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Numerical Integration

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  • First Online:
International Encyclopedia of Statistical Science

  • 373 Accesses

One of the most common mathematical operations in scientific computing is quadrature, the evaluation of a definite integral. It is used to determine volume, mass, or total charge, for example. In the evaluation of probabilities, of expectations, and of marginal or conditional densities, integration is the basic operation.

Most of the integrals and differential equations of interest in real-world applications do not have closed-form solutions; hence, their solutions must be approximated or estimated numerically.

The general problem is to approximate or estimate

$$I ={ \int \nolimits }_{D}f(x)\,\mathrm{d}x.$$
(1)

There are basically two approaches. One is based on sums of integrals of approximations of the integrand over subregions of the domain:

$${\int \nolimits }_{D}f(x)\,\mathrm{d}x \approx {\sum \limits_{i=0}^{n}}{ \int \nolimits }_{{D}_{i}}\tilde{{f}}_{i}(x)\,\mathrm{d}x,$$
(2)

where \({\cup }_{i=0}^{n}{D}_{i} = D\) and \(\tilde{{f}}_{i}(x) \approx f(x)\) within \({D}_{i}\).

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References and Further Reading

  • Evans M, Schwartz T (2000) Approximating integrals via Monte Carlo and deterministic methods. Oxford University Press, Oxford, UK

    MATH  Google Scholar 

  • Flournoy N, Tsutakawa RK (1991) Statistical multiple integration. American Mathematical Society, Providence, Rhode Island

    MATH  Google Scholar 

  • Gentle JE (2003) Random number generation and Monte Carlo methods. Springer, New York

    MATH  Google Scholar 

  • Gentle JE (2009) Computational statistics. Springer, New York

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. George Mason University, Fairfax, VA, USA

    James E. Gentle (Professor of Computational Statistics)

Authors
  1. James E. Gentle
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Editor information

Editors and Affiliations

  1. Department of Statistics and Informatics, Faculty of Economics, University of Kragujevac, City of Kragujevac, Serbia

    Miodrag Lovric

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© 2011 Springer-Verlag Berlin Heidelberg

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Gentle, J.E. (2011). Numerical Integration. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_45

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