Matrix Shanks Transformations

Shanks' transformation is a well know sequence transformation for accelerating the convergence of scalar sequences. It has been extended to the case of sequences of vectors and sequences of square matrices satisfying a linear difference equation with scalar coefficients. In this paper, a more general extension to the matrix case where the matrices can be rectangular and satisfy a difference equation with matrix coefficients is proposed and studied. In the particular case of square matrices, the new transformation can be recursively implemented by the matrix $arepsilon$-algorithm of Wynn. Then, the transformation is related to matrix Pad\ue9-type and Pad\ue9 approximants. Numerical experiments showing the interest of this transformation end the paper

Vector extrapolation methods with applications to solution of large systems of equations and to PageRank computations

AbstractAn important problem that arises in different areas of science and engineering is that of computing the limits of sequences of vectors {xn}, where xn∈CN 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 limn→∞xn 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 {xn} 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

Computer Science for Continuous Data:Survey, Vision, Theory, and Practice of a Computer Analysis System

Building on George Boole's work, Logic provides a rigorous foundation for the powerful tools in Computer Science that underlie nowadays ubiquitous processing of discrete data, such as strings or graphs. Concerning continuous data, already Alan Turing had applied "his" machines to formalize and study the processing of real numbers: an aspect of his oeuvre that we transform from theory to practice.The present essay surveys the state of the art and envisions the future of Computer Science for continuous data: natively, beyond brute-force discretization, based on and guided by and extending classical discrete Computer Science, as bridge between Pure and Applied Mathematics

Analysis and modification of Newton's method at singularities

For systems of nonlinear equations f=0 with singular Jacobian Vf(x*) at some solution x* E F-1(0) the behaviour of Newton's method is analysed. Under certain regularity condition Q-linear convergence is shown to be almost sure from all initial points that are sufficiently c,lose to x*. The possibility of significantly better performance by other nonlienar equation solvers is ruled out. Instead convergence acceleration is achieved by variation of the stepsize or Richardson extrapolation. If the Jacobian Vf of a possibly undetermined system is know to have a nullspace of a certain dimensional a solution of interest, and overdetermined system based on the QR or LU decomposition of Vf is used to obtain superlinear convergence