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Computation of GaussKronrod Quadrature Rules with NonPositive Weights
 Math. Comp
, 1999
"... Recently Laurie presented a fast algorithm for the computation of (2n + 1)point GaussKronrod quadrature rules with real nodes and positive weights. We describe modifications of this algorithm that allow the computation of GaussKronrod quadrature rules with complex conjugate nodes and weights or w ..."
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Cited by 16 (5 self)
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Recently Laurie presented a fast algorithm for the computation of (2n + 1)point GaussKronrod quadrature rules with real nodes and positive weights. We describe modifications of this algorithm that allow the computation of GaussKronrod quadrature rules with complex conjugate nodes and weights or with real nodes and positive and negative weights.
On Product Integration With GaussKronrod Nodes
, 1998
"... GaussKronrod product quadrature formulas for the numerical approximation of R 1 \Gamma1 k(x)f(x) dx are shown to converge for every Riemann integrable f , and to possess optimal stability. Similar results are proved for the product formulas based on the Kronrod nodes only. An application to the ..."
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Cited by 6 (2 self)
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GaussKronrod product quadrature formulas for the numerical approximation of R 1 \Gamma1 k(x)f(x) dx are shown to converge for every Riemann integrable f , and to possess optimal stability. Similar results are proved for the product formulas based on the Kronrod nodes only. An application to the uniform convergence of approximate solutions of integral equations is given.
Stieltjes polynomials and the error of GaussKronrod quadrature formulas
, 1998
"... . The GaussKronrod quadrature scheme, which is based on the zeros of Legendre polynomials and Stieltjes polynomials, is the standard method for automatic numerical integration in mathematical software libraries. For a long time, very little was known about the error of the GaussKronrod scheme. An ..."
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Cited by 4 (3 self)
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. The GaussKronrod quadrature scheme, which is based on the zeros of Legendre polynomials and Stieltjes polynomials, is the standard method for automatic numerical integration in mathematical software libraries. For a long time, very little was known about the error of the GaussKronrod scheme. An essential progress was made only recently, based on new bounds and asymptotic properties for the Stieltjes polynomials. The purpose of this paper is to give a survey on these results. In particular, the quality of the GaussKronrod formula for smooth and for nonsmooth functions is investigated and compared with other quadrature formulas. 1. Introduction The numerical evaluation of definite integrals is an important step in many applications of mathematics. From a practical point of view, great interest lies in automatic routines, whose only input are the integrand function, the domain of integration, and a tolerance for the error. In applications, such routines are expected to give quick and...
Stopping functionals for Gaussian quadrature formulas
 on Numerical Analysis in the 20th century
, 2000
"... Gaussian formulas are among the most often used quadrature formulas in practice. In this survey, an overview is given on stopping functionals for Gaussian formulas which are of the same type as quadrature formulas, i.e., linear combinations of function evaluations. In particular, methods based on ex ..."
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Cited by 3 (1 self)
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Gaussian formulas are among the most often used quadrature formulas in practice. In this survey, an overview is given on stopping functionals for Gaussian formulas which are of the same type as quadrature formulas, i.e., linear combinations of function evaluations. In particular, methods based on extended formulas like the important GaussKronrod and Patterson schemes, and methods which are based on Gaussian nodes, are presented and compared. Key words: Gaussian quadrature formulas, error estimates, GaussKronrod formulas, stopping functionals 1 Introduction 1.1 Motivation The problem of approximating definite integrals is of central importance in many applications of mathematics. In practice, a mere approximation of an integral very often will not be satisfactory unless it is accompanied by an estimate of the error. For most quadrature formulas of practical interest, error bounds are available in the literature which use, e.g., norms of higherorder derivatives or bounds for the integ...
Weighted convergence of Lagrange interpolation based on GaussKronrod nodes
 J. Comput. Anal. Appl
"... The GaussKronrod quadrature scheme, which is based on the zeros of Legendre polynomials and Stieltjes polynomials, is the standard method for automatic numerical integration in mathematical software libraries. For a long time, very little was known about the underlying Lagrange interpolation proces ..."
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Cited by 1 (1 self)
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The GaussKronrod quadrature scheme, which is based on the zeros of Legendre polynomials and Stieltjes polynomials, is the standard method for automatic numerical integration in mathematical software libraries. For a long time, very little was known about the underlying Lagrange interpolation processes. Recently, the authors proved new bounds and asymptotic properties for the Stieltjes polynomials, and subsequently applied these results to investigate the associated interpolation processes. The purpose of this paper is to give a survey on the quality of these interpolation processes, with additional results that extend and complete the existing ones. The principal new results in this paper are necessary and sufficient conditions for weighted convergence. In particular, we show that the Lagrange interpolation polynomials are equivalent to the polynomials of best approximation in certain weighted Besov spaces. Key Words and Phrases: Lagrange interpolation, GaussKronrod nodes, weighted ...
