Family Of Fourth-order Methods Research Articles

Higher Order Methods for Nonlinear Equations and Their Basins of Attraction

In this paper, we have presented a family of fourth order iterative methods, which uses weight functions. This new family requires three function evaluations to get fourth order accuracy. By the Kung–Traub hypothesis this family of methods is optimal and has an efficiency index of 1.587. Furthermore, we have extended one of the methods to sixth and twelfth order methods whose efficiency indices are 1.565 and 1.644, respectively. Some numerical examples are tested to demonstrate the performance of the proposed methods, which verifies the theoretical results. Further, we discuss the extraneous fixed points and basins of attraction for a few existing methods, such as Newton’s method and the proposed family of fourth order methods. An application problem arising from Planck’s radiation law has been verified using our methods.

A biparametric extension of King’s fourth-order methods and their dynamics

A class of two-point quartic-order simple-zero finders and their dynamics are investigated in this paper by extending King’s fourth-order family of methods. With the introduction of an error corrector having a weight function dependent on a function-to-function ratio, higher-order convergence is obtained. Through a variety of test equations, numerical experiments strongly support the theory developed in this paper. In addition, relevant dynamics of the proposed methods is successfully explored for a prototype quadratic polynomial as well as parameter spaces and dynamical planes.

On the optimality of some multi-point methods for finding multiple roots of nonlinear equation

This paper deals with the problem of determining the multiple roots of nonlinear equations, where the multiplicity of the roots is known. The paper contains some remarks on the optimality of the recently published methods [B. Liu, X. Zhou, A new family of fourth-order methods for multiple roots of nonlinear equations, Nonlinear Anal. Model. Control, 18(2):143–152, 2013] and [X. Zhou, X. Chen, Y. Song, Families of third- and fourth-order methods for multiple roots of nonlinear equations, Appl. Math. Comput., 219(11):6030–6038, 2013]. Separate analysis of odd and even multiplicity, has shown the cases where those methods lose their optimal convergence properties. Numerical experiments are made and they support theoretical analysis.

A family of higher order multi-point iterative methods based on power mean for solving nonlinear equations

In this paper, we have presented a family of two-point fourth order, three-point sixth order and four-point twelfth order iterative methods without memory based on power mean using weight function. The family of fourth order methods is optimal in the sense of Kung-Traub hypothesis. In terms of computational point of view, our methods require three evaluations (one function and two first derivatives) to get fourth order, four evaluations (two functions and two derivatives) to get sixth order and five evaluations (three functions and two derivatives) to get twelfth order. Hence, these methods have high efficiency indices 1.587, 1.565 and 1.644 respectively. Few known results can be regarded as particular cases of our family of methods. Some numerical examples are tested to know the efficiencies of the methods which verify the theoretical results.

On some efficient derivative-free iterative methods with memory for solving systems of nonlinear equations

We present derivative free methods with memory with increasing order of convergence for solving systems of nonlinear equations. These methods relied on the basic family of fourth order methods without memory proposed by Sharma et al. (Appl. Math. Comput. 235, 383---393, 2014). The order of convergence of new family is increased from 4 of the basic family to 2+5?4.24$2+\sqrt {5} \approx 4.24$ by suitable variation of a free self-corrected parameter in each iterative step. In a particular case of the family even higher order of convergence 2+6?4.45$2+\sqrt {6} \approx 4.45$ is achieved. It is shown that the new methods are more efficient in general. The presented numerical tests confirm the theoretical results.

Basins of attraction for Zhou–Chen–Song fourth order family of methods for multiple roots

There are very few optimal fourth order methods for solving nonlinear algebraic equations having roots of multiplicity m. In a previous paper we have compared 5 such methods, two of which require the evaluation of the (m−1)th root. We have used the basin of attraction idea to recommend the best optimal fourth order method. Here we suggest to improve on the best of those five, namely Zhou–Chen–Song method by showing how to choose the best weight function.

An efficient derivative free family of fourth order methods for solving systems of nonlinear equations

We present a derivative free two-step family of fourth order methods for solving systems of nonlinear equations using the well-known Traub–Steffensen method in the first step. In order to determine the local convergence order, we apply the first-order divided difference operator for functions of several variables and direct computation by Taylor’s expansion. Computational efficiencies of the methods of new family are considered and compared with existing methods of similar structure. It is showed that the new family is especially efficient in solving large systems. Four numerical examples are given to compare the proposed methods with existing methods and to confirm the theoretical results.

New modified optimal families of King’s and Traub-Ostrowski’s methods

Based on quadratically convergent Schroder’s method, we derive many new interesting families of fourth-order multipoint iterative methods without memory for obtaining simple roots of nonlinear equations by using the weight function approach. The classical King’s family of fourth-order methods and Traub-Ostrowski’s method are obtained as special cases. According to the Kung-Traub conjecture, these methods have the maximal efficiency index because only three functional values are needed per step. Therefore, the fourth-order family of King’s family and Traub-Ostrowski’smethod are the main findings of the present work. The performance of proposed multipoint methods is compared with their closest competitors, namely, King’s family, Traub-Ostrowski’s method, and Jarratt’s method in a series of numerical experiments. All the methods considered here are found to be effective and comparable to the similar robust methods available in the literature.

