This paper presents a general framework for Shanks transformations of sequences of elements in a vector space. It is shown that Minimal Polynomial Extrapolation (MPE), Modified Minimal Polynomial Extrapolation (MMPE), Reduced Rank Extrapolation (RRE), the Vector Epsilon Algorithm (VEA), the Topological Epsilon Algorithm (TEA), and Anderson Acceleration (AA), which are standard general techniques designed to accelerate arbitrary sequences and/or solve 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 “essentially” 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.

An important problem in Web search is determining the importance of each page. After introducing the main characteristics of this problem, we will see that, from the mathematical point of view, it could be solved by computing the left principal eigenvector (the PageRank vector) of a matrix related to the structure of the Web by using the power method. We will give expressions of the PageRank vector and study the mathematical properties of the power method. Various Padé‐style approximations of the PageRank vector will be given. Since the convergence of the power method is slow, it has to be accelerated. This problem will be discussed. Recently, several acceleration methods were proposed. We will give a theoretical justification for these methods. In particular, we will generalize the recently proposed Quadratic Extrapolation, and we interpret it on the basis of the method of moments of Vorobyev, and as a Krylov subspace method. Acceleration results are given for the various ε‐algorithms, and for the E‐algo...