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A reliable and computable a posteriori error bound is derived for the mixed Ciarlet–Raviart method for the equation , , with the first boundary condition and a piecewise constant . Several authors derived residual type a posteriori error bounds at assumptions that and the domain is polygonal, none of which is used in the paper. In case of a piecewise smooth boundary, we consider the mixed method with the triangular Lagrange finite elements of the order, which, in general, are curvilinear along the boundary. This provides an approximation to the boundary, matching the finite elements in accuracy. Our bounds belong to the class of a posteriori functional majorants and are evaluated with help of functions from the testing space. By the reasons of accuracy and simplicity, this space is generated by the finite elements with the domains, coinciding with the domains of the Lagrange elements, and singular rational coordinate functions. [ABSTRACT FROM AUTHOR]

In this article, we introduce a unified approach to constructing a higher-order derivative-free scheme based on the approximations of F'(zk)-1. A family of order p = 6,7 derivative-free method is proposed and compared to some well-known methods. The necessary and sufficient condition for p-th order of convergence are given in terms of parameter matrices ... and α(k). Some good choices of ... and α(k) are offered. Numerical experiments were carried out to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

We present a study of p -th degree immersed finite element (IFE) spaces based on pointwise interface conditions that lead to optimal convergence rates for typical second-order elliptic interface problems on interface-independent meshes. We study the existence and approximation properties of IFE spaces spanned by functions that satisfy the extended interface conditions in a pointwise manner on general curved interfaces. We present computational results to demonstrate the approximation capabilities of the proposed IFE spaces. [ABSTRACT FROM AUTHOR]

In this paper, we discover new techniques to construct efficient higher order iterative methods for the longstanding problem of solving systems of nonlinear equations. Given any iterative method with the extended Newton iteration being a predictor, a new efficient higher order method can be constructed without more matrix inversions, but it requires introduction of a new function. In terms of our main theorems, we find that several well-known existing results are just special cases of ours. Applying the new techniques to some classical third or fourth order methods, we make light work of developing some new efficient higher order methods. For further discussion, we analyze the computational efficiencies of the newly developed methods, and make comprehensive comparisons between the new higher order methods and the corresponding lower order ones. Finally, we take several numerical experiments of solving nonlinear problems, and the numerical results are consistent with the theoretical analysis, to a large extent. [ABSTRACT FROM AUTHOR]

We present an application of goal-oriented adaptive isotropic hp-refinement for the 2-D Maxwell eigenvalue problem. We apply a simplified goal-oriented error expression for improving the accuracy of the eigenvalues, which, when combined with indicators derived from the solution, enables highly targeted discretization tuning. Furthermore, we introduce an hp-refinement/coarsening optimizer coupled with smoothness estimation for refinement classification and execution. These enhancements yield cost-effective resource allocations that reach extremely high accuracy rapidly even for eigenvalues of singular eigenfunctions. Finally, we provide numerical benchmarks more accurate than existing numerical reference values, along with new benchmarks for higher order modes that will facilitate the comparison and development of new approaches to adaptivity and hp finite elements in computational electromagnetics (CEM). Our implementation is based on the open-source finite element library deal.II. [ABSTRACT FROM AUTHOR]

The suitability of six higher order root solvers is examined for solving the nonlinear equilibrium equations in large deformation analysis of structures. The applied methods have a better convergence rate than the quadratic Newton-Raphson method. These six methods do not require higher order derivatives to achieve a higher convergence rate. Six algorithms are developed to use the higher order methods in place of the Newton- Raphson method to solve the nonlinear equilibrium equations in geometrically nonlinear analysis of structures. The higher order methods are applied to both continuum and discrete problems (spherical shell and dome truss). The computational cost and the sensitivity of the higher order solution methods and the Newton-Raphson method with respect to the load increment size are comparatively investigated. The numerical results reveal that the higher order methods require a lower number of iterations that the Newton-Raphson method to converge. It is also shown that these methods are less sensitive to the variation of the load increment size. As it is indicated in numerical results, the average residual reduces in a lower number of iterations by the application of the higher order methods in the nonlinear analysis of structures. [ABSTRACT FROM AUTHOR]

Abstract There are a very small number of high quality derivative free methods with memory for solving a system of nonlinear equations numerically. Motivated and inspired by the fact, we propose a more efficient general class of Steffensen type multipoint methods with memory. This proposed class requires one divided difference and inverse of only one matrix per full iteration. We also demonstrate their applicability and illustrate that these methods produce approximations of greater accuracy and remarkably reduce the computational time for solving systems of nonlinear equations numerically. For quantitative comparison, we have also computed total number of convergent points and convergent percentage along with basins of attractions on a number of test problems to recommend the best quality algorithm. [ABSTRACT FROM AUTHOR]

