Shanks function transformations in a vector space

In this paper, we show how to construct various extensions of Shanks transformation for functions in a vector space. They are aimed at transforming a function tending slowly to its limit when the argument tends to infinity into another function with better convergence properties. Their expressions as ratio of determinants and recursive algorithms for their implementation are given. A simplified form of one of them is derived. It allows us to obtain a convergence result for an important class of functions. An application to integrable systems is discussed

Generalizations of Aitken's process for accelerating the convergence of sequences

When a sequence or an iterative process is slowly converging, a convergence acceleration process has to be used. It consists in transforming the slowly converging sequence into a new one which, under some assumptions, converges faster to the same limit. In this paper, new scalar sequence transformations having a kernel (the set of sequences transformed into a constant sequence) generalizing the kernel of the Aitken's delta2 process are constructed. Then, these transformations are extended to vector sequences. They also lead to new fixed point methods which are studied

Shanks sequence transformations and Anderson acceleration

This paper presents a general framework for Shanks transformations of sequences of elements in a vector space. It is shown that the Minimal Polynomial Extrapolation (MPE), the Modified Minimal Polynomial Extrapolation (MMPE), the Reduced Rank Extrapolation (RRE), the Vector Epsilon Algorithm (VEA), the Topological Epsilon Algorithm (TEA), and Anderson Acceleration (AA), which are standard general techniques designed for accelerating arbitrary sequences and/or solving nonlinear equations, all fall into this framework. Their properties and their connections with quasi-Newton and Broyden methods are studied. The paper then exploits this framework to compare these methods. In the linear case, it is known that AA and GMRES are \u2018essentially\u2019 equivalent in a certain sense while GMRES and RRE are mathematically equivalent. This paper discusses the connection between AA, the RRE, the MPE, and other methods in the nonlinear case