1,721,019 research outputs found
Extrapolation and rational approximation: the works of the main contributors
No full textThis book paints a fresco of the field of extrapolation and rational approximation over the last several centuries to the present through the works of their primary contributors. It can serve as an introduction to the topics covered, including extrapolation methods, Padé approximation, orthogonal polynomials, continued fractions, Lanczos-type methods, etc.; it also provides in depth discussion of the many links between these subjects. A highlight of this book is the presentation of the human side of the fields discussed via personal testimonies from contemporary researchers, their anecdotes, and their exclusive remembrances of some of the “actors.” This book shows how research in this domain started and evolved. Biographies of other scholars encountered have also been included. An important branch of mathematics is described in its historical context, opening the way to new developments. After a mathematical introduction, the book contains a precise description of the mathematical landscape of these fields spanning from the 19th century to the first part of the 20th. After an analysis of the works produced after that period (in particular those of Richardson, Aitken, Shanks, Wynn, and others), the most recent developments and applications are reviewed. Claude Brezinski is professor emeritus of mathematics at the University of Lille (France), where he has been head of the Laboratory of Numerical Analysis and Optimization for 30 years. He was the advisor of 60 doctoral students. Prof. Brezinski is founder and Editor-in-Chief of the Numerical Algorithms journal and author of over 240 papers and several books. Michela Redivo-Zaglia is professor of numerical analysis at the University of Padua (Italy). She has been vice-director of the Department of Mathematics for three years. She is a member of the Editorial Board of several journals. She published software packages, 7 scientific and didactic books, and about 80 papers. She was the organizer of many international congresses, and an invited speaker at several ones
Shanks function transformations in a vector space
Get PDFIn 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
The simplified topological ε-algorithms: software and applications
Get PDFIn this paper, we describe the Matlab toolbox EPSfun for implementing and using the simplified topological ε-algorithms for accelerating the convergence of sequences of elements of a vector space. The functions for other similar algorithms are also provided. We give applications to the solution of linear and nonlinear systems of equations and to the computation of matrix functions
Some unusual results on extrapolation methods
No full textThis paper is devoted to properties of sequence transformations and the corresponding recursive algorithms for their implementation, which were never considered. We first give necessary conditions that are satisfied if the transformed sequence converges faster than the initial one. These conditions can be used for deciding if a method is worth to be used. They also serve as the basis for defining criteria for stopping the acceleration algorithm when the best possible precision is obtained. Then, prescribing the transformed sequence, we show how to obtain the initial sequence which produces it via the transformation or via its recursive algorithm. These results show that almost any behavior is possible for the transformed sequence. A similar problem about Padé-type approximants is studied
Hirota’s bilinear method, Shanks’ transformation, and the ε-algorithms
No full textHirota's bilinear method can be quite useful in the solution of nonlinear differential and difference equations. In this paper, we show how this method can lead to a novel proof that the epsilon-algorithm of Wynn implements the Shanks' sequence transformation and, reciprocally, that the quantities it computes are expressed as ratios of Hankel determinants as given by Shanks. New identities between Hankel determinants and the quantities involved in Hirota's method are obtained, and they form the basis of our proof. Then, the same bunch of results is showed to hold also for the confluent form of the epsilon-algorithm. This treatment could also be useful for other sequence transformations and the corresponding recursive algorithms
A rational Arnoldi approach for ill-conditioned linear systems
No full textFor the solution of full-rank ill-posed linear systems a new approach based on the Arnoldi algorithm is presented. Working with regularized systems, the method theoretically reconstructs the true solution by means of the computation of a suitable function of matrix. In this sense, the method can be referred to as an iterative refinement process. Numerical experiments arising from integral equations and interpolation theory are presented. Finally, the method is extended to work in connection with the standard Tikhonov regularization with the right-hand side contaminated by noise
A Family of New Generating Functions for the Chebyshev Polynomials, Based on Works by Laplace, Lagrange and Euler
Get PDFAnalyzing, developing and exploiting results obtained by Laplace in 1785 on the Fourier-series expansion of the function (1−2αcosθ+α2)−s, we obtain a family of new expansions and generating functions for the Chebyshev polynomials. A relation between the generating functions of the Chebyshev polynomials Tn and the Gegenbauer polynomials Cn(2) is given
The genesis and early developments of Aitken’s process, Shanks’ transformation, the ε–algorithm, and related fixed point methods
Get PDFIn this paper, we trace back the genesis of Aitken’s Δ2 process and Shanks’ sequence transformation. These methods, which are extrapolation methods, are used for accelerating the convergence of sequences of scalars, vectors, matrices, and tensors. They had, and still have, many important applications in numerical analysis and in applied mathematics. They are related to continued fractions and Padé approximants. We go back to the roots of these methods and analyze the original contributions. New and detailed explanations on the building and properties of Shanks’ transformation and its kernel are provided. We then review their historical algebraic and algorithmic developments. We also analyze how they were involved in the solution of systems of linear and nonlinear equations, in particular in the methods of Steffensen, Pulay, and Anderson. Testimonies by various actors of the domain are given. The paper can also serve as an introduction to this domain of numerical analysis
Zeros of quadratic quasi-orthogonal order 2 polynomials
Get PDFCorollary 2 in [1] states that for , the quasi-orthogonal order 2 Gegenbauer polynomial has real, distinct zeros in one zero larger than and one zero smaller than This is correct provided but does not hold when for every in the range An elementary calculation shows that the quasi-orthogonal order Gegenbauer polynomial has real, distinct zeros with one zero larger than and one zero smaller than when and two distinct pure imaginary zeros when A similar error occurs in the proof of Corollary 4(i) in [1] relating to the location of the zeros of the quadratic quasi-orthogonal order Jacobi polynomial , Each error arises from a different incorrect application of Theorem VII due to Shohat (cf. [8, p. 472]). We discuss the Hilbert-Klein formulas (cf. [9, p. 145]) and indicate the overlap between two different stages of the migration process of the zeros of from the real axis to the imaginary axis (see [4] Section 3) that occurs when $n=2.
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