Sanjay Kumar Khattri

Two new classes of optimal Jarratt-type fourth-order methods

Abstract In this paper, we investigate the construction of some two-step without memory iterative classes of methods for finding simple roots of nonlinear scalar equations. The classes are built through the approach of weight functions and these obtained classes reach the optimal order four using one function and two first derivative evaluations per full cycle. This shows that our classes can be considered as Jarratt-type schemes. The accuracy of the classes is tested on a number of numerical examples. And eventually, it is observed that our contributions take less number of iterations than the compared existing methods of the same type to find more accurate approximate solutions of the nonlinear equations.

Sixth order derivative free family of iterative methods

In this study, we develop a four-parameter family of sixth order convergent iterative methods for solving nonlinear scalar equations. Methods of the family require evaluation of four functions per iteration. These methods are totally free of derivatives. Convergence analysis shows that the family is sixth order convergent, which is also verified through the numerical work. Though the methods are independent of derivatives, computational results demonstrate that family of methods are efficient and demonstrate equal or better performance as compared with other six order methods, and the classical Newton method.

On the Secant method

We present a new semilocal convergence analysis for the Secant method in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Our analysis is based on the weaker center-Lipschitz concept instead of the stronger Lipschitz condition which has been ubiquitously employed in other studies such as Amat et al. (2004) [2], Bosarge and Falb (1969) [9], Dennis (1971) [10], Ezquerro et al. (2010) [11], Hernandez et al. (2005, 2000) [13,12], Kantorovich and Akilov (1982) [14], Laasonen (1969) [15], Ortega and Rheinboldt (1970) [16], Parida and Gupta (2007) [17], Potra (1982, 1984-1985, 1985) [18-20], Proinov (2009, 2010) [21,22], Schmidt (1978) [23], Wolfe (1978) [24] and Yamamoto (1987) [25] for computing the inverses of the linear operators. We also provide lower and upper bounds on the limit point of the majorizing sequences for the Secant method. Under the same computational cost, our error analysis is tighter than that proposed in earlier studies. Numerical examples illustrating the theoretical results are also given in this study.

Constructing third-order derivative-free iterative methods

In this work, we develop nine derivative-free families of iterative methods from the three well-known classical methods: Chebyshev, Halley and Euler iterative methods. Methods of the developed families consist of two steps and they are totally free of derivatives. Convergence analysis shows that the methods of these families are cubically convergent, which is also verified through the computational work. Apart from being totally free of derivatives, numerical comparison demonstrates that the developed methods perform better than the three classical methods.

Unifying fourth-order family of iterative methods

In this work, we develop a new two-parameter family of iterative methods for solving nonlinear scalar equations. One of the parameters is defined through an infinite power series consisting of real coefficients while the other parameter is a real number. The methods of the family are fourth-order convergent and require only three evaluations during each iteration. It is shown that various fourth-order iterative methods in the published literature are special cases of the developed family. Convergence analysis shows that the methods of the family are fourth-order convergent which is also verified through the numerical work. Computations are performed to explore the efficiency of various methods of the family.

Derivative free algorithm for solving nonlinear equations

In this work, we develop a simple yet practical algorithm for constructing derivative free iterative methods of higher convergence orders. The algorithm can be easily implemented in software packages for achieving desired convergence orders. Convergence analysis shows that the algorithm can develop methods of various convergence orders which is also supported through the numerical work. Computational results ascertain that the developed algorithm is efficient and demonstrate equal or better performance as compared with other well known methods.

Algorithm for forming derivative-free optimal methods

We develop a simple yet effective and applicable scheme for constructing derivative free optimal iterative methods, consisting of one parameter, for solving nonlinear equations. According to the, still unproved, Kung-Traub conjecture an optimal iterative method based on k+1 evaluations could achieve a maximum convergence order of

Optimal Eighth Order Iterative Methods

2^{k}

OPTIMIZATION OF MAINTENANCE SCHEDULING OF SHIP BORNE MACHINERY FOR IMPROVED RELIABILITY AND REDUCED COST

. Through the scheme, we construct derivative free optimal iterative methods of orders two, four and eight which request evaluations of two, three and four functions, respectively. The scheme can be further applied to develop iterative methods of even higher orders. An optimal value of the free-parameter is obtained through optimization and this optimal value is applied adaptively to enhance the convergence order without increasing the functional evaluations. Computational results demonstrate that the developed methods are efficient and robust as compared with many well known methods.

Weak convergence conditions for the Newton's method in Banach space using general majorizing sequences

We develop an eighth order family of methods, consisting of three steps and three parameters, for solving nonlinear equations. Per iteration the methods require four evaluations (three function evaluations and one evaluation of the first derivative). Convergence analysis shows that the family is eighth-order convergent which is also substantiated through the numerical work. Computational results ascertain that family of methods are efficient and demonstrate equal or better performance as compared with other well known methods.