Multipoint Iterative Methods Research Articles

Two Novel With and Without Memory Multi-Point Iterative Methods for Solving Non-Linear Equations

Two Novel With and Without Memory Multi-Point Iterative Methods for Solving Non-Linear Equations

Hybrid methods for solving structural geometric nonlinear dynamic equations: Implementation of fifth-order iterative procedures within composite time integration methods

This paper studies a class of hybrid methods that implement multi-point iterative procedures as the nonlinear solver within optimized composite implicit methods. These multi-point methods are employed to enhance convergence order and accuracy without requiring higher-order derivatives for solving structural geometric nonlinear dynamic equations. The promising approach provides an opportunity to delve deeper into its implications and explore the potential applications of multi-point techniques and composite methods across the analysis of dynamical systems. This study involves the utilization of two-point and five-point iterative methods with fifth-order convergence within three-step and four-step optimized composite implicit time integration methods. Moreover, a comprehensive hybrid scheme is introduced by integrating a family of optimized composite implicit methods with single-point and multi-point iterative methods. Then, the performance of the hybrid scheme is analyzed through structural examples, considering factors such as time step size, tolerance threshold, and the degrees of freedom of structures. The findings and numerical results illustrate that multi-point methods outperform the single-point method in terms of the number of iterations. Simultaneously, the four-step time integration method exhibits superior performance compared to the three-step time integration method, albeit with a marginal difference. It is observed that the hybrid method, consisting of the four-step time integration method and the five-point iterative method, performs better than other methods in terms of the number of iterations and errors for solving geometric nonlinear dynamic equations, especially in large structures. Thus, it stands out as a favorable choice for practical analyses and a possible candidate for generalization to other types of nonlinear problems in computational mechanics.

Combination of Optimal Three-Step Composite Time Integration Method with Multi-Point Iterative Methods for Geometric Nonlinear Structural Dynamics

This study focuses on solving the geometric nonlinear dynamic equations of structures using the multi-point iterative methods within the optimal three-step composite time integration method (OTCTIM). The OTCTIM, initially devised for linear dynamic systems, is now proposed to encompass nonlinear dynamic systems in such a way that the semi-static nonlinear equations in time sub-steps can be solved using multi-point methods. The Weerakoon–Fernando method (WFM), Homeier method (HM), Jarrat method (JM), and Darvishi–Barati method (DBM) have been extended as multi-point solvers for nonlinear equations in OTCTIM, which exhibit a higher convergence order than the Newton–Raphson method (NRM), without requiring the calculation of second and higher derivatives. Several structural examples were solved to examine the performance of these methods in the OTCTIM approach. The results demonstrated that the multi-point iterative methods outperform NRM (in terms of the number of iterations) within the OTCTIM for geometric nonlinear structural dynamics and, among the multi-point methods, the JM and DBM converged with fewer number of iterations and lower error levels. Furthermore, it has been observed that when solving nonlinear dynamic equations for structures with a high number of degrees of freedom, the incorporation of the DBM into the OTCTIM mitigates the convergence iterations and the average elapsed time for iterative sub-steps.

Efficient Families of Multi-Point Iterative Methods and Their Self-Acceleration with Memory for Solving Nonlinear Equations

In this paper, we have constructed new families of derivative-free three- and four-parametric methods with and without memory for finding the roots of nonlinear equations. Error analysis verifies that the without-memory methods are optimal as per Kung–Traub’s conjecture, with orders of convergence of 4 and 8, respectively. To further enhance their convergence capabilities, the with-memory methods incorporate accelerating parameters, elevating their convergence orders to 7.5311 and 15.5156, respectively, without introducing extra function evaluations. As such, they exhibit exceptional efficiency indices of 1.9601 and 1.9847, respectively, nearing the maximum efficiency index of 2. The convergence domains are also analysed using the basins of attraction, which exhibit symmetrical patterns and shed light on the fascinating interplay between symmetry, dynamic behaviour, the number of diverging points, and efficient root-finding methods for nonlinear equations. Numerical experiments and comparison with existing methods are carried out on some nonlinear functions, including real-world chemical engineering problems, to demonstrate the effectiveness of the new proposed methods and confirm the theoretical results. Notably, our numerical experiments reveal that the proposed methods outperform their existing counterparts, offering superior precision in computation.

