyze the corresponding convergence properties. Numerical results demonstrate the effectiveness of the modified successive approximation methods. [ABSTRACT FROM AUTHOR]

Given a finite sequence of vectors in a finite dimensional Hilbert space, we present a method to obtain a tight frame by appending some vectors with prescribed norms. The approach, based on the alternating projection technique, is an iterative process. Von Neumann trace inequality and the polar decomposition of the matrix are employed to derive the method. The approach can be easily implemented and is numerically stable. The local convergence of the approach is achieved. Numerical examples are given to show the performance of the method. [ABSTRACT FROM AUTHOR]

There is a plethora of k‐step solvers for equations involving operators on Banach spaces. Their convergence is estimated by adopting hypotheses on high‐order derivatives which are not even in these iterative solvers. In addition, no computable error bounds or information on the uniqueness of the solution based on Lipschitz‐type functions are given. Moreover, the choice of the initial guess is like shooting in the dark. Finally, the criteria of convergence differ from solver to solver, so no comparison can be made between their convergence domains. The novelty of our work is that we address these problems by introducing a generalized k‐step solver containing all previous k‐step solvers. Moreover, we address all previously stated problems in the earlier papers and utilizing weaker conditions that require only the continuity of the involved operator. Applications are also presented where we test the convergence criteria. [ABSTRACT FROM AUTHOR]

In this paper, an efficient sixth order multi step method is proposed in Banach spaces and its local convergence is studied in order to approximate a locally unique solution of a nonlinear or a system of nonlinear equations. Based on Lipschitz's continuity condition, new convergence analysis provides radius of convergence, bounds on error and estimates on the uniqueness of solution under assumptions on the first derivative only. Hence, the method can be applied to a wider class of functions that expands its applicability. Numerical examples are presented to illustrate the theoretical results. In the end, the basins of attraction are given to show the convergence behavior in complex plane. [ABSTRACT FROM AUTHOR]

In the paper, we consider the local‐in‐time and the global‐in‐time convergence of infinity‐ion‐mass limit for bipolar Euler–Maxwell systems by setting the mass of an electron me=1 and letting the mass of an ion mi → +∞. We use the method of asymptotic expansions to handle the local‐in‐time convergence problem and find that the limiting process from bipolar models to unipolar models is actually decoupling but not the vanishing of equations for the corresponding the other particle. Moreover, when the initial data are sufficiently close to the constant equilibrium state, we also establish the corresponding global‐in‐time convergence. [ABSTRACT FROM AUTHOR]

We present a new theoretical analysis of local superlinear convergence of classical quasi-Newton methods from the convex Broyden class. As a result, we obtain a significant improvement in the currently known estimates of the convergence rates for these methods. In particular, we show that the corresponding rate of the Broyden–Fletcher–Goldfarb–Shanno method depends only on the product of the dimensionality of the problem and the logarithm of its condition number. [ABSTRACT FROM AUTHOR]

Quantum annealing algorithm is a classical natural computing method for skeuomorphs, and its algorithm design and application research have achieved fruitful results, so it is widely integrated into the research of modern intelligent optimization algorithm. This paper attempts to use the spiking neural network (SNN) dynamic system model to simulate the operation mechanism and convergence of the quantum annealing algorithm, and compares the process of searching the optimal solution to the elastic motion in the quantum tunneling field, and the change of function value during the operation of the algorithm is the simple harmonic vibration or damped vibration of quantum. Spiking neural network dynamic system model simulates the human brain by incorporating synaptic state and time components into their operational models, which represents the process of quantum fluctuations. The local convergence in the early stage and the global convergence in the late stage of the algorithm are proved by using the qualitative theory of ordinary differential equations to solve and analyze the dynamic system model, and a reasonable theoretical explanation is given for its operation mechanism. Several typical test problems are selected for experimental verification. The experimental results show that the numerical convergence curve is consistent with the convergence conclusion of theoretical analysis. Finally, both theoretical and experimental analyses show that the SNN dynamic system model established in this paper is suitable to describe the quantum annealing algorithm for optimization. [ABSTRACT FROM AUTHOR]