Treating discontinuities at element boundaries is a significant problem in understanding high-order FEM simulation data since the physics used to model the simulation is often continuous. Recently, the family of SIAC filters, especially the L-SIAC filter, has been gaining popularity for its use in postprocessing. The computational math community, with its focus on improving the theoretical aspects of the SIAC filter, has applied the filter only on simple, fairly uniform unstructured meshes, where the largest element in the mesh is less than or equal to twice the smallest element. In many engineering applications, the unstructured meshes have varying orders of mesh resolution, but there is no literature for adapting the characteristic length of the SIAC filter to address these real-world simulation data. The central contribution of this paper is an algorithm used to calculate the characteristic length dynamically at any point in the mesh. We demonstrate that our approach has a lower error and is computationally faster than using maximum edge length over the mesh. [ABSTRACT FROM AUTHOR]

Abstract Steady state simulations of magnetized electron fluid equations with strong anisotropic diffusion based on the first-order hyperbolic approach is carried out using cell-centered higher order upwind schemes, linear and weighted essentially non-oscillatory (WENO). Along with the magnetized electrons, the diffusion equation is also simulated to demonstrate the implementation and design order of the accuracy of the approach due to their similar upwind structure. We show the adequacy of linear upwind schemes for diffusion equation and the use of shock-capturing scheme like WENO does not have any adverse effect on the solution, unlike the total-variation diminishing (TVD) methods. We further extended the approach to advection–diffusion equation, and appropriate boundary conditions have obtained a consistent design accuracy of the third and fifth order. We implemented the WENO approach to advection–diffusion equation by using the split hyperbolic method to demonstrate the advantage of non-oscillatory schemes to capture sharp gradients in boundary layer type problems without spurious oscillations. Finally, numerical results for magnetized electrons simulations indicate that with increasing strength of magnetic confinement it is possible to capture sharp gradients without oscillations by WENO scheme. Highlights • Fifth order accurate solutions are obtained for diffusion equation in hyperbolic approach. • WENO schemes can capture boundary layer type solutions in advection–diffusion without oscillations. • WENO approach improve magnetized electron simulations. • TVD type schemes can pollute the entire simulation and may not be recommended. [ABSTRACT FROM AUTHOR]

In this paper, for solving systems of nonlinear equations, we develop a family of two-step third order methods and introduce a technique by which the order of convergence of many iterative methods can be improved. Given an iterative method of order p  ≥ 2 which uses the extended Newton iteration as a predictor, a new method of order p + 2 is constructed by introducing only one additional evaluation of the function. In addition, for an iterative method of order p  ≥ 3 using the Newton iteration as a predictor, a new method of order p + 3 can be extended. Applying this procedure, we develop some new efficient methods with higher order of convergence. For comparing these new methods with the ones from which they have been derived, we discuss the computational efficiency in detail. Several numerical examples are given to justify the theoretical results by the asymptotic behaviors of the considered methods. [ABSTRACT FROM AUTHOR]

Time-fractional parabolic equations with a Caputo time derivative are considered. For such equations, we explore and further develop the new methodology of the a-posteriori error estimation and adaptive time stepping proposed in Kopteva (2022). We improve the earlier time stepping algorithm based on this theory, and specifically address its stable and efficient implementation in the context of high-order methods. The considered methods include an L1-2 method and continuous collocation methods of arbitrary order, for which adaptive temporal meshes are shown to yield optimal convergence rates in the presence of solution singularities. [ABSTRACT FROM AUTHOR]

The volume integral of Riemann flux in the discontinuous Galerkin (DG) method is introduced in this paper. The boundaries integrals of the fluxes (Riemann flux) are transformed into volume integral. The new family of DG method is accomplished by applying divergence theorem to the boundaries integrals of the flux. Therefore, the (DG) method is independent of the boundaries integrals of fluxes (Riemann flux) at the cell (element) boundaries as in classical (DG) methods. The modified streamline upwind Petrov-Galerkin method is used to capture the oscillation of unphysical flow for shocked flow problems. The numerical results of applying totally volume integral discontinuous Galerkin method (TVI-DG) are presented to unsteady scalar hyperbolic equations (linear convection equation, inviscid Burger's equation and Buckley-Leverett equation) for one dimensional case. The numerical finding of this scheme is very accurate as compared with other high order schemes as the weighted compact finite difference method WCOMP.

