A powerful numerical integration technique which uses 
refinements of the extended trapezoidal rule
to remove error terms less than order 
. The routine advocated by Press et al. (1992)
makes use of Neville's algorithm.
Romberg Integration
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References
Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., pp. 106-107, 1990.Dahlquist, G. and Bjorck, A. §7.4.1-7.4.2 in Numerical Methods. Englewood Cliffs, NJ: Prentice-Hall, 1974.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Romberg Integration." §4.3 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 134-135, 1992.Ralston, A. and Rabinowitz, P. §4.10 in A First Course in Numerical Analysis, 2nd ed. New York: McGraw-Hill, 1978.Stoer, J.; and Bulirsch, R. §3.4-3.5 in Introduction to Numerical Analysis. New York: Springer-Verlag, 1980.Ueberhuber, C. W. "Romberg Formulas." §12.3.4 in Numerical Computation 2: Methods, Software, and Analysis. Berlin: Springer-Verlag, pp. 110-111, 1997.Referenced on Wolfram|Alpha
Romberg IntegrationCite this as:
Weisstein, Eric W. "Romberg Integration." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RombergIntegration.html