A Method for Accelerating of Convergence of Numerical and Power Series

In this paper is examined the acceleration of convergence of alternative and non-alternative numerical and power series by means of an Euler-Abel type operator, that is defined earlier by G.A.Sorokin and I.Z.Milovanovic. Through this type of linear operator of generalized difference of sequence is achieved that alternative and non-alternative numerical and power series to be transformed into series with higher speed of convergence than the initial series. At the end of this paper is given the implementation of this method through a numeric example.

Construction of conformal maps based on the locations of singularities for improving the double exponential formula

The double exponential formula, or DE formula, is a high-precision integration formula using a change of variables called a DE transformation; it has the disadvantage that it is sensitive to singularities of an integrand near the real axis. To overcome this disadvantage, Slevinsky & Olver (2015, On the use of conformal maps for the acceleration of convergence of the trapezoidal rule and Sinc numerical methods. SIAM J. Sci. Comput., 37, A676–A700) attempted to improve the formula by constructing conformal maps based on the locations of singularities. Based on their ideas, we construct a new transformation formula. Our method employs special types of the Schwarz–Christoffel transformation for which we can derive the explicit form. The new transformation formula can be regarded as a generalization of DE transformations. We confirm its effectiveness by numerical experiments.

New Mono- and Biaccelerator Iterative Methods with Memory for Nonlinear Equations

Acceleration of convergence is discussed for some families of iterative methods in order to solve scalar nonlinear equations. In fact, we construct mono- and biparametric methods with memory and study their orders. It is shown that the convergence orders 12 and 14 can be attained using only 4 functional evaluations, which provides high computational efficiency indices. Some illustrations will also be given to reverify the theoretical discussions.