Walter Gautschi - Böcker

This is the first book on constructive methods for, and applications of orthogonal polynomials, and the first available collection of relevant Matlab codes. The book begins with a concise introduction to the theory of polynomials orthogonal on the real line (or a portion thereof), relative to a positive measure of integration. Topics which are particularly relevant to computation are emphasized. The second chapter develops computational methods for generating the coefficients in the basic three-term recurrence relation. The methods are of two kinds: moment-based methods and discretization methods. The former are provided with a detailed sensitivity analysis. Other topics addressed concern Cauchy integrals of orthogonal polynomials and their computation, a new discussion of modification algorithms, and the generation of Sobolev orthogonal polynomials. The final chapter deals with selected applications: the numerical evaluation of integrals, especially by Gauss-type quadrature methods, polynomial least squares approximation, moment-preserving spline approximation, and the summation of slowly convergent series. Detailed historic and bibliographic notes are appended to each chapter. The book will be of interest not only to mathematicians and numerical analysts, but also to a wide clientele of scientists and engineers who perceive a need for applying orthogonal polynomials.

Revised and updated, this second edition of Walter Gautschi's successful Numerical Analysis explores computational methods for problems arising in the areas of classical analysis, approximation theory, and ordinary differential equations, among others.

Approximation theory and numerical analysis are central to the creation of accurate computer simulations and mathematical models. Research in these areas can influence the computational techniques used in a variety of mathematical and computational sciences.This collection of contributed chapters, dedicated to renowned mathematician Gradimir V. Milovanović, represent the recent work of experts in the fields of approximation theory and numerical analysis. These invited contributions describe new trends in these important areas of research including theoretic developments, new computational algorithms, and multidisciplinary applications.Special features of this volume:- Presents results and approximation methods in various computational settings including: polynomial and orthogonal systems, analytic functions, and differential equations.- Provides a historical overview of approximation theory and many of its subdisciplines;- Contains new results from diverse areas of research spanning mathematics, engineering, and the computational sciences."Approximation and Computation" is intended for mathematicians and researchers focusing on approximation theory and numerical analysis, but can also be a valuable resource to students and researchers in the computational and applied sciences.

Approximation theory and numerical analysis are central to the creation of accurate computer simulations and mathematical models. Research in these areas can influence the computational techniques used in a variety of mathematical and computational sciences.This collection of contributed chapters, dedicated to renowned mathematician Gradimir V. Milovanović, represent the recent work of experts in the fields of approximation theory and numerical analysis. These invited contributions describe new trends in these important areas of research including theoretic developments, new computational algorithms, and multidisciplinary applications.Special features of this volume:- Presents results and approximation methods in various computational settings including: polynomial and orthogonal systems, analytic functions, and differential equations.- Provides a historical overview of approximation theory and many of its subdisciplines;- Contains new results from diverse areas of research spanning mathematics, engineering, and the computational sciences."Approximation and Computation" is intended for mathematicians and researchers focusing on approximation theory and numerical analysis, but can also be a valuable resource to students and researchers in the computational and applied sciences.

This companion piece to the author’s 2018 book, A Software Repository for Orthogonal Polynomials, focuses on Gaussian quadrature and the related Christoffel function. The book makes Gauss quadrature rules of any order easily accessible for a large variety of weight functions and for arbitrary precision. It also documents and illustrates known as well as original approximations for Gauss quadrature weights and Christoffel functions.The repository contains 60 datasets, each dealing with a particular weight function. Included are classical, quasi-classical, and, most of all, nonclassical weight functions and associated orthogonal polynomials.

The workshop on Applications and Computation of Orthogonal Polynomials took place March 22-28, 1998 at the Oberwolfach Mathematical Research Institute. It was the first workshop on this topic ever held at Oberwolfach. There were 46 participants from 13 countries, more than half coming from Germany and the United States, and a substantial number from Italy. A total of 23 plenary lectures were presented and 4 short informal talks. Open problems were discussed during an evening session. This volume contains refereed versions of 18 papers presented at, or submitted to, the conference. The theory of orthogonal polynomials, as a branch of classical analysis, is well established. But orthogonal polynomials play also an important role in many areas of scientific computing, such as least squares fitting, numerical integration, and solving linear algebraic systems. Though the basic tenets have their roots in 19th­ century mathematics, the use of modern computers has required the development and study of new algorithms that are accurate and robust. The computational methods and applications represented in this volume, of necessity, are incomplete, yet sufficiently varied to convey an impression of current activities in this area.