Orthogonal polynomials: applications and computation

Acta Numerica ◽  
1996 ◽  
Vol 5 ◽  
pp. 45-119 ◽  
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.

Orthogonal Polynomials

2004 ◽  
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.

scholarly journals Numerical construction of the generalized Hermite polynomials

2003 ◽  
pp. 49-63
Author(s):  
Gradimir Milovanovic ◽  
Aleksandar Cvetkovic
Keyword(s):  
Weight Function ◽  
Orthogonal Polynomials ◽  
Recurrence Relation ◽  
Asymptotic Expansions ◽  
Recurrence Relations ◽  
Hermite Polynomials ◽  
Stable Method ◽  
Numerical Construction ◽  
Three Term Recurrence Relation

In this paper we are concerned with polynomials orthogonal with respect to the generalized Hermite weight function w(x) = |x ? z|? exp(?x2) on R, where z?R and ? > ? 1. We give a numerically stable method for finding recursion coefficients in the three term recurrence relation for such orthogonal polynomials, using some nonlinear recurrence relations, asymptotic expansions, as well as the discretized Stieltjes-Gautschi procedure.

On Recurrence Relations for Sobolev Orthogonal Polynomials

1995 ◽  
Vol 26 (2) ◽  
pp. 446-467 ◽  
Author(s):  
W. D. Evans ◽  
Lance L. Littlejohn ◽  
Francisco Marcellan ◽  
Clemens Markett ◽  
Andre Ronveaux
Keyword(s):  
Orthogonal Polynomials ◽  
Recurrence Relations ◽  
Sobolev Orthogonal Polynomials

scholarly journals On Certain Properties and Applications of the Perturbed Meixner–Pollaczek Weight

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 955
Author(s):  
Abey S. Kelil ◽  
Alta S. Jooste ◽  
Appanah R. Appadu
Keyword(s):  
Integral Representation ◽  
Weight Function ◽  
Orthogonal Polynomials ◽  
Finite Order ◽  
Recurrence Relations ◽  
Pollaczek Polynomials ◽  
Perturbed Polynomials ◽  
Recursive Relations ◽  
Fisher's Information

This paper deals with monic orthogonal polynomials orthogonal with a perturbation of classical Meixner–Pollaczek measure. These polynomials, called Perturbed Meixner–Pollaczek polynomials, are described by their weight function emanating from an exponential deformation of the classical Meixner–Pollaczek measure. In this contribution, we investigate certain properties such as moments of finite order, some new recursive relations, concise formulations, differential-recurrence relations, integral representation and some properties of the zeros (quasi-orthogonality, monotonicity and convexity of the extreme zeros) of the corresponding perturbed polynomials. Some auxiliary results for Meixner–Pollaczek polynomials are revisited. Some applications such as Fisher’s information, Toda-type relations associated with these polynomials, Gauss–Meixner–Pollaczek quadrature as well as their role in quantum oscillators are also reproduced.

scholarly journals A pair of biorthogonal polynomials for the Szegö-Hermite weight function

1988 ◽  
Vol 11 (4) ◽  
pp. 763-767 ◽  
Author(s):  
N. K. Thakare ◽  
M. C. Madhekar
Keyword(s):  
Positive Integer ◽  
Weight Function ◽  
Orthogonal Polynomials ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Polynomial Sequences ◽  
Biorthogonal Polynomials

A pair of polynomial sequences{Snμ(x;k)}and{Tmμ(x;k)}whereSnμ(x;k)is of degreeninxkandTmμ(x;k)is of degreeminx, is constructed. It is shown that this pair is biorthogonal with respect to the Szegö-Hermite weight function|x|2μexp(−x2),(μ>−1/2)over the interval(−∞,∞)in the sense that∫−∞∞|x|2μexp(−x2)Snμ(x;k)Tmμ(x;k)dx=0,   ifm≠n                    ≠0,   ifm=nwherem,n=0,1,2,…andkis an odd positive integer.Generating functions, mixed recurrence relations for both these sets are obtained. Fork=1, both the above sets get reduced to the orthogonal polynomials introduced by professor Szegö.

scholarly journals On Semi-Classical Orthogonal Polynomials Associated with a Modified Sextic Freud-Type Weight

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1250
Author(s):  
Abey S. Kelil ◽  
Appanah R. Appadu
Keyword(s):  
Weight Function ◽  
Orthogonal Polynomials ◽  
Distribution Of Zeros ◽  
Classical Orthogonal Polynomials ◽  
Ladder Operators ◽  
Recurrence Coefficients ◽  
Freud Weight ◽  
General Deformation ◽  
Linear Differential ◽  
Electrostatic Interpretation

Polynomials that are orthogonal with respect to a perturbation of the Freud weight function by some parameter, known to be modified Freudian orthogonal polynomials, are considered. In this contribution, we investigate certain properties of semi-classical modified Freud-type polynomials in which their corresponding semi-classical weight function is a more general deformation of the classical scaled sextic Freud weight |x|αexp(−cx6),c>0,α>−1. Certain characterizing properties of these polynomials such as moments, recurrence coefficients, holonomic equations that they satisfy, and certain non-linear differential-recurrence equations satisfied by the recurrence coefficients, using compatibility conditions for ladder operators for these orthogonal polynomials, are investigated. Differential-difference equations were also obtained via Shohat’s quasi-orthogonality approach and also second-order linear ODEs (with rational coefficients) satisfied by these polynomials. Modified Freudian polynomials can also be obtained via Chihara’s symmetrization process from the generalized Airy-type polynomials. The obtained linear differential equation plays an essential role in the electrostatic interpretation for the distribution of zeros of the corresponding Freudian polynomials.

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