Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-10T17:20:07.109Z Has data issue: false hasContentIssue false
  • English
  • Français

Simple Fourth-Degree Cubature Formulae with Few Nodes over General Product Regions

Published online by Cambridge University Press:  28 May 2015

Ran Yu ,
Zhaoliang Meng  and
Zhongxuan Luo
Ran Yu*
Affiliation:
School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, P.R. China
Zhaoliang Meng*
Affiliation:
School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, P.R. China
Zhongxuan Luo*
Affiliation:
School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, P.R. China School of Software, Dalian University of Technology, Dalian, 116620, P.R. China
*
Corresponding author.Email address:ranyu0602@sina.com
Corresponding author.Email address:mzhl@dlut.edu.cn
Corresponding author.Email address:zxluo@dlut.edu.cn
Get access

Abstract

A simple method is proposed for constructing fourth-degree cubature formulae over general product regions with no symmetric assumptions. The cubature formulae that are constructed contain at most n2 + 7n + 3 nodes and they are likely the first kind of fourth-degree cubature formulae with roughly n2 nodes for non-symmetric integrations. Moreover, two special cases are given to reduce the number of nodes further. A theoretical upper bound for minimal number of cubature nodes is also obtained.

Keywords

65D3065D32Fourth-degree cubature formulacubature formulaproduct regionnon-symmetric regionnumerical integration
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 7 , Issue 2 , May 2014 , pp. 179 - 192
Copyright
Copyright © Global Science Press Limited 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Bungartz, H.-J. and Griebel, M., Sparse grids, Acta Numer., vol. 13, no. 1 (2004), pp. 147–269.CrossRefGoogle Scholar
[2] Cools, R., Constructing cubature formulae: the science behind the art, Acta Numer., vol. 6, no. 1 (1997), pp. 1–54.CrossRefGoogle Scholar
[3] Cools, R., Mysovskikh, I. P., and Schmid, H. J., Cubature formulae and orthogonal polynomials, J. Comput. Appl. Math., vol. 127 (2001), pp. 121–152.CrossRefGoogle Scholar
[4] Gerstner, T. and Griebel, M., Numerical integration using sparse grids, Numer. Algorithms, vol. 18, no. 3-4 (1998), pp. 209–232.CrossRefGoogle Scholar
[5] Lu, J. and Darmofal, D. L., Higher-dimensional integration with Gaussian weight for applications in probabilistic design, SIAM J. Sci. Comput., vol. 26, no. 2 (2004), pp. 613–624.CrossRefGoogle Scholar
[6] Möller, H. M., Kubaturformeln mit minimaler knotenzahl, Numer. Math., vol. 25, no. 2 (1976), pp. 185–200.CrossRefGoogle Scholar
[7] Möller, H. M., Lower bounds for the number of nodes in cubature formulae, Numerische Integration, ISNM, vol. 45 (1979), pp. 221–230.Google Scholar
[8] Mysovskikh, I. P., Interpolatory cubature formulas, “Nauka”, Moscow, 1981 (in Russian).Google Scholar
[9] Novak, E. and Ritter, K., Simple cubature formulas with high polynomial exactness, Con-str. Approx., vol. 15, no. 4 (1999), pp. 499–522.Google Scholar
[10] Petras, K., Smolyak cubature of given polynomial degree with few nodes for increasing dimension, Numer. Math., vol. 93, no. 4 (2003), pp. 729–753.CrossRefGoogle Scholar
[11] Smolyak, S. A., Quadrature and interpolation formulas for tensor products of certain classes of functions, Dokl. Akad. Nauk SSSR, vol. 4 (1963), pp. 240–243.Google Scholar
[12] Stroud, A. H., Some fifth degree integration formulas for symmetric regions II, Numer. Math., vol. 9, no. 5 (1967), pp. 460–468.CrossRefGoogle Scholar
[13] Stroud, A. H., Approximate calculation of multiple integrals, vol. 431. Prentice-Hall En-glewood Cliffs, NJ, 1971.Google Scholar