Numerical Analysis of Glauert Inflow Formula for Single-Rotor Helicopter in Steady-Level Flight below Stall-Flutter Limit

This article addresses the numerical computation problem of induced inflow ratio based on the helicopter momentum theory in forward flight. The Glauert inflow formula (equation) is a nonlinear equation usually solved by the Newton–Raphson method in a relatively small number of iterations. However, many high-order convergence multipoint iterative methods have been developed over the last decade. The study examines several selected methods in terms of finding ones that provide a solution in only one iteration with acceptable accuracy. Furthermore, the influence of initial guesses on the accuracy of the obtained solutions has been investigated. In this regard, the practical range of parameters of the Glauert inflow equation for helicopters in forward flight is roughly determined by simplified modeling of a power and stall-flutter limitation. For these purposes, a basic low-fidelity longitudinal trim model of a single-rotor helicopter in steady-level flight is modified and numerically solved by a symbolic transformation of a system of 20+ nonlinear equations into a single nonlinear equation.

A numerically stable high-order Chebyshev-Halley type multipoint iterative method for calculating matrix sign function

<abstract><p>A new eighth-order Chebyshev-Halley type iteration is proposed for solving nonlinear equations and matrix sign function. Basins of attraction show that several special cases of the new method are globally convergent. It is analytically proven that the new method is asymptotically stable and the new method has the order of convergence eight as well. The effectiveness of the theoretical results are illustrated by numerical experiments. In numerical experiments, the new method is applied to a random matrix, Wilson matrix and continuous-time algebraic Riccati equation. Numerical results show that, compared with some well-known methods, the new method achieves the accuracy requirement in the minimum computing time and the minimum number of iterations.</p></abstract>

An Optimal Thirty-Second-Order Iterative Method for Solving Nonlinear Equations and a Conjecture

Many multipoint iterative methods without memory for solving non-linear equations in one variable are found in the literature. In particular, there are methods that provide fourth-order, eighth-order or sixteenth-order convergence using only, respectively, three, four or five function evaluations per iteration step, thus supporting the Kung-Traub conjecture on the optimal order of convergence. This paper shows how to find optimal high order root-finding iterative methods by means of a general scheme based in weight functions. In particular, we explicitly give an optimal thirty-second-order iterative method; as long as we know, an iterative method with that order of convergence has not been described before. Finally, we give a conjecture about optimal order multipoint iterative methods with weights.

On the Convergence of a New Family of Multi-Point Ehrlich-Type Iterative Methods for Polynomial Zeros

In this paper, we construct and study a new family of multi-point Ehrlich-type iterative methods for approximating all the zeros of a uni-variate polynomial simultaneously. The first member of this family is the two-point Ehrlich-type iterative method introduced and studied by Trićković and Petković in 1999. The main purpose of the paper is to provide local and semilocal convergence analysis of the multi-point Ehrlich-type methods. Our local convergence theorem is obtained by an approach that was introduced by the authors in 2020. Two numerical examples are presented to show the applicability of our semilocal convergence theorem.

Local and Semilocal Convergence of a Family of Multi-point Weierstrass-Type Root-Finding Methods

Weierstrass (Sitzungsber Konigl Preuss Akad Wiss Berlin II:1085–1101, 1891) introduced his famous iterative method for numerical finding all zeros of a polynomial simultaneously. Kyurkchiev and Ivanov (Ann Univ Sofia Fac Math Mech 78:132–136, 1984) constructed a family of multi-point root-finding methods which are based on the Weierstrass method. The purpose of this research is threefold: (1) to develop a new simple approach for the study of the local convergence of the multi-point simultaneous iterative methods; (2) to present a new local convergence result for this family which improves in several directions the result of Kyurkchiev and Ivanov; (3) to provide semilocal convergence results for Kyurkchiev–Ivanov’s family of iterative methods.

A general class of optimal eighth-order derivative free methods for nonlinear equations

In this paper, we present a new optimal derivative free scheme of eighth-order methods without memory in a general way. The advantage of our scheme over the earlier iteration functions, it is applicable to every optimal fourth-order derivative free scheme whose first sub step should be Steffensen’s type method to develop more advanced optimal iteration techniques of order eight. In addition, the theoretical convergence properties of our schemes are fully explored with the help of main theorem that demonstrate the convergence order. Each member of the proposed scheme satisfies the classical Kung and Traub conjecture which is related to multi-point iterative methods without memory. On the basis of average number of iterations required per point and the number of points requiring 40 iterations, we confirmed that our methods are more effective and comparable to the existing robust optimal eighth-order derivative free methods. Further, the dynamical study of these methods also supports the theoretical aspects.

