Extra Function Evaluations Research Articles

A New Adaptive Eleventh-Order Memory Algorithm for Solving Nonlinear Equations

In this article, we introduce a novel three-step iterative algorithm with memory for finding the roots of nonlinear equations. The convergence order of an established eighth-order iterative method is elevated by transforming it into a with-memory variant. The improvement in the convergence order is achieved by introducing two self-accelerating parameters, calculated using the Hermite interpolating polynomial. As a result, the R-order of convergence for the proposed bi-parametric with-memory iterative algorithm is enhanced from 8 to 10.5208. Notably, this enhancement in the convergence order is accomplished without the need for extra function evaluations. Moreover, the efficiency index of the newly proposed with-memory iterative algorithm improves from 1.5157 to 1.6011. Extensive numerical testing across various problems confirms the usefulness and superior performance of the presented algorithm relative to some well-known existing algorithms.

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.

Differential Evolution With Underestimation-Based Multimutation Strategy.

As we know, the performance of differential evolution (DE) highly depends on the mutation strategy. However, it is difficult to choose a suitable mutation strategy for a specific problem or different running stages. This paper proposes an underestimation-based multimutation strategy (UMS) for DE. In the UMS, a set of candidate offsprings are simultaneously generated for each target individual by utilizing multiple mutation strategies. Then a cheap abstract convex underestimation model is built based on some selected individuals to obtain the underestimation value of each candidate offspring. According to the quality of each candidate offspring measured by the underestimation value, the most promising candidate solution is chosen as the offspring. Compared to the existing probability-based multimutation techniques, no mutation strategies are lost during the search process as each mutation strategy has the same probability to generate a candidate solution. Moreover, no extra function evaluations are produced because the candidate solutions are filtered by the underestimation value. The UMS is integrated into some DE variants and compared with their original algorithms and several advanced DE approaches over the CEC 2013 and 2014 benchmark sets. Additionally, a well-known real-world problem is employed to evaluate the performance of the UMS. Experimental results show that the proposed UMS can improve the performance of the advanced DE variants.

Developing a new family of Newton–Secant method with memory based on a weight function

In this paper, we construct an iterative method with memory based on the Newton–Secants method to solve nonlinear equations. This proposed method has fourth order convergence and costs only three functions evaluation per iteration and without any evaluation of the derivative function. Acceleration of the convergence speed is achieved by an appropriate variation of a free parameter in each step. This self accelerator parameter is estimated using Newtons interpolation polynomial of third degree. The order of convergence is increased from 4 to 5.23 without any extra function evaluation. This method has the efficiency index equal to $$5.23^{\frac{1}{3}}\approx 1.7358$$ . We describe the analysis of the proposed method along with numerical experiments including comparison with the existing methods. Finally, the attraction basins of the proposed method are shown and compared with other existing methods.

Improved Newton-like methods for solving systems of nonlinear equations

We present the iterative methods of fifth and eighth order of convergence for solving systems of nonlinear equations. Fifth order method is composed of two steps namely, Newton’s and Newton-like steps and requires the evaluations of two functions, two first derivatives and one matrix inversion in each iteration. The eighth order method is composed of three steps, of which the first two steps are that of the proposed fifth order method whereas the third is Newton-like step. This method requires one extra function evaluation in addition to the evaluations of fifth order method. Computational efficiency of proposed techniques is discussed and compared with the existing methods. Some numerical examples are considered to test the performance of the new methods. Moreover, theoretical results concerning order of convergence and computational efficiency are confirmed in numerical examples. Numerical results have confirmed the robust and efficient character of the proposed techniques.

New two-parameter Chebyshev–Halley-like family of fourth and sixth-order methods for systems of nonlinear equations

The two-parameter Chebyshev–Halley-like family of optimal two-point fourth-order methods proposed by Babajee (2015), is further extended to solve systems of nonlinear equations. This two-step fourth-order family is further extended to obtain a two-parameter family of sixth-order methods which requires only one extra function evaluation. The performance of some special members of the proposed families using only single inverse per iteration have been tested through numerical examples and the results show that these are effective and comparable to existing methods both in order and efficiency.

