Orthogonal Polynomials Computation and Approximation 1st edition by Walter Gautschi ISBN 0198506724 978-0198506720

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ISBN 10: 0198506724
ISBN 13: 978-0198506720
Author: Walter Gautschi

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.

Orthogonal Polynomials Computation and Approximation 1st Table of contents:

1 Basic Theory

1.1 Orthogonal polynomials

1.1.1 Definition and existence

1.1.2 Examples

1.2 Properties of orthogonal polynomials

1.2.1 Symmetry

1.2.2 Zeros

1.2.3 Discrete orthogonality

1.2.4 Extremal properties

1.3 Three-term recurrence relation

1.3.1 Monic orthogonal polynomials

1.3.2 Orthonormal polynomials

1.3.3 Christoffel–Darboux formulae

1.3.4 Continued fractions

1.3.5 The recurrence relation outside the support interval

1.4 Quadrature rules

1.4.1 Interpolatory quadrature rules and beyond

1.4.2 Gauss-type quadrature rules

1.5 Classical orthogonal polynomials

1.5.1 Classical orthogonal polynomials of a continuous variable

1.5.2 Classical orthogonal polynomials of a discrete variable

1.6 Kernel polynomials

1.6.1 Existence and elementary properties

1.6.2 Recurrence relation

1.7 Sobolev orthogonal polynomials

1.7.1 Definition and properties

1.7.2 Recurrence relation and zeros

1.8 Orthogonal polynomials on the semicircle

1.8.1 Definition, existence, and representation

1.8.2 Recurrence relation

1.8.3 Zeros

1.9 Notes to Chapter 1

2 Computational Methods

2.1 Moment-based methods

2.1.1 Classical approach via moment determinants

2.1.2 Condition of nonlinear maps

2.1.3 The moment maps G[sub(n)] and K[sub(n)]

2.1.4 Condition of G[sub(n)] : μ → γ

2.1.5 Condition of G[sub(n)] : m → γ

2.1.6 Condition of K[sub(n)] : m → ρ

2.1.7 Modified Chebyshev algorithm

2.1.8 Finite expansions in orthogonal polynomials

2.1.9 Examples

2.2 Discretization methods

2.2.1 Convergence of discrete orthogonal polynomials to continuous ones

2.2.2 A general-purpose discretization procedure

2.2.3 Computing the recursion coeffcients of a discrete measure

2.2.4 A multiple-component discretization method

2.2.5 Examples

2.2.6 Discretized modified Chebyshev algorithm

2.3 Computing Cauchy integrals of orthogonal polynomials

2.3.1 Characterization in terms of minimal solutions

2.3.2 A continued fraction algorithm

2.3.3 Examples

2.4 Modification algorithms

2.4.1 Christoffel and generalized Christoffel theorems

2.4.2 Linear factors

2.4.3 Quadratic factors

2.4.4 Linear divisors

2.4.5 Quadratic divisors

2.4.6 Examples

2.5 Computing Sobolev orthogonal polynomials

2.5.1 Algorithm based on moment information

2.5.2 Stieltjes-type algorithm

2.5.3 Zeros

2.5.4 Finite expansions in Sobolev orthogonal polynomials

2.6 Notes to Chapter 2

3 Applications

3.1 Quadrature

3.1.1 Computation of Gauss-type quadrature formulae

3.1.2 Gauss–Kronrod quadrature formulae and their computation

3.1.3 Gauss–Turán quadrature formulae and their computation

3.1.4 Quadrature formulae based on rational functions

3.1.5 Cauchy principal value integrals

3.1.6 Polynomials orthogonal on several intervals

3.1.7 Quadrature estimation of matrix functionals

3.2 Least squares approximation

3.2.1 Classical least squares approximation

3.2.2 Constrained least squares approximation

3.2.3 Least squares approximation in Sobolev spaces

3.3 Moment-preserving spline approximation

3.3.1 Approximation on the positive real line

3.3.2 Approximation on a compact interval

3.4 Slowly convergent series

3.4.1 Series generated by a Laplace transform

3.4.2 “Alternating” series generated by a Laplace transform

3.4.3 Series generated by the derivative of a Laplace transform

3.4.4 “Alternating” series generated by the derivative of a Laplace transform

3.4.5 Slowly convergent series occurring in plate contact problems

3.5 Notes to Chapter 3

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Tags: Walter Gautschi, Orthogonal Polynomials, Computation and Approximation