Summary
We consider the problem of optimal quadratures for integrandsf: [−1,1]→ℝ which have an analytic extension\(\bar f\) to an open diskD r of radiusr about the origin such that\(\left| {\bar f} \right|\)≦1 on\(\bar D_r \). Ifr=1, we show that the penalty for sampling the integrand at zeros of the Legendre polynomial of degreen rather than at optimal points, tends to infinity withn. In particular there is an “infinite” penalty for using Gauss quadrature. On the other hand, ifr>1, Gauss quadrature is almost optimal. These results hold for both the worst-case and asymptotic settings.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bakhvalov, N.S.: On the optimal speed of integrating analytic functions (in Russian). Zh. Vychisl. Mat. Mat. Fiz.7, 1011–1020 (1967). (English Transl.): U.S.S.R. Comput. Math. Math. Phys.7, 63–75 (1967)
Bakhvalov, N.S.: On the optimality of linear methods for operator approximation in convex classes of functions (in Russian). Zh. Vychisl. Mat. Mat. Fiz.11, 1014–1018 (1971). (English Transl.): U.S.S.R. Comput. Math. Math. Phys.11, 244–249 (1971)
Barnhill, R.E.: Asymptotic properties of minimum norm and optimal quadratures. Numer. Math.12, 384–393 (1968)
Bojanov, B.D.: Best quadrature formula for a certain class of analytic functions. Zastosow. Mat.14, 441–447 (1974)
Bojanov, B.D.: Optimal rate of integration and ε-entropy of a class of analytic functions (in Russian). Mat. Zametki14, 3–10 (1973). (English Transl.): Math. Notes19, 551–556 (1973)
Chawla, M.M.: Asymptotic estimates for the error in the Gauss-Legendre quadrature formula. Comput. J.11, 339–340 (1968)
Chawla, M.M., Jain, M.K.: Error estimates for Gauss quadrature formulas for analytic functions. Math. Comput.22, 82–90 (1968)
Chawla, M.M., Jain, M.K.: Asymptotic error estimates for the Gauss quadrature formula. Math. Comput.22, 91–97 (1968)
Gautschi, W., Varga, R.S.: Error Bounds for Gaussian Quadrature of Analytic Functions. SIAM J. Numer. Anal.20, 1170–1186 (1983)
Larkin, F.M.: Optimal approximation in Hilbert spaces with reproducing kernel functions. Math. Comput.24, 911–921 (1970)
Pinkus, A.: Asymptotic minimum norm quadrature formulae. Numer. Math.24, 163–175 (1975)
Stenger, F.: Bounds on the error of Gauss-type quadratures. Numer. Math.8, 150–160 (1966)
Szegö, G.: Orthogonal Polynomials. New York: Am. Math. Soc. 1939
Traub, J.F., Woźniakowski, H.: A General Theory of Optimal Algorithms. New York: Academic Press 1980
Trojan, J.M.: Asymptotic model for linear problems (In preparation)
Author information
Authors and Affiliations
Additional information
This research was supported in part by the National Science Foundation under Grants MCS-8203271 and MCS-8303111
This research was supported in part by the National Science Foundation under Grant MCS-8923676
Rights and permissions
About this article
Cite this article
Kowalski, M.A., Werschulz, A.G. & Woźniakowski, H. Is Gauss quadrature optimal for analytic functions?. Numer. Math. 47, 89–98 (1985). https://doi.org/10.1007/BF01389877
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01389877