In this work, we construct a family of seventh order iterative methods for finding multiple roots of a nonlinear function. The scheme consists of three steps, of which the first is Newton’s step and last two are the weighted-Newton steps. Hence, the name of the scheme is ‘weighted-Newton methods’. Theoretical results are studied exhaustively along with the main theorem describing convergence analysis. Stability and convergence domain of the proposed class are also demonstrated by means of using a graphical technique, namely, basins of attraction. Boundaries of these basins are fractal like shapes through which basins are symmetric. Efficacy is demonstrated through numerical experimentation on variety of different functions that illustrates good convergence behavior. Moreover, the theoretical result concerning computational efficiency is verified by computing the elapsed CPU time. The overall comparison of numerical results including accuracy and CPU-time shows that the new methods are strong competitors for the existing methods.

With the aim of addressing the issue of sample damage during electron tomography data acquisition, we propose a number of new reconstruction strategies based on subsampling (which uses only a subset of a full image) and inpainting (recovery of a full image from subsampled one). We point out that the total-variation (TV) inpainting model commonly used to inpaint subsampled images may be inappropriate for 2D projection images of typical TEM specimens. Thus, we propose higher-order TV (HOTV) inpainting, which accommodates the fact that projection images may be inherently smooth, as a more suitable image inpainting scheme. We also describe how the HOTV method can be extended to 3D, a scheme which makes use of both image data and sinogram data. Additionally, we propose gradient subsampling as a more efficient scheme than random subsampling. We make a rigorous comparison of our proposed new reconstruction schemes with existing ones. The new schemes are demonstrated to perform better than or as well as existing schemes, and we show that they outperform existing schemes at low subsampling rates. [ABSTRACT FROM AUTHOR]

In this paper, we develop a class of third order methods which is a generalization of the existing ones and a method of fourth order method, then introduce a technique that improves the order of convergence of any given iterative method for solving systems of nonlinear equations. Based on a given iterative method of order p ≥ 2 which uses the extended Newton iteration as a predictor, a new method of order p + 2 is proposed with only one additional evaluation of the function. Moreover, if the given iterative method of order p ≥ 3 uses the Newton iteration as a predictor, then a new method of order p + 3 can be developed. Applying this procedure, we obtain some new methods with higher order of convergence. Moreover, computational efficiency is analyzed and comparisons are made between these new methods and the ones from which have been derived. Finally, several numerical tests are performed to show the asymptotic behaviors which confirm the theoretical results. [ABSTRACT FROM AUTHOR]

We present a three-step two-parameter family of derivative free methods with seventh-order of convergence for solving systems of nonlinear equations numerically. The proposed methods require evaluation of two central divided differences and inversion of only one matrix per iteration. As a result, the proposed family is more efficient as compared with the existing methods of same order. Numerical examples show that the proposed methods produce approximations of greater accuracy and remarkably reduce the computational time for solving systems of nonlinear equations. [ABSTRACT FROM AUTHOR]

A systematic way is presented for the construction of multi-step iterative method with frozen Jacobian. The inclusion of an auxiliary function is discussed. The presented analysis shows that how to incorporate auxiliary function in a way that we can keep the order of convergence and computational cost of Newton multi-step method. The auxiliary function provides us the way to overcome the singularity and ill-conditioning of the Jacobian. The order of convergence of proposed p-step iterative method is p + 1. Only one Jacobian inversion in the form of LU-factorization is required for a single iteration of the iterative method and in this way, it offers an efficient scheme. For the construction of our proposed iterative method, we used a decomposition technique that naturally provides different iterative schemes. We also computed the computational convergence order that confirms the claimed theoretical order of convergence. The developed iterative scheme is applied to large scale problems, and numerical results show that our iterative scheme is promising. [ABSTRACT FROM AUTHOR]