Local Convergence of an Efficient Multipoint Iterative Method in Banach Space

We discuss the local convergence of a derivative-free eighth order method in a Banach space setting. The present study provides the radius of convergence and bounds on errors under the hypothesis based on the first Fréchet-derivative only. The approaches of using Taylor expansions, containing higher order derivatives, do not provide such estimates since the derivatives may be nonexistent or costly to compute. By using only first derivative, the method can be applied to a wider class of functions and hence its applications are expanded. Numerical experiments show that the present results are applicable to the cases wherein previous results cannot be applied.

What Can Students Learn While Solving Colebrook’s Flow Friction Equation?

Even a relatively simple equation such as Colebrook’s offers a lot of possibilities to students to increase their computational skills. The Colebrook’s equation is implicit in the flow friction factor and, therefore, it needs to be solved iteratively or using explicit approximations, which need to be developed using different approaches. Various procedures can be used for iterative methods, such as single the fixed-point iterative method, Newton–Raphson, and other types of multi-point iterative methods, iterative methods in a combination with Padé polynomials, special functions such as Lambert W, artificial intelligence such as neural networks, etc. In addition, to develop explicit approximations or to improve their accuracy, regression analysis, genetic algorithms, and curve fitting techniques can be used too. In this learning numerical exercise, a few numerical examples will be shown along with the explanation of the estimated pedagogical impact for university students. Students can see what the difference is between the classical vs. floating-point algebra used in computers.

Short Overview of Early Developments of the Hardy Cross Type Methods for Computation of Flow Distribution in Pipe Networks

Hardy Cross originally proposed a method for analysis of flow in networks of conduits or conductors in 1936. His method was the first really useful engineering method in the field of pipe network calculation. Only electrical analogs of hydraulic networks were used before the Hardy Cross method. A problem with flow resistance versus electrical resistance makes these electrical analog methods obsolete. The method by Hardy Cross is taught extensively at faculties, and it remains an important tool for the analysis of looped pipe systems. Engineers today mostly use a modified Hardy Cross method that considers the whole looped network of pipes simultaneously (use of these methods without computers is practically impossible). A method from a Russian practice published during the 1930s, which is similar to the Hardy Cross method, is described, too. Some notes from the work of Hardy Cross are also presented. Finally, an improved version of the Hardy Cross method, which significantly reduces the number of iterations, is presented and discussed. We also tested multi-point iterative methods, which can be used as a substitution for the Newton–Raphson approach used by Hardy Cross, but in this case this approach did not reduce the number of iterations. Although many new models have been developed since the time of Hardy Cross, the main purpose of this paper is to illustrate the very beginning of modeling of gas and water pipe networks and ventilation systems. As a novelty, a new multi-point iterative solver is introduced and compared with the standard Newton–Raphson iterative method.

A Multi-Point Iterative Analysis Method for Vibration Control of a Steering Wheel at Idle Speed

Steering wheel vibration control is of great significance to improve the ride comfort of vehicles. However, the coupling effect of the multi-source vibration transfer path makes it difficult to screen out the abnormal vibration transfer intervals. Therefore, a multi-point iterative analysis method (MIAM) is presented in this paper. In addition, this paper researches the characteristics of the main vibration transmission path of a truck and proposes a new vibration isolation structure. Considering the structural characteristics of the modal deformation of the car body, the installation orientation of the measuring point is further adjusted according to the area formed by modal deformation nodes, and a simplified vibration transfer path is presented. After weighing the vibration data of the sensor and considering the vibration transfer rate as the analysis index, the abnormal interval of vibration transmission on the main path was detected. The effectiveness of the improved structure is verified with the modal verification analysis of an extracted steering system model. In addition, the experimental results show that the improved measures reduce the total vibration of the steering wheel by 34% and offsets the peak corresponding frequency by 4.8%, showing the effectiveness of the proposed method and proving its advantages for other vibration control problems.

