Correction to: From Halpern’s fixed-point iterations to Nesterov’s accelerated interpretations for root-finding problems

A novel approach regarding the fixed points of repelling nature

The high order Newton iteration formulas are revisited in this paper. Translating the nonlinear root finding problem into a fixed point iteration involving an unknown general function whose root is searched, a double Taylor series is undertaken regarding the root and the root finding function. Based on the error analysis of the expansion, a simple algorithm is later proposed to construct Newton iteration formulae of any order commencing from the traditional linearly convergent fixed point iteration method and quadratically convergent Newton-Raphson method of frequently at the disposal of the scientific community. It is shown that the well-known variants like the Halley's method or Haouseholder's methods of high order can be reproduced from the general case outlined here. Some further rare single-step classes of any order are shown to be derivable from the presented algorithm. Finally, some new higher order accurate variants are also offered taking into account multi-step compositions which demand less computation of higher derivatives. The efficiency, accuracy and performance of the proposed methods and also their potential advantages over the classical ones are numerically demonstrated and discussed on some well-documented examples from the open literature.

A numerical assessment of partitioned implicit methods for thermomechanical problems

The Extrapolation Algorithm is a technique devised in 1962 for accelerating the rate of convergence of slowly converging Picard iterations for fixed point problems. Versions to this technique are now called Anderson Acceleration in the applied mathematics community and Anderson Mixing in the physics and chemistry communities, and these are related to several other methods extant in the literature. We seek here to broaden and deepen the conceptual foundations for these methods, and to clarify their relationship to certain iterative methods for root-finding problems. For this purpose, the Extrapolation Algorithm will be reviewed in some detail, and selected papers from the existing literature will be discussed, both from conceptual and implementation perspectives.

From Halpern’s fixed-point iterations to Nesterov’s accelerated interpretations for root-finding problems