For solving nonlinear systems, we introduce a technique that improves the order of convergence of any given iterative method which uses the extended Newton iteration as a predictor. Based on a given iterative method of order $$p\ge 2$$ , a new method of order $$p+2$$ is developed by introducing just one evaluation of the function. We obtain some new methods with higher order of convergence by applying this procedure to some known methods. Computational efficiency in the general form is discussed and comparisons are made between these new methods and the ones from which have been derived. We also perform several numerical tests to show the asymptotic behaviors which confirm the theoretical results. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The Newton method is one of the most used methods for solving nonlinear system of equations when the Jacobian matrix is nonsingular. The method converges to a solution with Q-order two for initial points sufficiently close to the solution. The method of Halley and the method of Chebyshev are among methods that have local and cubic rate of convergence. Combining these methods with a backtracking and curvilinear strategy for unconstrained optimization problems these methods have been shown to be globally convergent. The backtracking forces a strict decrease of the function of the unconstrained optimization problem. It is shown that no damping of the step in the backtracking routine is needed close to a strict local minimizer and the global method behaves as a local method. The local behavior for the unconstrained optimization problem is investigated by considering problems with two unknowns and it is shown that there are no significant differences in the region where the global method turn into a local method for second and third order methods. Further, the final steps to reach a predefined tolerance are investigated. It is shown that the region where the higher order methods terminate in one or two iteration is significantly larger than the corresponding region for Newton's method. [ABSTRACT FROM AUTHOR]

By studying the commonness of some fifth order methods modified from third order ones for solving systems of nonlinear equations, we propose a new class of three-step methods of convergence order five by modifying a class of two-step methods with cubic convergence. Next, for a given method of order p ≥ 2 which uses the extended Newton iteration y k = x k − aF ′( x k ) −1 F ( x k ) as a predictor, a new method of order p + 2 is proposed. For example, we construct a class of m + 2-step methods of convergence order 2 m + 3 by introducing only one evaluation of the function to each of the last m steps for any positive integer m . In this paper, we mainly focus on the class of fifth order methods when m = 1. Computational efficiency in the general form is considered. Several examples for numerical tests are given to show the asymptotic behavior and the computational efficiency of these higher order methods. [ABSTRACT FROM AUTHOR]

We propose an m + 1 -step modified Newton method of convergence order m + 2 to solve systems of nonlinear equations which are third Fréchet differentiable in a convex set containing the zero. Computational efficiency in the general form for a positive integer m is discussed, which shows that the efficiency increases with m when applied to large systems. Moreover, a comparison between the efficiency of this technique and some existing efficient methods is made, which implies that the present method is more efficient particularly for solving large systems of equations. Theoretical results about order of convergence and computational efficiency are largely verified in numerical examples. [ABSTRACT FROM AUTHOR]

We propose a high order explicit finite difference method for fractional advection diffusion equations. These equations can be obtained from the standard advection diffusion equations by replacing the second order spatial derivative by a fractional operator of order α with 1 < α ≤ 2 . This operator is defined by a combination of the left and right Riemann–Liouville fractional derivatives. We study the convergence of the numerical method through consistency and stability. The order of convergence varies between two and three and for advection dominated flows is close to three. Although the method is conditionally stable, the restrictions allow wide stability regions. The analysis is confirmed by numerical examples. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a three-step iterative method of convergence order five for solving systems of nonlinear equations. The methodology is based on the two-step Homeier’s method with cubic convergence (Homeier, 2004). Computational efficiency in its general form is discussed and a comparison between the efficiency of proposed technique and existing ones is made. The performance is tested through numerical examples. Moreover, theoretical results concerning order of convergence and computational efficiency are verified in the examples. It is shown that the present method has an edge over existing methods, particularly when applied to large systems of equations. [Copyright &y& Elsevier]

The Powell singular function was introduced 1962 by M.J.D. Powell as an unconstrained optimization problem. The function is also used as nonlinear least squares problem and system of nonlinear equations. The function is a classic test function included in collections of test problems in optimization as well as an example problem in text books. In the global optimization literature the function is stated as a difficult test case. The function is convex and the Hessian has a double singularity at the solution. In this paper we consider Newton's method and methods in Halley class and we discuss the relationship between these methods on the Powell Singular Function. We show that these methods have global but linear rate of convergence. The function is in a subclass of unary functions and results for Newton's method and methods in the Halley class can be extended to this class. Newton's method is often made globally convergent by introducing a line search. We show that a full Newton step will satisfy many of standard step length rules and that exact line searches will yield slightly faster linear rate of convergence than Newton's method. We illustrate some of these properties with numerical experiments. [ABSTRACT FROM AUTHOR]

