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420
Calculation of Gauss–Kronrod Quadrature Rules
 Mathematics of Computation
, 1997
"... Abstract. The Jacobi matrix of the (2n+1)point GaussKronrod quadrature rule for a given measure is calculated efficiently by a fiveterm recurrence relation. The algorithm uses only rational operations and is therefore also useful for obtaining the JacobiKronrod matrix analytically. The nodes and ..."
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Cited by 19 (0 self)
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Abstract. The Jacobi matrix of the (2n+1)point GaussKronrod quadrature rule for a given measure is calculated efficiently by a fiveterm recurrence relation. The algorithm uses only rational operations and is therefore also useful for obtaining the JacobiKronrod matrix analytically. The nodes
COMPUTATION OF GAUSSKRONROD QUADRATURE RULES
"... Abstract. Recently Laurie presented a new algorithm for the computation of (2n+1)point GaussKronrod quadrature rules with real nodes and positive weights. This algorithm first determines a symmetric tridiagonal matrix of order 2n + 1 from certain mixed moments, and then computes a partial spectral ..."
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Abstract. Recently Laurie presented a new algorithm for the computation of (2n+1)point GaussKronrod quadrature rules with real nodes and positive weights. This algorithm first determines a symmetric tridiagonal matrix of order 2n + 1 from certain mixed moments, and then computes a partial
COMPUTATION OF GAUSSKRONROD QUADRATURE RULES
"... Abstract. Recently Laurie presented a new algorithm for the computation of (2n+1)point GaussKronrod quadrature rules with real nodes and positive weights. This algorithm first determines a symmetric tridiagonal matrix of order 2n + 1 from certain mixed moments, and then computes a partial spectral ..."
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Abstract. Recently Laurie presented a new algorithm for the computation of (2n+1)point GaussKronrod quadrature rules with real nodes and positive weights. This algorithm first determines a symmetric tridiagonal matrix of order 2n + 1 from certain mixed moments, and then computes a partial
Generalized Stieltjes Polynomials and Rational GaussKronrod Quadrature
"... Generalized Stieltjes polynomials are introduced and their asymptotic properties outside the support of the measure studied. As applications, we prove the convergence of sequences of interpolating rational functions, whose poles are partially fixed, to Markov functions and give an asymptotic estimat ..."
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estimate of the error of rational GaussKronrod quadrature formulae when functions which are analytic on some neighbourhood of the set of integration are considered.
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 5 (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
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 20 (3 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
Ultraspherical GaussKronrod Quadrature is not possible for lambda > 3
"... . With the help of a new representation of the Stieltjes polynomial it is shown by using Bessel functions that the Stieltjes polynomial with respect to the ultraspherical weight function with parameter has only few real zeros for > 3 and suciently large n . Since the nodes of the GaussKronrod ..."
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Cited by 10 (1 self)
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quadrature formulae subdivide into the zeros of the Stieltjes polynomial and the Gaussian nodes, it follows immediately that GaussKronrod quadrature is not possible for > 3 . On the other hand, for = 3 and suciently large n , even partially positive GaussKronrod quadrature is possible. Key Words
ASYMPTOTICS FOR STIELTJES POLYNOMIALS, PADÉTYPE APPROXIMANTS, AND GAUSSKRONROD QUADRATURE
, 2002
"... We study the asymptotic properties of Stieltjes polynomials outside the support of the measure as well as the asymptotic behaviour of their zeros. These properties are used to estimate the rate of convergence of sequences of rational functions, whose poles are partially fixed, which approximate M ..."
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Markovtype functions. An estimate for the speed of convergence of GaussKronrod quadrature formula in the case of analytic functions is also given.
Multivariate GaussKronrod Quadrature for Simultaneous Integrands in Octave/MATLAB
, 2014
"... ar ..."
CONSTRUCTION OF GAUSSKRONRODHERMITE QUADRATURE AND CUBATURE FORMULAS
"... Abstract. We study GaussKronrod quadrature formula for Hermite weight function for the particular cases n = 1, 2, 3, we introduce a new GaussKronrodHermite cubature formula and we describe the form of the weights and nodes. 1. Introduction. Quadrature ..."
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Abstract. We study GaussKronrod quadrature formula for Hermite weight function for the particular cases n = 1, 2, 3, we introduce a new GaussKronrodHermite cubature formula and we describe the form of the weights and nodes. 1. Introduction. Quadrature
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