A note on three numerical procedures to solve Volterra integro-differential equations in structural analysis

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New Eighth-Order Derivative-Free Methods for Solving Nonlinear Equations

A new family of eighth-order derivative-free methods for solving nonlinear equations is presented. It is proved that these methods have the convergence order of eight. These new methods are derivative-free and only use four evaluations of the function per iteration. In fact, we have obtained the optimal order of convergence which supports the Kung and Traub conjecture. Kung and Traub conjectured that the multipoint iteration methods, without memory based on n evaluations could achieve optimal convergence order of . Thus, we present new derivative-free methods which agree with Kung and Traub conjecture for . Numerical comparisons are made to demonstrate the performance of the methods presented

Two Optimal Eighth-Order Derivative-Free Classes of Iterative Methods

Optimization problems defined by (objective) functions for which derivatives are unavailable or available at an expensive cost are emerging in computational science. Due to this, the main aim of this paper is to attain as high as possible of local convergence order by using fixed number of (functional) evaluations to find efficient solvers for one-variable nonlinear equations, while the procedure to achieve this goal is totally free from derivative. To this end, we consider the fourth-order uniparametric family of Kung and Traub to suggest and demonstrate two classes of three-step derivative-free methods using only four pieces of information per full iteration to reach the optimal order eight and the optimal efficiency index 1.682. Moreover, a large number of numerical tests are considered to confirm the applicability and efficiency of the produced methods from the new classes

A New Family of Iterative Methods Based on an Exponential Model for Solving Nonlinear Equations

We present two new families of iterative methods for obtaining simple roots of nonlinear equations. The first family is developed by fitting the model m(x)=epx(Ax2+Bx+C) to the function f(x) and its derivative f′(x), f″(x) at a point xn. In order to remove the second derivative of the first methods, we construct the second family of iterative methods by approximating the equation f(x)=0 around the point (xn,f(xn)) by the quadratic equation. Analysis of convergence shows that the new methods have third-order or higher convergence. Numerical experiments show that new iterative methods are effective and comparable to those of the well-known existing methods