Highly efficient and versatile computational electromagnetic analysis of 3-D transformation-based metamaterial cloaking structures based on a hybridization of a higher order finite element method for discretization of the cloaking region and a higher order method of moments for numerical termination of the computational domain is proposed and demonstrated. The technique allows for an effective modeling of the continuously inhomogeneous anisotropic cloaking region, for cloaks based on both linear and nonlinear coordinate transformations, using a very small number of large curved finite elements with continuous spatial variations of permittivity and permeability tensors and high-order p-refined field approximations throughout their volumes, with a very small total number of unknowns. In analysis, there is no need for a discretization of the permittivity and permeability profiles of the cloak, namely for piecewise homogeneous (layered) approximate models, with material tensors replaced by appropriate piecewise constant approximations. Numerical results show a very significant reduction (three to five orders of magnitude for the simplest possible 6-element model and five to seven orders of magnitude for an h-refined 24-element model) in the scattering cross section of a perfectly conducting sphere with a metamaterial cloak, in a broad range of wavelengths. Given the introduced explicit approximations in modeling of the spherical geometry and continuous material tensor profiles (both by fourth-order Lagrange interpolating functions), and inherent numerical approximations involved in the finite element and moment method techniques and codes, the cloaking effects are shown to be predicted rather accurately by the full-wave numerical analysis method. [ABSTRACT FROM AUTHOR]

Third-order methods will, in most cases, use fewer iterations than a second-order method to reach the same accuracy. However, the number of arithmetic operations per iteration is higher for third-order methods than for a second-order method. Newton's method is the most commonly used second order method and Halley's method is the most well-known third-order method. Newton's method is more used in practical applications than any third-order method. We will show that for a large class of problems, the ratio of the number of arithmetic operations of Halley's method and Newton's method is constant per iteration. It is shown that [image omitted] We show that the zero elements in the Hessian matrix induce zero elements in the tensor (third derivative). The sparsity structure in the Hessian matrix we consider is the skyline or envelope structure. This is a very convenient structure for solving linear systems of equations with a direct method. The class of matrices that have a skyline structure includes banded and dense matrices. Numerical testing confirms that the ratio of the number of arithmetic operations of a third-order method and Newton's method is constant per iteration, and is independent of the number of unknowns. [ABSTRACT FROM AUTHOR]

A new set of multiresolution curvilinear Rao-Wilton-Glisson (MR-CRWG) basis functions is proposed for the method of moments (MoM) solution of integral equations for three-dimensional (3-D) electromagnetic (EM) problems. The MR-CRWG basis functions are constructed as linear combinations of curvilinear Rao-Wilton-Glisson (CRWG) basis functions which are defined over curvilinear triangular patches, thus allowing direct application on the existing MoM codes that using CRWG basis. The multiresolution property of the MR-CRWG basis can lead to the fast convergence of iterative solvers merely by a simple diagonal preconditioning to the corresponding MoM matrices. Moreover, the convergence of iterative solvers can be further improved by introducing a perturbation from the principle value term of the magnetic field integral equation (MFIE) operator to construct diagonal preconditioners for efficient iterative solution of the electric field integral equation (EFIE). Another important property of the MR-CRWG basis is that the MoM matrices using the MR-CRWG basis can be highly sparsified without loss of accuracy. The MR-CRWG basis has been applied to the 3-D electromagnetic scattering problems and the numerical results indicate that the MR-CRWG basis performs much better than the CRWG basis. [ABSTRACT FROM AUTHOR]

Abstract: In this manuscript we present a p-th degree immersed finite element method for solving boundary value problems with discontinuous coefficients. In this method, interface jump conditions are employed in the finite element basis functions, and the mesh does not have to be aligned with coefficient discontinuity. We show that under h refinement the immersed finite element solution converges to the true solution at the optimal and rates in the and norms, respectively. Furthermore, numerical results suggest that the immersed finite element solution converges exponentially fast under p refinement. Numerical examples are provided to illustrate features of this immersed finite element method. [Copyright &y& Elsevier]

This paper addresses the role of centrality in the implementation of interior point methods. We provide theoretical arguments to justify the use of a symmetric neighbourhood, and translate them into computational practice leading to a new insight into the role of re-centering in the implementation of interior point methods. Second-order correctors, such as Mehrotra’s predictor–corrector, can occasionally fail: we derive a remedy to such difficulties from a new interpretation of multiple centrality correctors. Through extensive numerical experience we show that the proposed centrality correcting scheme leads to noteworthy savings over second-order predictor–corrector technique and previous implementations of multiple centrality correctors. [ABSTRACT FROM AUTHOR]

