Jovana Džunić

On optimal fourth-order iterative methods free from second derivative and their dynamics

In this paper new fourth order optimal root-finding methods for solving nonlinear equations are proposed. The classical Jarratt’s family of fourth-order methods are obtained as special cases. We then present results which describe the conjugacy classes and dynamics of the presented optimal method for complex polynomials of degree two and three. The basins of attraction of existing optimal methods and our method are presented and compared to illustrate their performance.

Three-point methods with and without memory for solving nonlinear equations

Abstract A new family of three-point derivative free methods for solving nonlinear equations is presented. It is proved that the order of convergence of the basic family without memory is eight requiring four function-evaluations, which means that this family is optimal in the sense of the Kung–Traub conjecture. Further accelerations of convergence speed are attained by suitable variation of a free parameter in each iterative step. This self-accelerating parameter is calculated using information from the current and previous iteration so that the presented methods may be regarded as the methods with memory. The self-correcting parameter is calculated applying the secant-type method in three different ways and Newton’s interpolatory polynomial of the second degree. The corresponding R -order of convergence is increased from 8 to 4 ( 1 + 5 / 2 ) ≈ 8.472 , 9, 10 and 11. The increase of convergence order is attained without any additional function calculations, providing a very high computational efficiency of the proposed methods with memory. Another advantage is a convenient fact that these methods do not use derivatives. Numerical examples and the comparison with existing three-point methods are included to confirm theoretical results and high computational efficiency.

Interpolatory multipoint methods with memory for solving nonlinear equations

Abstract A general way to construct multipoint methods for solving nonlinear equations by using inverse interpolation is presented. The proposed methods belong to the class of multipoint methods with memory. In particular, a new two-point method with memory with the order ( 5 + 17 ) / 2 ≈ 4.562 is derived. Computational efficiency of the presented methods is analyzed and their comparison with existing methods with and without memory is performed on numerical examples. It is shown that a special choice of initial approximations provides a considerably great accuracy of root approximations obtained by the proposed interpolatory iterative methods.

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.

On generalized biparametric multipoint root finding methods with memory

A general family of biparametric n -point methods with memory for solving nonlinear equations is proposed using an original accelerating procedure with two parameters. This family is based on derivative free classes of n -point methods without memory of interpolatory type and Steffensen-like method with two free parameters. The convergence rate of the presented family is considerably increased by self-accelerating parameters which are calculated in each iteration using information from the current and previous iteration and Newtons interpolating polynomials with divided differences. The improvement of convergence order is achieved without any additional function evaluations so that the proposed family has a high computational efficiency. Numerical examples are included to confirm theoretical results and demonstrate convergence behavior of the proposed methods.

On an efficient family of derivative free three-point methods for solving nonlinear equations

Abstract New three-step derivative free families of three-point methods for solving nonlinear equations are presented. First, a new family without memory of optimal order eight, consuming four function evaluations per iteration, is proposed by using two weight functions. The improvement of the convergence rate of this basic family, even up to 50%, is obtained without any additional function evaluation using a self-accelerating parameter. This varying parameter is calculated in each iterative step employing only information from the current and the previous iteration, defining in this way a family with memory. The self-accelerating parameter is calculated applying Newton’s interpolating polynomials of degree scaling from 1 to 4. The corresponding R-orders of convergence are increased from 8 to 10, 11, 6 + 4 2 ≈ 11.66 and 12, providing very high computational efficiency of the proposed methods with memory. Another convenient fact is that these methods do not use derivatives. Numerical examples and comparison with the existing three-point methods are included to confirm theoretical results.

A Family of Three-Point Methods of Ostrowski's Type for Solving Nonlinear Equations

A class of three-point methods for solving nonlinear equations of eighth order is constructed. These methods are developed by combining two-point Ostrowskis fourth-order methods and a modified Newtons method in the third step, obtained by a suitable approximation of the first derivative using the product of three weight functions. The proposed three-step methods have order eight costing only four function evaluations, which supports the Kung-Traub conjecture on the optimal order of convergence. Two numerical examples for various weight functions are given to demonstrate very fast convergence and high computational efficiency of the proposed multipoint methods.

A cubically convergent Steffensen-like method for solving nonlinear equations

Abstract A derivative free method for solving nonlinear equations of Steffensen’s type is presented. Using a self-correcting parameter, calculated by using Newton’s interpolatory polynomial of second degree, the R -order of convergence is increased from 2 to 3. This acceleration of the convergence rate is attained without any additional function calculations, which provides a very high computational efficiency of the proposed method. Another advantage is the convenient fact that this method does not use derivatives. Numerical examples are included to confirm the theoretical results and high computational efficiency.

Traub’s accelerating generator of iterative root-finding methods

Abstract An accelerating generator of iterative methods for finding multiple roots, based on Traub’s differential–difference recurrence relation, is presented. It is proved that this generator yields an iteration function of order r + 1 starting from arbitrary iteration function of order r . In this way, it is possible to construct various iterative formulas of higher order for finding single roots of nonlinear equations and all simple or multiple roots of algebraic polynomials, simultaneously. For demonstration, two iterative methods of the fourth order in ordinary (real or complex) arithmetic and an iterative method in interval arithmetic are presented.

On the similarity of some three-point methods for solving nonlinear equations

Abstract In this short note we discuss certain similarities between some three-point methods for solving nonlinear equations. In particular, we show that the recent three-point method published in [R. Thukral, A new eighth-order iterative method for solving nonlinear equations, Appl. Math. Comput. 217 (2010) 222–229] is a special case of the family of three-point methods proposed previously in [R. Thukral, M.S. Petkovic, Family of three-point methods of optimal order for solving nonlinear equations, J. Comput. Appl. Math. 233 (2010) 2278–2284].