Vinay Kanwar

An optimal fourth-order family of methods for multiple roots and its dynamics

There are few optimal fourth-order methods for solving nonlinear equations when the multiplicity m of the required root is known in advance. Therefore, the principle focus of this paper is on developing a new fourth-order optimal family of iterative methods. From the computational point of view, the conjugacy maps and the strange fixed points of some iterative methods are discussed, their basins of attractions are also given to show their dynamical behavior around the multiple roots. Further, using Mathematica with its high precision compatibility, a variety of concrete numerical experiments and relevant results are extensively treated to confirm the theoretical development.

Optimal equi-scaled families of Jarratt's method

In this paper, we present many new fourth-order optimal families of Jarratts method and Ostrowskis method for computing simple roots of nonlinear equations numerically. The proposed families of Jarratts method having the same scaling factor of functions as that of Jarratts method (i.e. quadratic scaling factor of functions in the numerator and denominator of the correction factor) are the main finding of this paper. It is observed that the body structures of our proposed families of Jarratts method are simpler than those of the original families of Jarratts method. The efficiency of these methods is tested on a number of relevant numerical problems. Furthermore, numerical examples suggest that each member of the proposed families can be competitive to other similar robust methods available in the literature.

On some modified families of multipoint iterative methods for multiple roots of nonlinear equations

In this paper, we propose a new one-parameter family of Schroder’s method for finding the multiple roots of nonlinear equations numerically. Further, we derive many new cubically convergent families of Schroder-type methods. Proposed families are derived from the modified Newton’s method for multiple roots and one-parameter family of Schroder’s method. Furthermore, we introduce new families of third-order multipoint iterative methods for multiple roots free from second-order derivative by semi discrete modifications of the above proposed methods. One of the families requires two evaluations of the function and one evaluation of its first-order derivative and the other family requires one evaluation of the function and two evaluations of its first-order derivative per iteration. Numerical examples are also presented to demonstrate the performance of proposed iterative methods.

Simply constructed family of a Ostrowski's method with optimal order of convergence

In this paper, we propose a simple modification over Chuns method for constructing iterative methods with at least cubic convergence [5]. Using iteration formulas of order two, we now obtain several new interesting families of cubically or quartically convergent iterative methods. The fourth-order family of Ostrowskis method is the main finding of the present work. Per iteration, this family of Ostrowskis method requires two evaluations of the function and one evaluation of its first-order derivative. Therefore, the efficiency index of this Ostrowskis family is E=43~1.587, which is better than those of most third-order iterative methods E=33~1.442 and Newtons method E=2~1.414. The performance of Ostrowskis family is compared with its closest competitors, namely Ostrowskis method, Jarratts method and Kings family in a series of numerical experiments.

Geometrically Constructed Families of Newton's Method for Unconstrained Optimization and Nonlinear Equations

One-parameter families of Newtons iterative method for the solution of nonlinear equations and its extension to unconstrained optimization problems are presented in the paper. These methods are derived by implementing approximations through a straight line and through a parabolic curve in the vicinity of the root. The presented variants are found to yield better performance than Newtons method, in addition that they overcome its limitations.

Simple geometric constructions of quadratically and cubically convergent iterative functions to solve nonlinear equations

In this paper, we derive one-parameter families of Newton, Halley, Chebyshev, Chebyshev-Halley type methods, super-Halley, C-methods, osculating circle and ellipse methods respectively for finding simple zeros of nonlinear equations, permitting f ′ (x) = 0 at some points in the vicinity of the required root. Halley, Chebyshev, super-Halley methods and, as an exceptional case, Newton method are seen as the special cases of the family. All the methods of the family and various others are cubically convergent to simple roots except Newton’s or a family of Newton’s method.

Modified families of multi-point iterative methods for solving nonlinear equations

We further present some semi-discrete modifications to the cubically convergent iterative methods derived by Kanwar and Tomar (Modified families of Newton, Halley and Chebyshev methods, Appl. Math. Comput. http://dx.doi.org/10.1016/j.amc.2007.02.119) and derived a number of interesting new classes of third-order multi-point iterative methods free from second derivatives. Furthermore, several functions have been tested and all the methods considered are found to be effective and compared to the well-known existing third and fourth-order multi-point iterative methods.

Another Simple Way of Deriving Several Iterative Functions to Solve Nonlinear Equations

We present another simple way of deriving several iterative methods for solving nonlinear equations numerically. The presented approach of deriving these methods is based on exponentially fitted osculating straight line. These methods are the modifications of Newtons method. Also, we obtain well-known methods as special cases, for example, Halleys method, super-Halley method, Ostrowskis square-root method, Chebyshevs method, and so forth. Further, new classes of third-order multipoint iterative methods free from a second-order derivative are derived by semidiscrete modifications of cubically convergent iterative methods. Furthermore, a simple linear combination of two third-order multipoint iterative methods is used for designing new optimal methods of order four.

Exponentially fitted variants of Newton's method with quadratic and cubic convergence

In this paper, we present some new families of Newton-type iterative methods, in which f′(x)=0 is permitted at some points. The presented approach of deriving these iterative methods is different. They have well-known geometric interpretation and admit their geometric derivation from an exponential fitted osculating parabola. Cubically convergent methods require the use of the first and second derivatives of the function as Eulers, Halleys, Chebyshevs and other classical methods do. Furthermore, new classes of third-order multipoint iterative methods free from second derivative are derived by semi-discrete modifications of cubically convergent iterative methods. Further, the approach has been extended to solve a system of non-linear equations.

A stable class of improved second-derivative free Chebyshev-Halley type methods with optimal eighth order convergence

In this paper, we present a uniparametric family of modified Chebyshev-Halley type methods with optimal eighth-order of convergence. In terms of computational cost, each member of the family requires only four functional evaluations per step, and hence is optimal in the sense of Kung-Traub conjecture. Moreover, in order to have additional information to choose some elements of the class, in particular some stable enough, we use complex dynamics tools to analyze their stability. Then, some ranges of values of the parameter are found to be avoided but we show that the region of stable members of this family is vast. It is found by way of illustration that these proposed methods are very useful in high precision computations.