Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-28T03:56:59.986Z Has data issue: false hasContentIssue false
  • English
  • Français

A New Composite Quadrature Rule

Published online by Cambridge University Press:  03 June 2015

Weiwei Sun  and
Qian Zhang
Weiwei Sun*
Affiliation:
Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong
Qian Zhang*
Affiliation:
College of Mathematics and Computational Science, Shenzhen University, Shenzhen 518060, Guangdong, China
*
Corresponding author. Email: [email protected]
Email: [email protected]
Get access

Abstract

We present a new composite quadrature rule which is exact for polynomials of degree 2N + K – 1 with N abscissas at each subinterval and K boundary conditions. The corresponding orthogonal polynomials are introduced and the analytic formulae for abscissas and weight functions are presented. Numerical results show that the new quadrature rule is more efficient, compared with classical ones.

Keywords

65M1078A48Composite quadratureorthogonal polynomial
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 5 , Special Issue 4 , August 2013 , pp. 595 - 606
Copyright
Copyright © Global-Science Press 2013

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]Beardon, A. F., Driver, K. A. and Redivo-Zaglia, M., Quasi-orthogonality with applications to some families of classical orthogonal polynomials, Appl. Numer. Math., 48 (2004), pp. 157–168.Google Scholar
[2]Casaletto, J., Pickett, M. and Rice, J. R., A comparison of some integration problems, SIGNUM Newslett., 4 (1969), pp. 30–40.CrossRefGoogle Scholar
[3]Curtis, O., Diran, S. and Leroy, D., Generalization of the Euler-Maclaurin formula for Gauss, Laobatoo and other quadrature formulas, Rend. Mat. 7(2) (1982), pp. 523–530.Google Scholar
[4]Davis, P. J. and Rabinowitz, P., Methods of Numerical Integration, Academic Press, New York, 1975.Google Scholar
[5]Elliott, D. and Venturino, E., Sigmoidal transformations and the Euler-Maclaurin expansion for evaluating certain Hadamard finite-part integrals, Numer. Math., 77 (1997), pp. 453–465.Google Scholar
[6]Elliott, D., Sigmoidal transformations and the trapezoidal rule, J. Austral. Math. Soc. Ser. B, 40 (1998/99), pp. 77–137.Google Scholar
[7]Evans, G., Practical Numerical Intergation, John Wiley, England, 1993.Google Scholar
[8]Keast, P. and Fairweather, G., Numerical Integration, D. Reidel Prblishing, Dordrecht, Holland, 1986.Google Scholar
[9]Kress, R., Numerical Analysis, Springer-Verlag, New York, 1998.Google Scholar
[10]Leroy, D., Curtis, O. and Diran, S., Generalization of the Euler-Maclaurin sum formula to the six-point Newton-Cotes quadrature formula, Rend. Mat., 6(12) (1979), pp. 597–507.Google Scholar
[11]Szegö, G., Orthogonal polynomials, American Mathematics Society, Providence, RI, 1975.Google Scholar
[12]Smith, F., Quadrature methods based on the Euler-Maclaurin formula ans on the Clenshaw-Curtis method of integration, Numer. Math., 7 (1965), pp. 406–411.Google Scholar
[13]Xiang, S., On quadrature of Bessel transformations, J. Comput. Appl. Math., 177 (2005), pp. 231–239.CrossRefGoogle Scholar
[14]Stancu, D. D. and Sroud, A. H., Quadrature formulas with simple Gaussian nodes and multiple fixed nodes, Math. Comput., 17 (1963), pp. 384–394.Google Scholar
[15]Xu, Y., Quasi-orthogonal polynomials, quadrature and interpolation, J. Math. Anal. Appl., 182 (1994), pp. 779–799.CrossRefGoogle Scholar