40 research outputs found

    Special Classes of Orthogonal Polynomials and Corresponding Quadratures of Gaussian Type

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    MSC 2010: 33C47, 42C05, 41A55, 65D30, 65D32In the first part of this survey paper we present a short account on some important properties of orthogonal polynomials on the real line, including computational methods for constructing coefficients in the fundamental three-term recurrence relation for orthogonal polynomials, and mention some basic facts on Gaussian quadrature rules. In the second part we discuss our Mathematica package Orthogonal Polynomials (see [2]) and show some applications to problems with strong nonclassical weights on (0;+1), including a conjecture for an oscillatory weight on [Ā”1; 1]. Finally, we give some new results on orthogonal polynomials on radial rays in the complex plane

    Programs to Compute Distribution Functions and Critical Values for Extreme Value Ratios for Outlier Detection

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    A set of FORTRAN subprograms is presented to compute density and cumulative distribution functions and critical values for the range ratio statistics of Dixon (1951, The Annals of Mathematical Statistics ) These statistics are useful for detection of outliers in small samples.

    Weighted quadrature formulas for semi-infinite range integrals

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    Weighted quadrature formulas on the half line (a,+āˆž)(a,+\infty), a>0a>0, for non-exponentially decreasing integrandsĀ are developed. Such nn-point quadrature rules are exact for all functions of the form xā†¦xāˆ’2P(xāˆ’1)x\mapsto x^{-2}P(x^{-1}), where PP is an arbitrary algebraic polynomial of degree at most 2nāˆ’12n-1. In particular, quadrature formulas with respect to the weight function xā†¦w(x)=xĪ²logā”mxx\mapsto w(x)=x^\beta\log^m x (0ā‰¤Ī²<10\le \beta<1, māˆˆN0m\in \mathbb{N}_0)Ā are considered and several numerical examples are included

    Stratified nested and related quadrature rules

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    AbstractThe stratified nested quadrature procedure due to Laurie is discussed together with an alternative computational procedure which leads to the concept of hybrid GKP rules. In the context of the approximation of stratified nested sequences the work of Krogh and Van Snyder on the representation of the GKP rules is considered and a generalisation of this employing hybrid rules of special form is discussed

    Design of quadrature rules for MĆ¼ntz and MĆ¼ntz-logarithmic polynomials using monomial transformation

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    A method for constructing the exact quadratures for MĆ¼ntz and MĆ¼ntz-logarithmic polynomials is presented. The algorithm does permit to anticipate the precision (machine precision) of the numerical integration of MĆ¼ntz-logarithmic polynomials in terms of the number of Gauss-Legendre (GL) quadrature samples and monomial transformation order. To investigate in depth the properties of classical GL quadrature, we present new optimal asymptotic estimates for the remainder. In boundary element integrals this quadrature rule can be applied to evaluate singular functions with end-point singularity, singular kernel as well as smooth functions. The method is numerically stable, efficient, easy to be implemented. The rule has been fully tested and several numerical examples are included. The proposed quadrature method is more efficient in run-time evaluation than the existing methods for MĆ¼ntz polynomial

    A new family of extended Gauss quadratures with an interior interval constraint

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    AbstractStarting from two sequences {GĢ‚a,c,n} and {GĢ‚d,b,n} of ordinary Gauss quadrature formulae with an orthogonality measure dĻƒ on the open intervals (a,c) and (d,b), respectively. We construct a new sequence {GĢ‚a,b,e(n)} of extended Gaussian quadrature formulae for dĻƒ on (a,b), which is based on some preassigned points, the nodes of GĢ‚a,c,n, GĢ‚d,b,n and the e(n) zeros contained in (c,d) of a nonclassical orthogonal polynomial on [a,b] with respect to a linear functional. The principal result gives explicit formulae relating these polynomials and shows how their recurrence coefficients in the three-term recurrence formulae are related. Thus, a new class of Gaussian quadratures, having some nodes contained in a given interior interval, can be computed directly by standard software for ordinary Gauss quadrature formulae
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