This paper is concerned with the periodic problem to the two-fluid non-isentropic Euler–Maxwell (N-E-M) equations. The equations arises in the modeling of magnetic plasma, in which appear two physical parameters, the mass of an electron m e and the mass of an ion m i . With the help of methods of asymptotic expansions, we prove the local-in-time convergence of smooth solutions to this problem by setting m e = 1 and letting m i → + ∞ . Moreover, when the initial data are near constant equilibrium states, by means of uniform energy estimates and compactness arguments, we rigorously prove the infinity-ion-mass convergence of the system for all time. The limit system is the one-fluid N-E-M system. [ABSTRACT FROM AUTHOR]

In this paper, a new Gauss-Newton-like method that is based on a rational approximation model with linear numerator is proposed for solving nonlinear equations. The new method revises the JkT Jk matrix by a rank-one matrix at each iteration. Furthermore, we design a new iterative algorithm for nonlinear equations and prove that it is locally q-quadratically convergent. The numerical results show that the new proposed method has better performance than the classical Gauss-Newton method. [ABSTRACT FROM AUTHOR]

This paper is devoted to convergence study and analysis of a fifthorder iterative method to solve nonlinear equations in Banach spaces. The idea of the Lipschitz condition on the second Fréchet derivative has been used to obtain semilocal convergence balls, R-order of convergence and error bounds by following the main theorem. The local convergence follows under weak-Lipschitz-type conditions. Theoretical results are verified through numerical examples, including integral equation and boundary value problem. It is observed that better results have been obtained in terms of accuracy and number of iterations in comparison with the well-known existing algorithms using similar information. The basins of attraction of the presented method show good performance as compared to already established methods which enhance the applicability of our approach. [ABSTRACT FROM AUTHOR]

We study the convergence issue for the gradient algorithm (employing general step sizes) for optimization problems on general Riemannian manifolds (without curvature constraints). Under the assumption of the local convexity/quasi-convexity (resp., weak sharp minima), local/global convergence (resp., linear convergence) results are established. As an application, the linear convergence properties of the gradient algorithm employing the constant step sizes and the Armijo step sizes for finding the Riemannian Lp (p ∈ [1, + ∞)) centers of mass are explored, respectively, which in particular extend and/or improve the corresponding results in [B. Afsari, R. Tron, and R. Vidal, SIAM J. Control Optim., 51 (2013), pp. 2230-2260; G. C. Bento et al., J. Optim. Theory Appl., 183 (2019), pp. 977-992]. [ABSTRACT FROM AUTHOR]

We introduce a new faster two-step King–Werner-type iterative method for solving nonlinear equations. The methodology is based on rational Hermite interpolation. The local as well as semi-local convergence analyses are presented under weak center Lipschitz and Lipschitz conditions. The convergence order is increased from 1 + 2 to 3 without any additional function calculations. Another advantage is the convenient fact that this method does not use derivatives. Numerical examples further validate the theoretical results. [ABSTRACT FROM AUTHOR]

We propose a gradient-based Jacobi algorithm for a class of maximization problems on the unitary group, with a focus on approximate diagonalization of complex matrices and tensors by unitary transformations. We provide weak convergence results, and prove local linear convergence of this algorithm. The convergence results also apply to the case of real-valued tensors. [ABSTRACT FROM AUTHOR]

In this paper, the local convergence analysis of the family of Kung-Traub's two-point method and the convergence ball for this family are obtained and the dynamical behavior on quadratic and cubic polynomials of the resulting family is studied. We use complex dynamic tools to analyze their stability and show that the region of stable members of this family is vast. Numerical examples are also presented in this study. This method is compared with several widely used solution methods by solving test problems from different chemical engineering application areas, e.g. Planck's radiation law problem, natch distillation at infinite reflux, van der Waal's equation, air gap between two parallel plates and flow in a smooth pipe, in order to check the applicability and effectiveness of our proposed methods. [ABSTRACT FROM AUTHOR]

