A Survey of Gauss-Christoffel Quadrature Formulae

Gautschi, Walter

doi:10.1007/978-3-0348-5452-8_6

Cited by 160 publications

(94 citation statements)

References 238 publications

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“…Of all quadrature rules that have received attention, Gaussian quadrature is the most investigated (cf. Gautschi [19] and the bibliography cited therein). The Gaussian quadrature formula Qn (1-6) *(¿1)/2."…”

Section: Introductionmentioning

confidence: 99%

On estimates for the weights in Gaussian quadrature in the ultraspherical case

Förster¹,

Petras²

1990

Math. Comp.

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Abstract.In this paper the Christoffel numbers av n for ultraspherical weight functions wk , wx(x) = (\ -x ) ~ ' , are investigated. Using only elementary functions, we state new inequalities, monotonicity properties and asymptotic approximations, which improve several known results. In particular, denoting by dv " the trigonometric representation of the Gaussian nodes, we obtain for À e [0, 1] the inequalities and similar results for X <£ (0, 1). Furthermore, assuming that a\ ' remains in a fixed closed interval, lying in the interior of (0, n) as n -► oo , we show that, for every fixed A > -1/2 ,

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Section: Introductionmentioning

confidence: 99%

On estimates for the weights in Gaussian quadrature in the ultraspherical case

Förster¹,

Petras²

1990

Math. Comp.

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“…As shown by Gautschi (see [16], [17] or [18]) among others, here the basic fact is the three-term recurrence relation satisfied by the sequence of orthogonal polynomials for the measure σ giving rise to certain tridiagonal matrices (Jacobi matrices) so that the eigenvalues of the n-th principal submatrix coincide with the nodes {x j } n j=1 i.e., with the zeros of the n-th orthogonal polynomial. Furthermore, the weights {A j } n j=1 can be easily expressed in terms of the first component of the normalized eigenvectors.…”

Section: Introductionmentioning

confidence: 98%

A matrix approach to the computation of quadrature formulas on the unit circle

Cantero¹,

Cruz-Barroso²,

González-Vera³

2008

Applied Numerical Mathematics

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In this paper we consider a general sequence of orthogonal Laurent polynomials on the unit circle and we first study the equivalences between recurrences for such families and Szegő's recursion and the structure of the matrix representation for the multiplication operator in Λ when a general sequence of orthogonal Laurent polynomials on the unit circle is considered. Secondly, we analyze the computation of the nodes of the Szegő quadrature formulas by using Hessenberg and five-diagonal matrices. Numerical examples concerning the family of Rogers-Szegő q-polynomials are also analyzed.

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“…Extended interpolation processes have been proposed to find the numerical solution of functional equations [ 1,2], and they are used especially for numerical quadrature (extended quadrature formulae). Quadratures of this type have been studied by several authors (see [5,6,9]). …”

Section: Introductionmentioning

confidence: 99%