×

zbMATH — the first resource for mathematics

Error bounds for Gauss-Kronrod quadrature formulae. (English) Zbl 0790.41020
Summary: The Gauss-Konrod quadrature formula \(Q^{GK}_{2n+1}\) is used for a practical estimate of the error \(R^ G_ n\) of an approximate integration using the Gaussian quadrature formula \(Q^ G_ n\). Studying an often-used theoretical quality measure, for \(Q^{GK}_{2n+1}\) we prove best presently known bounds for the error constants \[ c_ s(R^{GK}_{2n+1})=\sup_{\| f^{(s)}\|_ \infty\leq 1} | R^{GK}_{2n+1}[f]| \] in the case \(s= 3n+ 2+ \kappa\), \(\kappa=\Bigl\lfloor{n+1\over 2}\Bigr\rfloor-\Bigl\lfloor{n\over 2}\Bigr\rfloor\). A comparison with the Gaussian quadrature formula \(Q^ G_{2n+1}\) shows that there exist quadrature formulae using the same number of nodes but having considerably better error constants.

MSC:
41A55 Approximate quadratures
Software:
QUADPACK
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Helmut Brass, Quadraturverfahren, Vandenhoeck & Ruprecht, Göttingen, 1977 (German). Studia Mathematica, Skript 3. · Zbl 0368.65014
[2] Helmut Brass and Klaus-Jürgen Förster, On the estimation of linear functionals, Analysis 7 (1987), no. 3-4, 237 – 258. · Zbl 0639.41019
[3] H. Brass and G. Schmeisser, Error estimates for interpolatory quadrature formulae, Numer. Math. 37 (1981), no. 3, 371 – 386. · Zbl 0462.41019 · doi:10.1007/BF01400316 · doi.org
[4] S. Ehrich, Error estimates for Gauss-Kronrod quadrature formulae, Hildesheimer Informatik-Berichte 14 (1991). · Zbl 0806.41018
[5] Walter Gautschi, Gauss-Kronrod quadrature — a survey, Numerical methods and approximation theory, III (Niš, 1987) Univ. Niš, Niš, 1988, pp. 39 – 66. · Zbl 0691.41027
[6] Th. Kaluza, Über die Koeffizienten reziproker Potenzreihen, Math. Z. 28 (1928), no. 1, 161 – 170 (German). · JFM 54.0335.03 · doi:10.1007/BF01181155 · doi.org
[7] Giovanni Monegato, Some remarks on the construction of extended Gaussian quadrature rules, Math. Comp. 32 (1978), no. 141, 247 – 252. · Zbl 0378.65018
[8] Giovanni Monegato, Stieltjes polynomials and related quadrature rules, SIAM Rev. 24 (1982), no. 2, 137 – 158. · Zbl 0494.33010 · doi:10.1137/1024039 · doi.org
[9] Sotirios E. Notaris, Gauss-Kronrod quadrature formulae for weight functions of Bernstein-Szegö type. II, J. Comput. Appl. Math. 29 (1990), no. 2, 161 – 169. · Zbl 0697.41017 · doi:10.1016/0377-0427(90)90355-4 · doi.org
[10] Franz Peherstorfer, Weight functions admitting repeated positive Kronrod quadrature, BIT 30 (1990), no. 1, 145 – 151. · Zbl 0693.41028 · doi:10.1007/BF01932139 · doi.org
[11] R. Piessens, E. de Doncker, C. Überhuber, and D. K. Kahaner, QUADPACK–A subroutine package for automatic integration, Springer Ser. in Comput. Math., vol. 1, Springer-Verlag, Berlin, 1983. · Zbl 0508.65005
[12] Philip Rabinowitz, The exact degree of precision of generalized Gauss-Kronrod integration rules, Math. Comp. 35 (1980), no. 152, 1275 – 1283. · Zbl 0461.65019
[13] Philip Rabinowitz, On the definiteness of Gauss-Kronrod integration rules, Math. Comp. 46 (1986), no. 173, 225 – 227. · Zbl 0619.41024
[14] G. Szegö, Über gewisse orthogonale Polynome, die zu einer oszillierenden Belegungsfunktion gehören, Math. Ann. 110 (1935), no. 1, 501 – 513 (German). · JFM 60.1038.01 · doi:10.1007/BF01448041 · doi.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.