We consider solving the unconstrained minimization problem using an iterative method derived from the third order Super Halley method. The Super Halley method requires solution of two linear systems of equations. We show a practical implementation using an iterative method to solve the linear systems. This paper introduces an array of arrays (jagged) data structure for storing the second and th...

A one-parameter family of iteration functions for finding the simple roots of nonlinear equations is presented. The iteration process is based on one-point approximation by the quadratic equation x + ay + bx + cy + d = 0, where the unknowns b, c and d are determined in terms of a. Different choices of a correspond to different approximating quadratic curves, viz. parabola, circle, ellipse and h...

348 © 2015 Guilford Publications, Inc. Halley E. Woodward thanks the National Science Foundation and the University of Iowa Presidential Graduate Research Fund for funding through Graduate Research Fellowships. This study was also funded by the University of Iowa’s Social Sciences Funding Program. Address correspondence to Halley Woodward, Department of Psychology, University of Iowa, 11 Seasho...

In this study, we use a convex acceleration of Newton's method (or super-Halley method) to approximate solutions of nonlinear equations. We provide sufficient convergence conditions for this method in three space settings: real line, complex plane, and Banach space. Several applications of our results are also provided.

Further to the proposal and application of a stochastic methodology and the resulting first exit time distribution function to life table data we introduce a theoretical framework for the estimation of the maximum deterioration age and to explore on how “vitality,” according to Halley and Strehler and Mildvan, changes during the human lifetime. The mortality deceleration or mortality leveling-o...

Matrix fixed-point iterations zn+1 = ψ(zn) defined by a rational function ψ are considered. For these iterations a new proof is given that matrix convergence is essentially reduced to scalar convergence. It is shown that the principal Padé family of iterations for the matrix sign function and the matrix square root is a special case of a family of rational iterations due to Ernst Schröder. This...

From a dynamical point of view applied to complex polynomials, we study a number of root–finding iterative methods. We consider Newton’s method, Newton’s method for multiple roots, Jarratt’s method, the super–Halley method, the convex as well as the double convex acceleration of Whittaker’s method, the methods of Chebyshev, Stirling, and Steffensen, among others. Since all of the iterative root...