An efficient predictor-corrector method for solving nonlinear equations

This paper proposes an efficient continuation method for solving nonlinear equations. The proposed method belongs to a class of predictor-corrector methods and uses modified Euler's predictors in order that larger step-sizes may be accepted. Some numerical results show that the method obtains a solution with less computational effort than the ordinary Euler's method, especially when the starting point is far from the solution.     

New eighth-order iterative methods for solving nonlinear equations

In this paper, three new families of eighth-order iterative methods for solving simple roots of nonlinear equations are developed by using weight function methods. Per iteration these iterative methods require three evaluations of the function and one evaluation of the first derivative. This implies that the efficiency index of the developed methods is 1.682, which is optimal according to Kung and Traub’s conjecture [7] for four function evaluations per iteration. Notice that Bi et al.’s method in [2] and [3] are special cases of the developed families of methods. In this study, several new examples of eighth-order methods with efficiency index 1.682 are provided after the development of each family of methods. Numerical comparisons are made with several other existing methods to show the performance of the presented methods.     

Steffensen type methods for solving nonlinear equations

In the present paper, by approximating the derivatives in the well known fourth-order Ostrowski’s method and in a sixth-order improved Ostrowski’s method by central-difference quotients, we obtain new modifications of these methods free from derivatives. We prove the important fact that the methods obtained preserve their convergence orders 4 and 6, respectively, without calculating any derivatives. Finally, numerical tests confirm the theoretical results and allow us to compare these variants with the corresponding methods that make use of derivatives and with the classical Newton’s method.     

Combined bracketing methods for solving nonlinear equations

Several methods based on combinations of bisection, regula falsi, and parabolic interpolation has been developed. An interval bracketing ensures the global convergence while the combination with the parabolic interpolation increases the speed of the convergence. The proposed methods have been tested on a series of examples published in the literature and show good results.     

Accurate fourteenth-order methods for solving nonlinear equations

We establish new iterative methods of local order fourteen to approximate the simple roots of nonlinear equations. The considered three-step eighth-order construction can be viewed as a variant of Newton’s method in which the concept of Hermite interpolation is used at the third step to reduce the number of evaluations. This scheme includes three evaluations of the function and one evaluation of the first derivative per iteration, hence its efficiency index is 1.6817. Next, the obtained approximation for the derivative of the Newton’s iteration quotient is again taken into consideration to furnish novel fourteenth-order techniques consuming four function and one first derivative evaluations per iteration. In providing such new fourteenth-order methods, we also take a special heed to the computational burden. The contributed four-step methods have 1.6952 as their efficiency index. Finally, various numerical examples are given to illustrate the accuracy of the developed techniques.     

New higher order methods for solving nonlinear equations with multiple roots

For a nonlinear equation f(x)=0 having a multiple root we consider Steffensen’s transformation, T. Using the transformation, say, F_q(x)=T^qf(x) for integer q≥2, repeatedly, we develop higher order iterative methods which require neither derivatives of f(x) nor the multiplicity of the root. It is proved that the convergence order of the proposed iterative method is 1+2^q−2 for any equation having a multiple root of multiplicity m≥2. The efficiency of the new method is shown by the results for some numerical examples.     

On efficient two-parameter methods for solving nonlinear equations

Derivative free methods for solving nonlinear equations of Steffensen’s type are presented. Using two self-correcting parameters, calculated by Newton’s interpolatory polynomials of second and third degree, the order of convergence is increased from 2 to 3.56. This method is used as a corrector for a family of biparametric two-step derivative free methods with and without memory with the accelerated convergence rate up to order 7. Significant acceleration of convergence is attained without any additional function calculations, which provides very high computational efficiency of the proposed methods. Another advantage is a convenient fact that the proposed methods do not use derivatives. Numerical examples are given to demonstrate excellent convergence behavior of the proposed methods and good coincidence with theoretical results.     