Moment‐independent regional sensitivity analysis of complicated models with great efficiency

SummaryMoment‐independent regional sensitivity analysis (RSA) is a very useful guide tool for assessing the effect of a specific range of an individual input on the uncertainty of model output, while large computational burden is involved to perform RSA, which would certainty lead to the limitation of engineering application. Main tasks for performing RSA are to estimate the probability density function (PDF) of model output and the joint PDF of model output and the input variable by some certain smart techniques. Firstly, a method based on the concepts of maximum entropy, fractional moment and sparse grid integration is utilized to estimate the PDF of the model output. Secondly, Nataf transformation is applied to obtain the joint PDF of model output and the input variable. Finally, according to an integral transformation, those regional sensitivity indices can be easily computed by a Monte Carlo procedure without extra function evaluations. Because all the PDFs can be estimated with great efficiency, and only a small amount of function evaluations are involved in the whole process, the proposed method can greatly decrease the computational burden. Several examples with explicit or implicit input–output relations are introduced to demonstrate the accuracy and efficiency of the proposed method. Copyright © 2015 John Wiley & Sons, Ltd.

Solving nonlinear equations by a derivative-free form of the King’s family with memory

In this paper, we present an iterative three-point method with memory based on the family of King's methods to solve nonlinear equations. This proposed method has eighth order convergence and costs only four function evaluations per iteration which supports the Kung-Traub conjecture on the optimal order of convergence. An acceleration of the convergence speed is achieved by an appropriate variation of a free parameter in each step. This self accelerator parameter is estimated using Newton's interpolation polynomial of fourth degree. The order of convergence is increased from 8 to 12 without any extra function evaluation. Consequently, this method, possesses a high computational efficiency. Finally, a numerical comparison of the proposed method with related methods shows its effectiveness and performance in high precision computations.

A novel hybrid adaptive collaborative approach based on particle swarm optimization and local search for dynamic optimization problems

This paper proposes a novel hybrid approach based on particle swarm optimization and local search, named PSOLS, for dynamic optimization problems. In the proposed approach, a swarm of particles with fuzzy social-only model is frequently applied to estimate the location of the peaks in the problem landscape. Upon convergence of the swarm to previously undetected positions in the search space, a local search agent (LSA) is created to exploit the respective region. Moreover, a density control mechanism is introduced to prevent too many LSAs crowding in the search space. Three adaptations to the basic approach are then proposed to manage the function evaluations in the way that are mostly allocated to the most promising areas of the search space. The first adapted algorithm, called HPSOLS, is aimed at improving PSOLS by stopping the local search in LSAs that are not contributing much to the search process. The second adapted, algorithm called CPSOLS, is a competitive algorithm which allocates extra function evaluations to the best performing LSA. The third adapted algorithm, called CHPSOLS, combines the fundamental ideas of HPSOLS and CPSOLS in a single algorithm. An extensive set of experiments is conducted on a variety of dynamic environments, generated by the moving peaks benchmark, to evaluate the performance of the proposed approach. Results are also compared with those of other state-of-the-art algorithms from the literature. The experimental results indicate the superiority of the proposed approach.

Solving Nondifferentiable Nonlinear Equations by New Steffensen-Type Iterative Methods with Memory

It is attempted to present two derivative-free Steffensen-type methods with memory for solving nonlinear equations. By making use of a suitable self-accelerator parameter in the existing optimal fourth- and eighth-order without memory methods, the order of convergence has been increased without any extra function evaluation. Therefore, its efficiency index is also increased, which is the main contribution of this paper. The self-accelerator parameters are estimated using Newton’s interpolation. To show applicability of the proposed methods, some numerical illustrations are presented.

On a new efficient Steffensen-like iterative class by applying a suitable self-accelerator parameter.

