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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.
Basic Theory
Orthogonal polynomials
Properties of orthogonal polynomials
Three-term recurrence relation
Quadrature rules
Classical orthogonal polynomials
Kernal polynomials
Sobolev orthogonal polynomials
Orthogonal polynomials on the semicircle
Notes to chapter 1
Computional Methods
Moment-based methods
Discretization methods
Computing Cauchy integrals of orthogonal polynomials
Modification algorithms
Computing Sobolev orthogonal polynomials
Notes to chapter 2
Applications
Quadrature
Least squares approximation
Moment-preserving spline approximation
Slowly convergent series
Notes to chapter 3