Acceleration of convergence of vector sequences
SIAM Journal on Numerical Analysis
Convergence and stability properties of minimal polynomial and reduced rank extrapolation algorithms
SIAM Journal on Numerical Analysis
GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
Extrapolation methods for vector sequences
SIAM Review
Journal of Computational and Applied Mathematics
Extrapolation vs. projection methods for linear systems of equations
Journal of Computational and Applied Mathematics
Recursive algorithms for vector extrapolation methods
Applied Numerical Mathematics
Application of vector extrapolation methods to consistent singular linear systems
Applied Numerical Mathematics
Efficient implementation of minimal polynomial and reduced rank extrapolation methods
Journal of Computational and Applied Mathematics
Rational approximations from power series of vector-valued meromorphic functions
Journal of Approximation Theory
SIAM Journal on Matrix Analysis and Applications
Extrapolation methods for accelerating PageRank computations
WWW '03 Proceedings of the 12th international conference on World Wide Web
The PageRank Vector: Properties, Computation, Approximation, and Acceleration
SIAM Journal on Matrix Analysis and Applications
Calcolo: a quarterly on numerical analysis and theory of computation
Accelerating the Arnoldi-Type Algorithm for the PageRank Problem and the ProteinRank Problem
Journal of Scientific Computing
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An important problem that arises in different areas of science and engineering is that of computing the limits of sequences of vectors {x"n}, where x"n@?C^N with N very large. Such sequences arise, for example, in the solution of systems of linear or nonlinear equations by fixed-point iterative methods, and lim"n"-"~x"n are simply the required solutions. In most cases of interest, however, these sequences converge to their limits extremely slowly. One practical way to make the sequences {x"n} converge more quickly is to apply to them vector extrapolation methods. In this work, we review two polynomial-type vector extrapolation methods that have proved to be very efficient convergence accelerators; namely, the minimal polynomial extrapolation (MPE) and the reduced rank extrapolation (RRE). We discuss the derivation of these methods, describe the most accurate and stable algorithms for their implementation along with the effective modes of usage in solving systems of equations, nonlinear as well as linear, and present their convergence and stability theory. We also discuss their close connection with the method of Arnoldi and with GMRES, two well-known Krylov subspace methods for linear systems. We show that they can be used very effectively to obtain the dominant eigenvectors of large sparse matrices when the corresponding eigenvalues are known, and provide the relevant theory as well. One such problem is that of computing the PageRank of the Google matrix, which we discuss in detail. In addition, we show that a recent extrapolation method of Kamvar et al. that was proposed for computing the PageRank is very closely related to MPE. We present a generalization of the method of Kamvar et al. along with a very economical algorithm for this generalization. We also provide the missing convergence theory for it.