Applications of Higher-Order Optimal Newton Secant Iterative Methods in Ocean Acidification and Investigation of Long-Run Implications of CO2 Emissions on Alkalinity of Seawater

The Newton secant method is a third-order iterative nonlinear solver. It requires two function and one first derivative evaluations. However, it is not optimal as it does not satisfy the Kung-Traub conjecture. In this work, we derive an optimal fourth-order Newton secant method with the same number of function evaluations using weight functions and we show that it is a member of the King family of fourth-order methods. We also obtain an eighth-order optimal Newton-secant method. We prove the local convergence of the methods. We apply the methods to solve a fourth-order polynomial arising in ocean acidifications and study their dynamics. We use the data of CO2 available from the National Oceanic and Atmospheric Administration from 1959 to 2012 and calculate the pH of the oceans for these years. Finally we further investigate the long-run implications of CO2 emissions on alkalinity of seawater using fully modified ordinary least squares (FMOLS) and dynamic OLS (DOLS). Our findings reveal that a one-percent increase in CO2 emissions will lead to a reduction in seawater alkalinity of 0.85 percent in the long run.

Families of Newton-like methods with fourth-order convergence

Families of fourth-order methods are presented which are obtained by existing third-order methods applied in succession with the secant method.

Improved King’s methods with optimal order of convergence based on rational approximations

In this paper, we present three-point and four-point methods for solving nonlinear equations. The methodology is based on King’s family of fourth order methods [R.F. King, A family of fourth order methods for nonlinear equations, SIAM J. Numer. Anal. 10 (1973) 876–879] and further developed by using rational function approximations. The three-point method requires four function evaluations and has the order of convergence eight, whereas the four-point method requires five function evaluations and has the order of convergence sixteen. Therefore, the methods are optimal in the sense of Kung–Traub hypothesis. The proposed schemes are compared with closest competitors in a series of numerical examples. Moreover, theoretical order of convergence is verified in the examples.

Runge–Kutta methods and Banach algebras

The equations defining both the exact and the computed solution to an initial value problem are related to a single functional equation, which can be regarded as prototypical. The functional equation can be solved in terms of a formal Taylor series, which can also be generated using an iteration process. This leads to the formal Taylor expansions of the solution and approximate solutions to initial value problems. The usual formulation, using rooted trees, can be modified to allow for linear combinations of trees, and this gives an insight into the nature of order conditions for explicit Runge–Kutta methods. A short derivation of the family of fourth order methods with four stages is given.

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.

A family of fourth-order methods for solving non-linear equations

In this paper, we present a family of new methods with the order of convergence four. Per iteration these methods require two evaluations of the function and one evaluation of its first derivative and therefore this family of methods has the efficiency index equal to 1.587. These methods can be of practical interest, as we show in some examples.

P-stable modification of the numerov method

A fourth order family of methods to approximate numerical integration of periodic initial value problem x″ = ƒ (t, x) is presented. The method can be regarded as a P-stable modification of the Numerov method which originally has interval of periodicity (0, 6).

A family of fourth order iterative methods for solving nonlinear operator equations

In this paper a one parameter family of fourth order methods for the iterative solution of nonlinear operator equations of the form (1) P(x) = 0 where P maps the Banach space X into a Banach space Y, is described. For non-zero values of the parameter, each application of each family member requires no explicit evaluations of derivatives. The convergence of these methods is proved under several assumptions, and a numerical example is given.

Phase properties of high order, almostP-stable formulae

Cash [3] and Chawla [4] derive families of two-step, symmetric,P-stable (hybrid) methods for solving periodic initial value problems numerically. Chawla demonstrates the existence of a family of fourth order methods while Cash derives both fourth order and sixth order methods. In this paper, we demonstrate that these methods, which are dependent on certain free parameters, havein phase particular solutions. We consider more general families of 2-step symmetric methods, including those derived by Cash and Chawla, and show that some members of these families have higher order phase lag and are almostP-stable. We also consider how the free parameters can be chosen so as to lead to an efficient implementation of the fourth order methods for large periodic systems.

TheA-stability of a family of fourth order methods

A proof is outlined showing that a fourth order exponentially fitted method isA-stable provided that the fitting parameterQ satisfiesQ≦z06≈ −4.12779 wherez06 is the negative root of the equation $$\exp (Q)(2Q^4 + 3Q^3 - 18Q^2 + 72) + (Q^4 + 3Q^3 - 18Q^2 - 72Q - 72) = 0.$$