A. C. Aitken

Bio: A. C. Aitken is an academic researcher. The author has contributed to research in topics: Bernoulli's principle & Bernoulli polynomials. The author has an hindex of 1, co-authored 1 publications receiving 477 citations.

XXV.—On Bernoulli's Numerical Solution of Algebraic Equations

Abstract: The aim of the present paper is to extend Daniel Bernoulli's method of approximating to the numerically greatest root of an algebraic equation. On the basis of the extension here given it now becomes possible to make Bernoulli's method a means of evaluating not merely the greatest root, but all the roots of an equation, whether real, complex, or repeated, by an arithmetical process well adapted to mechanical computation, and without any preliminary determination of the nature or position of the roots. In particular, the evaluation of complex roots is extremely simple, whatever the number of pairs of such roots. There is also a way of deriving from a sequence of approximations to a root successive sequences of ever-increasing rapidity of convergence.

Non‐linear Transformations of Divergent and Slowly Convergent Sequences

Abstract: This paper discusses a family of non-linear sequence-to-sequence transformations designated as ek, ekm, ẽk, and ed. A brief history of the transforms is related and a simple motivation for the transforms is given. Examples are given of the application of these transformations to divergent and slowly convergent sequences. In particular the examples include numerical series, the power series of rational and meromorphic functions, and a wide variety of sequences drawn from continued fractions, integral equations, geometry, fluid mechanics, and number theory. Theorems are proven which show the effectiveness of the transformations both in accelerating the convergence of (some) slowly convergent sequences and in inducing convergence in (some) divergent sequences. The essential unity of these two motives is stressed. Theorems are proven which show that these transforms often duplicate the results of well-known, but specialized techniques. These special algorithms include Newton's iterative process, Gauss's numerical integration, an identity of Euler, the Pade Table, and Thiele's reciprocal differences. Difficulties which sometimes arise in the use of these transforms such as irregularity, non-uniform convergence to the wrong answer, and the ambiguity of multivalued functions are investigated. The concepts of antilimit and of the spectra of sequences are introduced and discussed. The contrast between discrete and continuous spectra and the consequent contrasting response of the corresponding sequences to the e1 transformation is indicated. The characteristic behaviour of a semiconvergent (asymptotic) sequence is elucidated by an analysis of its spectrum into convergent components of large amplitude and divergent components of small amplitude.

Extrapolation methods for accelerating PageRank computations

Abstract: We present a novel algorithm for the fast computation of PageRank, a hyperlink-based estimate of the ''importance'' of Web pages. The original PageRank algorithm uses the Power Method to compute successive iterates that converge to the principal eigenvector of the Markov matrix representing the Web link graph. The algorithm presented here, called Quadratic Extrapolation, accelerates the convergence of the Power Method by periodically subtracting off estimates of the nonprincipal eigenvectors from the current iterate of the Power Method. In Quadratic Extrapolation, we take advantage of the fact that the first eigenvalue of a Markov matrix is known to be 1 to compute the nonprincipal eigenvectors using successive iterates of the Power Method. Empirically, we show that using Quadratic Extrapolation speeds up PageRank computation by 25-300% on a Web graph of 80 million nodes, with minimal overhead. Our contribution is useful to the PageRank community and the numerical linear algebra community in general, as it is a fast method for determining the dominant eigenvector of a matrix that is too large for standard fast methods to be practical.

Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series

Abstract: Slowly convergent series and sequences as well as divergent series occur quite frequently in the mathematical treatment of scientific problems. In this report, a large number of mainly nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series are discussed. Some of the sequence transformations of this report as for instance Wynn's $\epsilon$ algorithm or Levin's sequence transformation are well established in the literature on convergence acceleration, but the majority of them is new. Efficient algorithms for the evaluation of these transformations are derived. The theoretical properties of the sequence transformations in convergence acceleration and summation processes are analyzed. Finally, the performance of the sequence transformations of this report are tested by applying them to certain slowly convergent and divergent series, which are hopefully realistic models for a large part of the slowly convergent or divergent series that can occur in scientific problems and in applied mathematics.

Studies in perturbation theory. x. lower bounds to energy eigenvalues in perturbation-theory ground state

Abstract: The eigenvalue problem $H\ensuremath{\Psi}=E\ensuremath{\Psi}$ in quantum theory is conveniently studied by means of the partitioning technique. It is shown that, if $\mathcal{E}$ is a real variable, one may construct a function ${\mathcal{E}}_{1}=f(\mathcal{E})$ such that each pair $\mathcal{E}$ and ${\mathcal{E}}_{1}$ always bracket at least one true eigenvalue $E$. If $\mathcal{E}$ is chosen as an upper bound by means of, e.g., the variation principle, the function ${\mathcal{E}}_{1}$ is hence going to provide a lower bound. The reaction operator $t$ associated with the perturbation problem $H={H}_{0}+V$ for a positive-definite perturbation $V$ is studied in some detail, and it is shown that a lower bound to $t$ may be constructed in a finite number of operations by using the idea of "inner projection" closely associated with the Aronszajn projection previously utilized in the method of intermediate Hamiltonians. By means of truncated basic sets one can now obtain not only upper bounds but also useful lower bounds which converge towards the correct eigenvalues when the set becomes complete. The method is applied to the Brillouin-type perturbation theory, and lower bounds may be obtained either by pure expansion methods, by inner projections, or by a combination of both approaches leading to perturbation expansions with estimated remainders. The applications to Schr\"odinger's perturbation theory are also outlined. The method is numerically illustrated a study of lower bounds to the ground-state energies of the helium-like ions: He, ${\mathrm{Li}}^{+}$, ${\mathrm{B}}^{+2}$, etc.