Numerical Construction of Orthogonal Polynomials from a General Recurrence Formula

Biometrics ◽  
1968 ◽  
Vol 24 (3) ◽  
pp. 695 ◽  
Author(s):  
Phillip L. Emerson
Keyword(s):  
Orthogonal Polynomials ◽  
Recurrence Formula ◽  
Numerical Construction

scholarly journals Quadrature with multiple nodes, power orthogonality, and moment-preserving spline approximation, part II

2019 ◽  
Vol 13 (1) ◽  
pp. 1-27 ◽  
Author(s):  
Gradimir Milovanovic ◽  
Miroslav Pranic ◽  
Miodrag Spalevic
Keyword(s):  
Numerical Analysis ◽  
Error Analysis ◽  
Orthogonal Polynomials ◽  
Spline Approximation ◽  
Gauss Quadrature ◽  
Quadrature Formulas ◽  
Numerical Construction ◽  
Appl Math

The paper deals with new contributions to the theory of the Gauss quadrature formulas with multiple nodes that are published after 2001, including numerical construction, error analysis and applications. The first part was published in Numerical analysis 2000, Vol. V, Quadrature and orthogonal polynomials (W. Gautschi, F. Marcellan, and L. Reichel, eds.) [J. Comput. Appl. Math. 127 (2001), no. 1-2, 267-286].

Small Oscillations, Sturm Sequences, and Orthogonal Polynomials

1994 ◽  
Vol 49 (3) ◽  
pp. 445-457
Author(s):  
Michael Baake
Keyword(s):  
Orthogonal Polynomials ◽  
Secular Equation ◽  
Recurrence Formula ◽  
Spin Systems ◽  
Particle Systems ◽  
Recurrence Coefficients ◽  
Small Oscillations ◽  
One Dimensional ◽  
Physical Problems ◽  
Sturm Sequences

Abstract The relation between small oscillations of one-dimensional mechanical n-particle systems and the theory of orthogonal polynomials is investigated. It is shown how the polynomials provide a natural tool to determine the eigenfrequencies and eigencoordinates completely, where the existence of a specific two-termed recurrence formula is essential. Physical and mathematical statements are formulated in terms of the recurrence coefficients which can directly be obtained from the corresponding secular equation. Several results on Sturm sequences and orthogonal polynomials are presented with respect to the treatment of small oscillations. The relation to the numerical treatment of the generalized eigenvalue problem is discussed and further applications to physical problems from quantum mechanics, statistical mechanics, and spin systems are briefly outlined.

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.

scholarly journals Orthogonal polynomials and generalized Gauss-Rys quadrature formulae

2021 ◽  
Vol 49 (1) ◽  
Author(s):  
Gradimir V. Milovanovic ◽  
◽  
Nevena Vasovic ◽  
Keyword(s):  
Orthogonal Polynomials ◽  
Quadrature Rules ◽  
Quadrature Formulas ◽  
Numerical Construction ◽  
Quadrature Formulae ◽  
Gaussian Type ◽  
Modified Moments

Orthogonal polynomials and the corresponding quadrature formulas of Gaussian type concerni λ ng > the 1 e / v 2 en wei x gh > t f 0 unction ω(t; x) = exp λ (−= xt 1 2) / ( 2 1 − t2)−1/2 on (−1, 1), with parameters − and , are considered. For these quadrature rules reduce to the socalled Gauss-Rys quadrature formulas, which were investigated earlier by several authors, e.g., Dupuis at al 1976 and 1983; Sagar 1992; Schwenke 2014; Shizgal 2015; King 2016; Milovanovic ´ 2018, etc. In this generalized case, the method of modified moments is used, as well as a transformation of quadratures on (−1, 1) with N nodes to ones on (0, 1) with only (N + 1)/2 nodes. Such an approach provides a stable and very efficient numerical construction.

scholarly journals Orthogonal Polynomials Represented by $CW$-Spheres

10.37236/1861 ◽  
2004 ◽  
Vol 11 (2) ◽  
Author(s):  
Gábor Hetyei
Keyword(s):  
Orthogonal Polynomials ◽  
Recurrence Formula ◽  
Testing Ground ◽  
Eulerian Posets ◽  
Direct Calculations ◽  
Limited Testing

Given a sequence $\{Q_n(x)\}_{n=0}^{\infty}$ of symmetric orthogonal polynomials, defined by a recurrence formula $Q_n(x)=\nu_n\cdot x\cdot Q_{n-1}(x)-(\nu_n-1)\cdot Q_{n-2}(x)$ with integer $\nu_i$'s satisfying $\nu_i\geq 2$, we construct a sequence of nested Eulerian posets whose $ce$-index is a non-commutative generalization of these polynomials. Using spherical shellings and direct calculations of the $cd$-coefficients of the associated Eulerian posets we obtain two new proofs for a bound on the true interval of orthogonality of $\{Q_n(x)\}_{n=0}^{\infty}$. Either argument can replace the use of the theory of chain sequences. Our $cd$-index calculations allow us to represent the orthogonal polynomials as an explicit positive combination of terms of the form $x^{n-2r}(x^2-1)^r$. Both proofs may be extended to the case when the $\nu_i$'s are not integers and the second proof is still valid when only $\nu_i>1$ is required. The construction provides a new "limited testing ground" for Stanley's non-negativity conjecture for Gorenstein$^*$ posets, and suggests the existence of strong links between the theory of orthogonal polynomials and flag-enumeration in Eulerian posets.

A Characterization and a Class of Distribution Functions for the Stieltjes-Wigert Polynomials

1970 ◽  
Vol 13 (4) ◽  
pp. 529-532 ◽  
Author(s):  
T. S. Chihara
Keyword(s):  
Orthogonal Polynomials ◽  
Moment Problem ◽  
Recurrence Formula ◽  
Distribution Functions ◽  
Moment Sequence ◽  
The Moment ◽  
Image Position ◽  
Stieltjes Moment Sequence

In his classic memoir on the moment problem that bears his name, Stieltjes [2] exhibited1as an example of an indeterminate (Stieltjes) moment sequence.Stieltjes also obtained the corresponding S-fraction and thus implicitly obtained the three-term recurrence formula satisfied by the corresponding orthogonal polynomials.

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