It is attempted to present an efficient and free derivative class of Steffensen-like methods for solving nonlinear equations. To this end, firstly, we construct an optimal eighth-order three-step uniparameter without memory of iterative methods. Then the self-accelerator parameter is estimated using Newton's interpolation in such a way that it improves its convergence order from 8 to 12 without any extra function evaluation. Therefore, its efficiency index is increased from 81/4 to 121/4 which is the main feature of this class. To show applicability of the proposed methods, some numerical illustrations are presented.

Efficient Jarratt-like methods for solving systems of nonlinear equations

We present the iterative methods of fourth and sixth order convergence for solving systems of nonlinear equations. Fourth order method is composed of two Jarratt-like steps and requires the evaluations of one function, two first derivatives and one matrix inversion in each iteration. Sixth order method is the composition of three Jarratt-like steps of which the first two steps are that of the proposed fourth order scheme and requires one extra function evaluation in addition to the evaluations of fourth order method. Computational efficiency in its general form is discussed. A comparison between the efficiencies of proposed techniques with existing methods of similar nature is made. The performance is tested through numerical examples. Moreover, theoretical results concerning order of convergence and computational efficiency are confirmed in the examples. It is shown that the present methods are more efficient than their existing counterparts, particularly when applied to the large systems of equations.

A new probabilistic distribution matching PATC formulation using polynomial chaos expansion

In the existing probabilistic hierarchical optimization approaches, such as probabilistic analytical target cascading (PATC), all the stochastic interrelated responses are characterized only by the first two statistical moments. However, due to the high nonlinear relation between the inputs and outputs, the interrelated responses are not necessarily normally distributed. The existing approaches, therefore, may not accurately quantify the probabilistic characteristics of the interrelated responses, and would further prevent achieving the real optimal solution. To overcome this deficiency, a novel PATC approach, named PATC-PCE is developed. By employing the polynomial chaos expansion (PCE) technique, the entire distribution of interrelated response can be characterized by a PCE coefficients vector, and then matched and propagated in the hierarchy. Comparative studies show that PATC-PCE outperforms PATC in terms of yielding more accurate optimal solutions and fewer design cycles when the interrelated response are random non-normal quantities, while at a sacrifice of extra function evaluations.

Approximate preservation of quadratic first integrals by explicit Runge–Kutta methods

The approximate preservation of quadratic first integrals (QFIs) of differential systems in the numerical integration with Runge–Kutta (RK) methods is studied. Conditions on the coefficients of the RK method to preserve all QFIs up to a given order are obtained, showing that the pseudo-symplectic methods studied by Aubry and Chartier (BIT 98(3):439–461, 1998) of algebraic order p preserve QFIs with order q = 2p. An expression of the error of conservation of QFIs by a RK method is given, and a new explicit six-stage formula with classical order four and seventh order of QFI-conservation is obtained by choosing their coefficients so that they minimize both local truncation and conservation errors. Several formulas with algebraic orders 3 and 4 and different orders of conservation have been tested with some problems with quadratic and general first integrals. It is shown that the new fourth-order explicit method preserves much better the qualitative properties of the flow than the standard fourth-order RK method at the price of two extra function evaluations per step and it is a practical and efficient alternative to the fully implicit methods required for a complete preservation of QFIs.

Numerical solution of neutral functional differential equations by Adams methods in divided difference form

A variable step and variable order algorithm for the numerical solution of neutral functional differential equations is described. This general purpose algorithm is based on the variable step formulation of the Adams methods represented in divided difference form. The Adams–Bashforth and Adams–Moulton methods are implemented in predictor–corrector mode. The detection of derivative discontinuities relies on the estimates of the local discretization errors. The restarting of the integration at each discontinuity point relies on the step size and order changing strategy based on the estimates of the local discretization errors. This algorithm reduces the overhead cost by implementing the Adams methods in divided difference form. This algorithm increases the reliability and efficiency by obtaining asymptotically correct estimates of the local discretization errors for the lower adjacent orders without any extra function evaluation by using local extrapolation. The results from three test examples are presented.

Recursive algorithm without extra function evaluations for the Jacobian matrix of Viéta's polynomial system with applications

This paper describes a simple recursion without extra function calculating for Jacobian matrix of Viéta's polynomial system and its applications in simultaneously approximating all the zeros of a polynomial with complex coefficient.

