Pade and Rational Approximation

Theory and Applications

1st Edition - January 1, 1977
Editor: E.B. Safe
Language: English
eBook ISBN:
9 7 8 - 0 - 3 2 3 - 1 4 7 7 7 - 4

Padé and Rational Approximation: Theory and Applications presents the proceedings of the Conference on Rational Approximation with Emphasis on Applications of Padé Approxima… Read more

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Padé and Rational Approximation: Theory and Applications presents the proceedings of the Conference on Rational Approximation with Emphasis on Applications of Padé Approximants, held in Tampa, Florida on December 15-17, 1976. The contributors focus on the interplay of theory, computation, and physical applications.

This book is composed of six parts encompassing 44 chapters. The introductory part discusses the general theory of orthogonal polynomials that is the mathematical basis of Padé approximants and related matters evaluation. This text also examines the connection between approximants on a stepline in the ordinary Padé table and certain continued fractions and the convergence of diagonal Padé approximants to a class of functions with an even number of branch points. The following parts deal with the special functions and continued fractions of Padé approximation and the theory of rational approximations. These parts also investigate the geometric convergence of Chebyshev rational approximation on the half line, the optimal approximation by “Almost Classical” interpolation, and the incomplete polynomials approximation. The discussion then shifts to the physical applications and computations of the Padé approximants. The concluding part presents the applications of rational approximation to gun fire control and to the White Sands Missile Range Computer Facility. This part also provides a list of some open problems and conjectures concerning polynomials and rational functions.

This book is of great benefit to mathematicians, physicists, and laboratory workers.

List of Contributors and Participants

Preface

Theory of Padé Approximants and Generalizations

Padé Approximants and Orthogonal Polynomials

Convergence along Steplines in a Generalized Padé Table

N-Variable Rational Approximants

Asymptotic Behavior of the Zeros of Sequences of Padé Polynomials

Confirming the Accuracy of Padé Table Approximants

Laurent, Fourier, and Chebyshev-Padé Tables

Generalisations of the Theorem of de Montessus Using Canterbury Approximants

Rational Approximation by an Interpolation Procedure in Several Variables

The Convergence of Padé Approximants to Functions with Branch Points

Padé Approximants and Indefinite Inner Product Spaces

The Transformation of Series by the Use of Padé Quotients and More General Approximants

Special Functions and Continued Fractions

A Relationship between Lie Theory and Continued Fraction Expansions for Special Functions

Note on a Theorem of Saff and Varga

Multiple-Point Padé Tables

Application of Stieltjes Fractions to Birth-Death Processes

On Geometric Characterizations of an Indeterminate Stieltjes Moment Sequence

Rational Approximation to the Exponential Function

On the Zeros and Poles of Padé Approximants to ez. II

Two-Point Padé Tables, T-Fractions, and Sequences of Schur

Three-Term Contiguous Relations and Some New Orthogonal Polynomials

Theory of Rational Approximations

On a Problem of Saff and Varga Concerning Best Rational Approximation

A Minimization Problem Related to Padé Synthesis of Recursive Digital Filters

A Contribution to Rational Approximation on the Whole Real Line

On the Possibility of Rational Approximation

Geometric Convergence of Chebyshev Rational Approximations on the Half Line

Optimal Approximation by "Almost Classical" Interpolation

Approximation by Incomplete Polynomials (Problems and Results)

On the Cardinality of a Set of Best Complex Rational Approximations to a Real Function

Physical Applications

The Application of Padé Approximants to Critical Phenomena

Variational Principles and Matrix Padé Approximants

Operator Padé Approximants for the Bethe-Salpeter Equation of Nucleon-Nucleon Scattering

Series Summation Methods

A Variational Approach to Operator and Matrix Padé Approximants. Applications to Potential Scattering and Field Theory

Operator Padé Approximants and Three-Body Scattering

Computations

Software for Approximations or Approximation Theory as an Experimental Science

Best Rational Approximation with Negative Poles to e-x on [0, ∞] 413

Algorithms for Rational Approximations for the Gaussian Hypergeometric Function

On the Limitation and Application of Padé Approximation to the Matrix Exponential

Statistical Roundoff Error Analysis of a Padé Algorithm for Computing the Matrix Exponential

Open Problems

Summary of Panel Discussion

Minimal Realization from Data Sets—A Problem of Rational Approximation

Some Directions for Gun Fire Control Predictors

Some Open Problems Concerning Polynomials and Rational Functions

Rational Approximation at White Sands Missile Range