Orthogonal Polynomials: Computation and Approximation (Numerical Mathematics and Scientific Computation)

✍ Scribed by Walter Gautschi

• Publisher: Oxford University Press
• Year: 2004
• Tongue: English
• Leaves: 314
• Series: Numerical Mathematics and Scientific Computation
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Cover......Page 1
Table of Contents......Page 6
Preface......Page 9
1.1 Orthogonal polynomials......Page 12
1.2 Properties of orthogonal polynomials......Page 17
1.3 Three-term recurrence relation......Page 21
1.4 Quadrature rules......Page 31
1.5 Classical orthogonal polynomials......Page 37
1.6 Kernel polynomials......Page 46
1.7 Sobolev orthogonal polynomials......Page 51
1.8 Orthogonal polynomials on the semicircle......Page 54
1.9 Notes to Chapter 1......Page 60
2.1 Moment-based methods......Page 63
2.2 Discretization methods......Page 101
2.3 Computing Cauchy integrals of orthogonal polynomials......Page 123
2.4 Modification algorithms......Page 132
2.5 Computing Sobolev orthogonal polynomials......Page 149
2.6 Notes to Chapter 2......Page 159
3.1 Quadrature......Page 163
3.2 Least squares approximation......Page 227
3.3 Moment-preserving spline approximation......Page 238
3.4 Slowly convergent series......Page 250
3.5 Notes to Chapter 3......Page 264
Bibliography......Page 272
Index......Page 294
Corrections......Page 313