Marcinkiewicz inequalities based on Stieltjes zeros
"... The authors find necessary and sufficient conditions for GDT weighted Marcinkiewicz inequalities based at Stieltjes zeros. Key words: Marcinkiewicz inequalities, Stieltjes polynomials 1 Introduction If P is an arbitrary polynomial of degree m \Gamma 1 and u is a weight function in [\Gamma1; 1]; then ..."
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The authors find necessary and sufficient conditions for GDT weighted Marcinkiewicz inequalities based at Stieltjes zeros. Key words: Marcinkiewicz inequalities, Stieltjes polynomials 1 Introduction If P is an arbitrary polynomial of degree m \Gamma 1 and u is a weight function in [\Gamma1; 1]; then the following identity is wellknown: 1 Z \Gamma1 jP (x)u(x)j 2 dx = m X k=1 m (u 2 ; y k )jP (y k )j 2 ; where m (u 2 ; t) is the m\Gammath Christoffel function with respect to the weight function u 2 and y k ; k = 1; : : : ; m; are the zeros of the m\Gammath orthogonal polynomial associated with the weight function u 2 : Preprint submitted to Elsevier Preprint 18 November 1998 The above identity is generally false if we replace 2 by an arbitrary p 2 (0; +1): Thus, we can investigate on the validity of the inequality 1 Z \Gamma1 jP (x)u(x)j p dx C m X k=1 m (u p ; y k )jP (y k )j p ; (1) for any absolute constant C ? 0 and for given \Gamma1 y 1 ! \Delt...
Stieltjes Polynomials
"... > n (x), the Chebyshev polynomial of the second kind, where E n+1 (x) = T n+1 (x), the Chebyshev polynomial of the first kind. For h(x) = (1 \Gamma x 2 ) \Gamma1=2 , P n (x) = T n (x) and E n+1 (x) = (1 \Gamma x 2 )U n\Gamma1 (x). A generalisation are the BernsteinSzego weight functions h(x ..."
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> n (x), the Chebyshev polynomial of the second kind, where E n+1 (x) = T n+1 (x), the Chebyshev polynomial of the first kind. For h(x) = (1 \Gamma x 2 ) \Gamma1=2 , P n (x) = T n (x) and E n+1 (x) = (1 \Gamma x 2 )U n\Gamma1 (x). A generalisation are the BernsteinSzego weight functions h(x) = (1 \Gamma x 2 ) \Sigma1=2 =ae m (x), where ae m is a polynomial of degree m that is p
STIELTJESTYPE POLYNOMIALS ON THE UNIT CIRCLE
, 2008
"... Stieltjestype polynomials corresponding to measures supported on the unit circle T are introduced and their asymptotic properties away from T are studied for general classes of measures. As an application, we prove the convergence of an associated sequence of interpolating rational functions to th ..."
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Stieltjestype polynomials corresponding to measures supported on the unit circle T are introduced and their asymptotic properties away from T are studied for general classes of measures. As an application, we prove the convergence of an associated sequence of interpolating rational functions to the corresponding Carathéodory function. In turn, this is used to give an estimate of the rate of convergence of certain quadrature formulae that resemble the GaussKronrod rule, provided that the integrand is analytic in a neighborhood of T.
Article electronically published on October 27, 2008 STIELTJESTYPE POLYNOMIALS ON THE UNIT CIRCLE
"... Abstract. Stieltjestype polynomials corresponding to measures supported on the unit circle T are introduced and their asymptotic properties away from T are studied for general classes of measures. As an application, we prove the convergence of an associated sequence of interpolating rational functi ..."
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Abstract. Stieltjestype polynomials corresponding to measures supported on the unit circle T are introduced and their asymptotic properties away from T are studied for general classes of measures. As an application, we prove the convergence of an associated sequence of interpolating rational functions to the corresponding Carathéodory function. In turn, this is used to give an estimate of the rate of convergence of certain quadrature formulae that resemble the GaussKronrod rule, provided that the integrand is analytic in a neighborhood of T. 1.
STIELTJESTYPE POLYNOMIALS ON THE UNIT CIRCLE
"... Abstract. Stieltjestype polynomials corresponding to measures supported on the unit circle T are introduced and their asymptotic properties away from T are studied for general classes of measures. As an application, we prove the convergence of an associated sequence of interpolating rational functi ..."
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Abstract. Stieltjestype polynomials corresponding to measures supported on the unit circle T are introduced and their asymptotic properties away from T are studied for general classes of measures. As an application, we prove the convergence of an associated sequence of interpolating rational functions to the corresponding Carathéodory function. In turn, this is used to give an estimate of the rate of convergence of certain quadrature formulae that resemble the GaussKronrod rule, provided that the integrand is analytic in a neighborhood of T. 1.