Various Newton-type iterative methods for solving nonlinear equations

The aim of the present paper is to introduce and investigate new ninth and seventh order convergent Newton-type iterative methods for solving nonlinear equations. The ninth order convergent Newton-type iterative method is made derivative free to obtain seventh-order convergent Newton-type iterative method. These new with and without derivative methods have efficiency indices 1.5518 and 1.6266, respectively. The error equations are used to establish the order of convergence of these proposed iterative methods. Finally, various numerical comparisons are implemented by MATLAB to demonstrate the performance of the developed methods.     

Construction of Newton-like iteration methods for solving nonlinear equations

In this paper, we present a simple, and yet powerful and easily applicable scheme in constructing the Newton-like iteration formulae for the computation of the solutions of nonlinear equations. The new scheme is based on the homotopy analysis method applied to equations in general form equivalent to the nonlinear equations. It provides a tool to develop new Newton-like iteration methods or to improve the existing iteration methods which contains the well-known Newton iteration formula in logic; those all improve the Newton method. The orders of convergence and corresponding error equations of the obtained iteration formulae are derived analytically or with the help of Maple. Some numerical tests are given to support the theory developed in this paper.     

Ostrowski type methods for solving systems of nonlinear equations

Four generalized algorithms builded up from Ostrowski’s method for solving systems of nonlinear equations are written and analyzed. A development of an inverse first-order divided difference operator for functions of several variables is presented, as well as a direct computation of the local order of convergence for these variants of Ostrowski’s method. Furthermore, a sequence that approximates the order of convergence is generated for the examples and it confirms in a numerical way that the order of the methods is well deduced.     

New predictor-corrector approach for nonlinear fractional differential equations: error analysis and stability

In this paper, the predictor-corrector approach is used to propose two algorithms for the numerical solution of linear and non-linear fractional differential equations (FDE). The fractional order derivative is taken to be in the sense of Caputo and its properties are used to transform FDE into a Volterra-type integral equation. Simpson''s 3/8 rule is used to develop new numerical schemes to obtain the approximate solution of the integral equation associated with the given FDE. The error and stability analysis for the two methods are presented. The proposed methods are compared with the ones available in the literature. Numerical simulation is performed to demonstrate the validity and applicability of both the proposed techniques. As an application, the problem of dynamics of the new fractional order non-linear chaotic system introduced by Bhalekar and Daftardar-Gejji is investigated by means of the obtained numerical algorithms.     

New efficient derivative free family of seventh-order methods for solving systems of nonlinear equations

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.     

Two- and three-point with memory methods for solving nonlinear equations

The main objective and inspiration in the construction of two- and three-point with memory method is to attain the utmost computational efficiency, without any additional function evaluations. At this juncture, we have modified the existing fourth and eighth order without memory method with optimal order of convergence by means of different approximations of self-accelerating parameters. The parameters have been calculated by Hermite interpolating polynomial, which accelerates the order of convergence of the without memory methods. In particular, the R-order convergence of the proposed two- and three-step with memory methods is increased from four to five and eight to ten. One more advantage of these methods is that the condition f′(x) ≠ 0, in the neighborhood of the required root, imposed on Newton’s method, can be removed. Numerical comparison is also stated to confirm the theoretical results.     

Quadrature-difference methods for solving linear and nonlinear singular integro-differential equations

Here we propose and justify quadrature-difference methods for solving different kinds (linear, nonlinear and multidimensional) of periodic singular integro-differential equations.     

On novel classes of iterative methods for solving nonlinear equations

In this paper, we establish two new classes of derivative-involved methods for solving single valued nonlinear equations of the form f(x) = 0. The first contributed two-step class includes two evaluations of the function and one of its first derivative where its error analysis shows a fourth-order convergence. Next, we construct a three-step high-order class of methods including four evaluations per full cycle to achieve the seventh-order of convergence. Numerical examples are included to re-verify the theoretical results and moreover put on show the efficiency of the new methods from our classes.