Orthogonal Polynomials

Mapping Intimacies ◽

10.1093/oso/9780198506720.001.0001 ◽

2004 ◽

Cited By ~ 42

Author(s):

Walter Gautschi

Keyword(s):

Orthogonal Polynomials ◽

Spline Approximation ◽

Convergent Series ◽

Quadrature Methods ◽

Cauchy Integrals ◽

Sobolev Orthogonal Polynomials ◽

Gauss Type ◽

Scientists And Engineers ◽

Three Term Recurrence Relation

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.

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Orthogonal polynomials: applications and computation

Acta Numerica ◽

10.1017/s0962492900002622 ◽

1996 ◽

Vol 5 ◽

pp. 45-119 ◽

Cited By ~ 52

Author(s):

Walter Gautschi

Keyword(s):

Numerical Methods ◽

Weight Function ◽

Orthogonal Polynomials ◽

Rational Function ◽

Recurrence Relations ◽

Recurrence Coefficients ◽

Basic Task ◽

Sobolev Orthogonal Polynomials ◽

Gauss Type ◽

Three Term Recurrence Relation

We give examples of problem areas in interpolation, approximation, and quadrature, that call for orthogonal polynomials not of the classical kind. We then discuss numerical methods of computing the respective Gauss-type quadrature rules and orthogonal polynomials. The basic task is to compute the coefficients in the three-term recurrence relation for the orthogonal polynomials. This can be done by methods relying either on moment information or on discretization procedures. The effect on the recurrence coefficients of multiplying the weight function by a rational function is also discussed. Similar methods are applicable to computing Sobolev orthogonal polynomials, although their recurrence relations are more complicated. The paper concludes with a brief account of available software.

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Computational Methods

Orthogonal Polynomials ◽

10.1093/oso/9780198506720.003.0004 ◽

2004 ◽

Author(s):

Walter Gautschi

Keyword(s):

Orthogonal Polynomials ◽

Fundamental Problem ◽

Simple Algorithm ◽

Large Integer ◽

Discrete Measure ◽

Cauchy Integrals ◽

Sobolev Orthogonal Polynomials ◽

Chebyshev Algorithm ◽

Original Measure ◽

Modified Moments

The fundamental problem is to compute the first n recursion coefficients αk (dλ), Βk(dλ), k = 0, 1, . . . , n − 1 (cf. §1.3.1), where n ≥ 1 is a (typically large) integer and dλ a positive measure given either implicitly via moment information or explicitly. In the former case, an important aspect is the sensitivity of the problem with respect to small perturbations in the data (the first 2n moments or modified moments); this is the question of conditioning. In principle, there is a simple algorithm, essentially due to Chebyshev, that produces the desired recursion coefficients from given moment information. The effectiveness of this algorithm, however, depends critically on the conditioning of the underlying problem. If the problem is ill-conditioned, as it often is, recourse has to be made either to symbolic computation or to the explicit form of the measure. A procedure applicable in the latter case is discretization of the measure and subsequent approximation of the desired recursion coefficients by those relative to a discrete measure. Other problems calling for numerical methods are the evaluation of Cauchy integrals of orthogonal polynomials and the problem of passing from the recursion coefficients of a measure to those of a modified measure—the original measure multiplied by a rational function. Finally, Sobolev orthogonal polynomials present their own problems of calculating recursion coefficients and zeros. Orthogonal polynomials as well as their recursion coefficients are expressible in determinantal form in terms of the moments of the underlying measure. Indeed, much of the classical theory of orthogonal polynomials is moment-oriented. This is true, in particular, of a classical algorithm due to Chebyshev, which generates the recursion coefficients directly from the moments, bypassing determinants. The use of moments, unfortunately, is numerically problematic inasmuch as they give rise to severe ill-conditioning. In many cases, particularly for measures with bounded support, it is possible, however, to work with the so-called “modified moments,” which lead to better conditioned problems and a more stable analog of the Chebyshev algorithm.

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