In this paper, we develop a set of two-dimensional finite-difference time-domain (FDTD) schemes, capable of minimizing dispersion errors according to problem-related requirements. Their performance is optimized within either a narrow or wide frequency band, by requiring the minimization of an error estimator at selected frequency points. By properly utilizing a series expansion of the estimator, accuracy improvement is ensured regardless of the propagation angle. Contrary to standard FDTD approaches, the design procedure is proven to efficiently exploit the potential of higher order operators, as the developed techniques attain significantly lower error levels, at no or low additional computational cost. [ABSTRACT FROM AUTHOR]

A new methodology that facilitates the control of the inherent dispersion error in the case of higher order finite-difference time-domain (FDTD) schemes is presented in this paper. The basic idea is to define suitable algebraic expressions that reflect numerical inaccuracies reliably. Then, finite-difference operators are determined via the minimization of the error estimators at selected frequencies. In order to apply this procedure, an error expansion in terms of cylindrical harmonic functions is performed, which also enables accuracy enhancement for all propagation angles. The design process produces a set of two-dimensional (2-D) FDTD algorithms with optimized frequency response. Contrary to conventional methodologies, the proposed techniques adjust their reliability range according to the requirements of the examined problem and can be, therefore, more efficient in computationally demanding simulations. [ABSTRACT FROM AUTHOR]

Abstract: In the numerical solution of ordinary differential systems, the method of fractional steps (also known as operator splitting) yields high-order accurate schemes based on separate, computationally convenient treatments of distinct physical effects. Such schemes are equally desirable but much less accurate for semi-explicit index-1 differential-algebraic equations (DAEs). In the first half of this note, it is shown that naı¨ve application to DAEs of standard splitting schemes suffers from order reduction: both first and second-order schemes are only first-order accurate for DAEs. In the second half of this note, a new family of higher-order splitting schemes for semi-explicit index-1 DAEs is developed. The new schemes are based on a deferred correction paradigm in which an error equation is solved numerically, and therefore inherit a simple computationally convenient structure. Higher-order convergence of the new schemes is proved, and numerical results confirm the expected order of accuracy in addition to establishing efficiency. [Copyright &y& Elsevier]

The radiation properties of several contemporary array structures in three-dimensional problems are numerically investigated by implementing an advanced second-order tangential vector finite element (TVFE) methodology. The choice and the development of the proposed strategy is justified, since it provides a more precise field representation compared to the first-order edge elements and enables its utilization within coarse meshes without reducing the accuracy of the calculated solution. The latter observation becomes more important when electrically large domains are modeled, as their study with low-order elements usually calls for a fine discretization and, consequently, results in a large number of unknowns. In this paper, we mainly focus on arrays of patch antennas with linear and planar configurations, while cases where beam steering is investigated are also taken into account. For comparison purposes, our study incorporates results obtained with a higher order nonstandard finite-difference time-domain (FDTD) method, which efficiently overcomes some of the deficiencies concerning the conventional Yee algorithm. [ABSTRACT FROM AUTHOR]

Abstract: The Rachford-Rice equation calculates the amounts and compositions of two equilibrium phases at constant K-factors. This single variable equation is a perfect candidate for applying higher order iteration methods because the additional cost for evaluating the derivatives with order higher than two is relatively low. This study compares the performance of Householder's high order iterations up to order seven. In addition, a method to improve the initial estimate is proposed for the situation where the Rachford-Rice function shows abrupt change close to the asymptotes and overshooting happens. The proposed method can largely reduce repeated use of the bisection adjustment in the subsequent iterations. The comparison shows that the Householder's iteration with order three or four gives the best performance on average. [Copyright &y& Elsevier]

• The main advantage of the proposed methods is that they work well for any value of parameter "a" in the first stage of iterations, while the existing Jarratt-like methods work only for some a=2/3, 1/2. • To develop new methods with the highest possible order of convergence which requires the smallest possible evaluation of the function F and its derivatives and matrix inversions. • To extend the domain of applicability of existing methods. • From numerical results, clearly show that proposed method with the selected parameters are faster than other cases. In this article, we propose a new family of methods, such as Jarratt, with the fifth and sixth order. This includes some popular methods as special cases. We propose four different selection for parameter matrix T k. The main advantage of the proposed methods is that they work well for any value of parameter " a " in the first stage of iterations, while the existing methods work only for some a (2 / 3 or 1 / 2). Thus, we extend essentially the domain of applicability of the original ones. Based on the computational efficiency analysis, we also made a selection of some high-efficiency ones among the families. [ABSTRACT FROM AUTHOR]