Fiza Zafar

A Generalized Ostrowski Type Inequality for a Random Variable Whose Probability Density Function Belongs to L ∞ [a,b]

Abstract We establish here an inequality of Ostrowski type for a random variable whose probability density function belongs to Lp[a, b], in terms of the cumulative distribution function and expectation. The inequality is then applied to generalized beta random variable.

Some Generalized Error Inequalities and Applications

We present a family of four-point quadrature rule, a generalization of Gauss-two point, Simpsons , and Lobatto four-point quadrature rule for twice-differentiable mapping. Moreover, it is shown that the corresponding optimal quadrature formula presents better estimate in the context of four-point quadrature formulae of closed type. A unified treatment of error inequalities for different classes of function is also given.

A General Class of Derivative Free Optimal Root Finding Methods Based on Rational Interpolation

We construct a new general class of derivative free n-point iterative methods of optimal order of convergence 2n−1 using rational interpolant. The special cases of this class are obtained. These methods do not need Newtons iterate in the first step of their iterative schemes. Numerical computations are presented to show that the new methods are efficient and can be seen as better alternates.

A New Jarratt-Type Fourth-Order Method for Solving System of Nonlinear Equations and Applications

Solving systems of nonlinear equations plays a major role in engineering problems. We present a new family of optimal fourth-order Jarratt-type methods for solving nonlinear equations and extend these methods to solve system of nonlinear equations. Convergence analysis is given for both cases to show that the order of the new methods is four. Cost of computations, numerical tests, and basins of attraction are presented which illustrate the new methods as better alternates to previous methods. We also give an application of the proposed methods to well-known Burgers equation.

A Family of Fourteenth-Order Convergent Iterative Methods for Solving Nonlinear Equations

We present a family of fourteenth-order convergent iterative methods for solving nonlinear equations involving a specific step which when combined with any two-step iterative method raises the convergence order by , if is the order of convergence of the two-step iterative method. This new class include four evaluations of function and one evaluation of the first derivative per iteration. Therefore, the efficiency index of this family is . Several numerical examples are given to show that the new methods of this family are comparable with the existing methods.

SOME NEW ČEBYŠEV TYPE INEQUALITIES

Some new Cebysev type inequalities have been developed by working on functions whose first derivatives are absolutely continuous and the second derivatives belong to the usual Lebesgue space L∞ [a, b] . A unified treatment of the special cases is also given.

Stability analysis of a family of optimal fourth-order methods for multiple roots

Complex dynamics tools applied on the rational functions resulting from a parametric family of roots solvers for nonlinear equations provide very useful results that have been stated in the last years. These qualitative properties allow the user to select the most efficient members from the family of iterative schemes, in terms of stability and wideness of the sets of convergent initial guesses. These tools have been widely used in the case of iterative procedures for finding simple roots and only recently are being applied on the case of multiplicity m >u20091. In this paper, by using weight function procedure, we design a general class of iterative methods for calculating multiple roots that includes some known methods. In this class, conditions on the weight function are not very restrictive, so a large number of different subfamilies can be generated, all of them are optimal with fourth-order of convergence. Their dynamical analysis gives us enough information to select those with better properties and test them on different numerical experiments, showing their numerical properties.

Efficient Four-Parametric with-and-without-Memory Iterative Methods Possessing High Efficiency Indices

We construct a family of derivative-free optimal iterative methods without memory to approximate a simple zero of a nonlinear function. Error analysis demonstrates that the without-memory class has eighth-order convergence and is extendable to with-memory class. The extension of new family to the with-memory one is also presented which attains the convergence order and a very high efficiency index . Some particular schemes of the with-memory family are also described. Numerical examples and some dynamical aspects of the new schemes are given to support theoretical results.

Optimal iterative methods for finding multiple roots of nonlinear equations using free parameters

In this paper, we propose a family of optimal eighth order convergent iterative methods for multiple roots with known multiplicity with the introduction of two free parameters and three univariate weight functions. Also numerical experiments have applied to a number of academical test functions and chemical problems for different special schemes from this family that satisfies the conditions given in convergence result.

Optimal iterative methods for finding multiple roots of nonlinear equations using weight functions and dynamics

Abstract In this paper, we propose a family of iterative methods for finding multiple roots, with known multiplicity, by means of the introduction of four univariate weight functions. With the help of these weight functions, that play an important role in the development of higher order convergent iterative techniques, we are able to construct three-point eight-order optimal multiple-root finders. Also, numerical experiments have been applied to a number of test equations for different special schemes from this family satisfying the conditions given in the convergence analysis. We have also compared the basins of attraction of some proposed and known methods in order to check the wideness of the sets of converging initial points for each problem.