Abstract
The objective of this chapter is to derive and then test methods that can be used to evaluate the definite integral In most calculus textbooks the examples and problems dedicated to integration are not particularly complicated, although some require a clever combination of methods to carry out the integration. In the real world the situation is much worse. As an example, to find the deformation of an elastic body when compressed by a rigid punch it is necessary to evaluate (Gladwell [1980] Moreover, it is relatively easy to find integrals even worse than the one above. To illustrate, in the study of the emissions from a pulsar it is necessary to evaluate (Gwinn et al. [2012] where K 2 is the modified Bessel function. The point here is that effective numerical methods for evaluating integrals are needed, and our objective is to determine what they are.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bogaert, I.: Iteration-free computation of Gauss–Legendre quadrature nodes and weights. SIAM J. Sci. Comput. 36 (3), A1008–A1026 (2014). doi:10.1137/140954969. http://dx.doi.org/10.1137/140954969
Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration. Dover, New York (2007). ISBN 9780486453392. https://books.google.com/books?id=gGCKdqka0HAC
Evans, G.A., Webster, J.R.: A comparison of some methods for the evaluation of highly oscillatory integrals. J. Comput. Appl. Math. 112, 55–69 (1999). ISSN 0377-0427. doi:http://dx.doi.org/10.1016/S0377-0427(99)00213-7
Gander, W., Gautschi, W.: Adaptive quadrature—revisited. BIT 40 (1), 84–101 (2000). ISSN 0006-3835. doi:10.1023/A:1022318402393
Gladwell, G.M.L.: Contact Problems in the Classical Theory of Elasticity. Sijthoff and Noordhoff, Germantown, MD (1980)
Golub, G.H., Welsch, J.H.: Calculation of Gauss quadrature rules. Math. Comput. 23 (106), 221–230 (1969). doi:10.3934/jcd.2014.1.391
Gonnet, P.: A review of error estimation in adaptive quadrature. ACM Comput. Surv. 44 (4), 22:1–22:36 (2012). ISSN 0360-0300. doi:10.1145/ 2333112.2333117
Gwinn, C.R., Johnson, M.D., Reynolds, J.E., Jauncey, D.L., Tzioumis, A.K., Dougherty, S., Carlson, B., Del Rizzo, D., Hirabayashi, H., Kobayashi, H., Murata, Y., Edwards, P.G., Quick, J.F.H., Flanagan, C.S., McCulloch, P.M.: Noise in the cross-power spectrum of the Vela pulsar. Astrophys. J. 758 (1), 6 (2012)
Holmes, M.H.: Connections between cubic splines and quadrature rules. Am. Math. Mon. 121 (7), 661–662 (2014)
Isaacson, E., Keller, H.B.: Analysis of Numerical Methods. Dover Publications, New York (1994). ISBN 9780486680293. https://books.google.com/books?id=y77n2ySMJHUC
Iserles, A., Nørsett, S.P., Olver, S.: Highly oscillatory quadrature: the story so far. In: de Castro, A.B., Gomez, D., Quintela, P., Salgado, P. (eds.) Numerical Mathematics and Advanced Applications, pp. 97–118. Springer, Berlin (2006). ISBN 978-3-540-34287-8. doi:10.1007/978-3-540-34288-5_6
Love, C.H.: Abscissas and Weights for Gaussian Quadrature for n=2 to 100, and n=125, 150, 175, and 200. National Bureau of Standards, U.S. Government Printing Office, Washington, DC (1966)
Nenad, U., Roberts, A.J.: A corrected quadrature formula and applications. ANZIAM J. 45, E41–E56 (2008)
Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W.: NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010)
Trefethen, L.N.: Is gauss quadrature better than Clenshaw–Curtis? SIAM Rev. 50 (1), 67–87 (2008). doi:10.1137/060659831. http://dx.doi.org/10.1137/060659831
Trefethen, L.N., Weideman, J.A.C.: The exponentially convergent trapezoidal rule. SIAM Rev. 56 (3), 385–458 (2014). doi:10.1137/130932132
Waldvogel, J.: Towards a general error theory of the trapezoidal rule. In: Gautschi, W., Mastroianni, G., Rassias, T.M. (eds.) Approximation and Computation. Springer Optimization and Its Applications, vol. 42, pp. 267–282. Springer, New York (2011)
Weideman, J.A.C.: Numerical integration of periodic functions: a few examples. Am. Math. Mon. 109 (1), 21–36 (2002). ISSN 00029890. http://www.jstor.org/stable/2695765
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Holmes, M.H. (2016). Numerical Integration. In: Introduction to Scientific Computing and Data Analysis. Texts in Computational Science and Engineering, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-319-30256-0_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-30256-0_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-30254-6
Online ISBN: 978-3-319-30256-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)