A novel hybrid genetic algorithm using local optimizer based on heuristic pattern move

A new hybrid genetic algorithm with the significant improvement of convergence performance is proposed in this study. This algorithm comes from the incorporation of a modified microgenetic algorithm with a local optimizer based on the heuristic pattern move. The hybridization process is implemented by replacing the two worst individuals in the offspring obtained from the conventional genetic operations with two new individuals generated from the local optimizer in each generation. Some implementation-related problems such as the selection of control parameters in the local optimizer are addressed in detail. This new algorithm has been examined using six benchmarking functions, and is compared with the conventional genetic algorithms without the local optimizer incorporated, as well as the hybrid algorithms incorporated with the hill-climbing method in terms of convergence performance. The results show that the proposed hybrid algorithm is more effective and efficient to obtain the global optimum. It takes about 6.4%-74.4% of the number of generations normally required by the conventional genetic algorithms to obtain the global optimum, while the computation cost for reproducing each new generation has hardly increased compared to the conventional genetic algorithms. Another advantage of this new algorithm is the implementation process is very simple and straightforward. There are no extra function evaluations and other complex calculations involved in the added local optimizer as well as in the hybridization process. This makes the new algorithm easy to be incorporated with the existing software packages of genetic algorithms so as to further improve their performance. As an engineering example, this new algorithm is applied for the detection of a crack in a composite plate, which demonstrates its effectiveness in solving engineering practical problems.

Real-Time Dynamic Simulation of Single Spool Turboj et Engines

A hybrid integration method is proposed in this paper for the real-time simulation of single spool turbojet engines. The nonlinear dynamic model of the J-85 turbojet engine is developed first. Three types of integration methods, including explicit integration methods, implicit integration methods, and the proposed hybrid integration method, are used to integrate the dynamic model. Simulation of the model by explicit integration methods fails to reach real time, since the stiff characteristic of the engine model requires small integration step size to maintain stability. On the other hand, al though use of an implicit integration algorithm can enlarge the step size, extra function evaluations are needed to calculate the system Jacobian for the Newton-Raphson algorithm, and the simulation still cannot reach real time. The proposed hybrid integration scheme uses an explicit method to in tegrate the non-stiff part of the system, and an implicit method to integrate the stiff part of the system. Real-time simulation can be achieved by using the proposed hybrid scheme.

Continuous extensions to high order runge-kutta methods

Computer assisted derivation and improved techniques have led to effective explicit Runge-Kutta methods of higher order. These methods become inefficient when the step size must be reduced often to produce approximations at specified points. Considerable effort has been devoted to providing Runge-Kutta methods with an interpolation capability, so that approximations can be produced inexpensively at intermediate points of a successful step. New high order Hermite interpolants for two well known embedded Runge-Kutta methods of orders 7 and 8 are presented. These interpolants are constructed using values from two successive integration steps, are locally of O(h 8) or O(h 9), and require only one or four extra function evaluations per step respectively.

Choosing the relaxation parameter for the solution of nonlinear magnetic field problems by the Newton-Raphson method

The use of the modified Newton method for the solution of nonlinear magnetostatic problems arising from the vector potential Finite Element analysis is investigated. In particular the optimum choice of the relaxation factor /spl alpha/ is investigated. A new method is developed for determining the relaxation factor which minimizes the energy functional in the direction along the solution update at each nonlinear iteration. This method is based on approximating the functional with a fourth order polynomial. In this way the optimum relaxation factor can be quickly determined with the minimum number of extra function evaluations. This choice of /spl alpha/ is compared to choosing the relaxation factor which minimizes the residual norm at each iteration. The modified Newton methods are compared to the standard Newton-Raphson method for the solution of 2D and 3D problems. For problems involving saturated iron parts convergence rates are greatly improved by use of the modified Newton methods. Of the two methods for choosing the relaxation factor the one which minimizes the functional is shown to be the better. Using the new algorithm to determine this relaxation factor results in substantial reductions in solution times. >