We present a local convergence analysis for inexact Newton methods in order to approximate a solution of a nonlinear equation in a Banach space. Our sufficient convergence conditions involve only hypotheses on the first Fréchet-derivative of the operator involved. Numerical examples are also provided in this study. [ABSTRACT FROM AUTHOR]

This paper dealt with the local convergence study of the parameter based sixth and eighth order iterative method. This analysis discuss under assumption that the first order Fréchet derivative satisfied the Lipschitz continuity condition. In this way, we also proposed the theoretical radius of convergence of these methods. Finally, some numerical examples demonstrate that our results apply to compute the radius of convergence ball of iterative method to solve nonlinear equations. We compare the results with the method in Kumar et al. (J Comput Appl Math 330:676–694, 2018) and observe that by our approach we get much larger balls as existing ones. [ABSTRACT FROM AUTHOR]

This paper deal with the study of local convergence of fourth and fifth order iterative method for solving nonlinear equations in Banach spaces. Only the premise that the first order Fréchet derivative fulfills the Lipschitz continuity condition is needed. Under these conditions, a convergence theorem is established to study the existence and uniqueness regions for the solution for each method. The efficacy of our convergence study is shown solving various numerical examples as a nonlinear integral equation and calculating the radius of the convergence balls. We compare the radii of convergence balls and observe that by our approach, we get much larger balls as existing ones. In addition, we also include the real and complex dynamic study of one of the methods applied to a generic polynomial of order two. [ABSTRACT FROM AUTHOR]

In this paper we propose an alternative for the study of local convergence radius and the uniqueness radius for some third-order methods for multiple roots whose multiplicity is known. The main goal is to provide an alternative that avoids the use of sophisticated properties of divided differences that are used in already published papers about local convergence for multiple roots. We defined the local study by using a technique taking into consideration a bounding condition for the (m + i) th derivative of the function f (x) with i=1,2. In the case that the method uses first and second derivative in its iterative expression and i=1 in case the method only uses first derivative. Furthermore we implement a numerical analysis in the following sense. Since the radius of local convergence for high-order methods decreases with the order, we must take into account the analysis of ITS behaviour when we introduce a new iterative method. Finally, we have used these iterative methods for multiple roots for the case where the multiplicity m is unknown, so we estimate this factor by different strategies comparing the behaviour of the corresponding estimations and how this fact affect to the original method. [ABSTRACT FROM AUTHOR]

The Levenberg–Marquardt method is widely used for solving nonlinear systems of equations, as well as nonlinear least-squares problems. In this paper, we consider local convergence properties of the method, when applied to nonzero-residue nonlinear least-squares problems under an error bound condition, which is weaker than requiring full rank of the Jacobian in a neighborhood of a stationary point. Differently from the zero-residue case, the choice of the Levenberg–Marquardt parameter is shown to be dictated by (i) the behavior of the rank of the Jacobian and (ii) a combined measure of nonlinearity and residue size in a neighborhood of the set of (possibly non-isolated) stationary points of the sum of squares function. [ABSTRACT FROM AUTHOR]

In this paper, we study the quadratic matrix equations. To improve the application of iterative schemes, we use a transform of the quadratic matrix equation into an equivalent fixed‐point equation. Then, we consider an iterative process of Chebyshev‐type to solve this equation. We prove that this iterative scheme is more efficient than Newton's method. Moreover, we obtain a local convergence result for this iterative scheme. We finish showing, by an application to noisy Wiener‐Hopf problems, that the iterative process considered is computationally more efficient than Newton's method. [ABSTRACT FROM AUTHOR]

In 1984, Kyurkchiev et al. constructed an infinite sequence of iterative methods for simultaneous approximation of polynomial zeros (with known multiplicity). The first member of this sequence of iterative methods is the famous root-finding method derived independently by Farmer and Loizou (1977) and Gargantini (1978). For every given positive integer N , the N th method of this family has the order of convergence 2 N + 1. In this paper, we prove two new local convergence results for this family of iterative methods. The first one improves the result of Kyurkchiev et al. (1984). We end the paper with a comparison of the computational efficiency, the convergence behavior and the computational order convergence of different methods of the family. [ABSTRACT FROM AUTHOR]