Construction and efficiency of multipoint root-ratio methods for finding multiple zeros

Several root-ratio multipoint methods for finding multiple zeros of univariate functions were recently presented. The characteristic of these methods is that they deal with mth root of ratio of two functions (hence the name root-ratio methods), where m is the multiplicity of the sought zero, known in advance. Some of these methods were presented without any derivation and motivation, it could be said, out of the blue. In this paper we present an easy and entirely natural way for constructing root-ratio multipoint iterative methods starting from multipoint methods for finding simple zeros. In this way, a vast number of root-ratio multipoint methods for multiple zeros, existing as well new ones, can be constructed. For demonstration, we derive four root-ratio methods for multiple zeros. Besides, we study computational cost of the considered methods and give a comparative analysis that involves CPU time needed for the extraction of the mth root. This analysis shows that root-ratio methods are pretty inefficient from the computational point of view and, thus, not suitable in practice. A general discussion on a practical need for multipoint methods of very high order is also considered.

Choosing the Optimal Multi-Point Iterative Method for the Colebrook Flow Friction Equation

The Colebrook equation is implicitly given in respect to the unknown flow friction factor λ; λ = ζ ( R e , ε * , λ ) which cannot be expressed explicitly in exact way without simplifications and use of approximate calculus. A common approach to solve it is through the Newton–Raphson iterative procedure or through the fixed-point iterative procedure. Both require in some cases, up to seven iterations. On the other hand, numerous more powerful iterative methods such as three- or two-point methods, etc. are available. The purpose is to choose optimal iterative method in order to solve the implicit Colebrook equation for flow friction accurately using the least possible number of iterations. The methods are thoroughly tested and those which require the least possible number of iterations to reach the accurate solution are identified. The most powerful three-point methods require, in the worst case, only two iterations to reach the final solution. The recommended representatives are Sharma–Guha–Gupta, Sharma–Sharma, Sharma–Arora, Džunić–Petković–Petković; Bi–Ren–Wu, Chun–Neta based on Kung–Traub, Neta, and the Jain method based on the Steffensen scheme. The recommended iterative methods can reach the final accurate solution with the least possible number of iterations. The approach is hybrid between the iterative procedure and one-step explicit approximations and can be used in engineering design for initial rough, but also for final fine calculations.

A multi-point iterative method for solving nonlinear equations with optimal order of convergence

In this study, a three-point iterative method for solving nonlinear equations is presented. The purpose is to upgrade a fourth order iterative method by adding one Newton step and using a proportional approximation for last derivative. Per iteration this method needs three evaluations of the function and one evaluation of its first derivatives. In addition, the efficiency index of the developed method is $$\root 4 \of {8}\approx 1.682$$ which supports the Kung-Traub conjecture on the optimal order of convergence. Moreover, numerical and graphical comparison of the proposed method with other existing methods with the same order of convergence are given.

An optimal and efficient general eighth-order derivative free scheme for simple roots

The main motivation of this study is to present an optimal scheme in a general way that can be applied to any existing optimal multipoint fourth-order iterative scheme whose first substep employs Steffensen’s method or Steffensen like method to further produce optimal eighth-order iterative schemes. A rational function approximation approach is used in the construction of proposed scheme. In addition, we also discussed the theoretical and computational properties of our scheme. Each member of the presented scheme satisfies the optimality conjecture for multipoint iterative methods without memory which was given by Kung and Traub in 1970. Finally, we also concluded on the basis of obtained numerical results that our methods have faster convergence in contrast to the existing methods of same order because they have minimum residual errors, minimum error difference between two consecutive iterations and minimum asymptotic error constants corresponding to the considered test function.

General approach to constructing optimal multipoint families of iterative methods using Hermite’s rational interpolation

We discuss accelerating convergence of multipoint iterative methods for solving scalar equations, using particular type rational interpolant. Both derivative-free and Newton-type methods are investigated simultaneously. As a conclusion a Theorem of König’s type for multipoint iterations is stated. A new optimal multipoint family of methods based on rational interpolation is constructed. The iteration uses n function evaluations per cycle and O(j) operations in jth step of a single iteration to obtain 2n−1 order of convergence. Several equivalent forms of the obtained iterates and development techniques are presented.

Another Newton-type method with (k+2) order convergence for solving quadratic equations

In this paper we define another Newton-type method for finding simple root of quadratic equations. It is proved that the new one-point method has the convergence order of k  2 requiring only three function evaluations per full iteration,where k is the number of terms in the generating series. The Kung and Traub conjecture states that the multipoint iteration methods, without memory based on n function evaluations, could achieve maximum convergence order 1 2nï€ ­but, the new method produces convergence order of nine, which is better than the expected maximum convergence order. Finally, we have demonstrated that our present method is very competitive with the similar methods.