Abstract
We study the trigonometric degree of pairs of embedded cubature rules for the approximation of two-dimensional integrals, where the basic cubature rule is a Fibonacci lattice rule. The embedded cubature rule is constructed by simply doubling the points which results in adding a shifted version of the basic Fibonacci rule. An explicit expression is derived for the trigonometric degree of this particular extension of the Fibonacci rule based on the index of the Fibonacci number.
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References
Beckers, M., Cools, R.: A relation between cubature formulae of trigonometric degree and lattice rules. In: H. Brass, G. Hämmerlin (eds.) Numerical Integration IV, pp. 13–24. Birkhäuser Verlag, Basel (1993)
Cools, R., Haegemans, A.: Optimal addition of knots to cubature formulae for planar regions. Numer. Math. 49, 269–274 (1986)
Cools, R., Haegemans, A.: A lower bound for the number of function evaluations in an error estimate for numerical integration. Constr. Approx. 6, 353–361 (1990)
Cools, R., Nuyens, D.: A Belgian view on lattice rules. In: A. Keller, et al. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2006, pp. 3–21. Springer (2008)
Cools, R., Sloan, I.: Minimal cubature formulae of trigonometric degree. Math. Comp. 65(216), 1583–1600 (1996)
Davis, P., Rabinowitz, P.: Methods of Numerical Integration. Academic Press, London (1984)
Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics, 2nd edn. Addison-Wesley (1994)
Niederreiter, H., Sloan, I.: Integration of nonperiodic functions of two variables by Fibonacci lattice rules. J. Comput. Appl. Math. 51, 57–70 (1994)
Sloan, I., Joe, S.: Lattice Methods for Multiple Integration. Oxford University Press (1994)
Sloan, I., Kachoyan, P.: Lattice methods for multiple integration: theory, error analysis and examples. SIAM J. Numer. Anal. 24, 116–128 (1987)
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© 2009 Springer-Verlag Berlin Heidelberg
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Cools, R., Nuyens, D. (2009). Extensions of Fibonacci Lattice Rules. In: L' Ecuyer, P., Owen, A. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2008. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04107-5_15
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DOI: https://doi.org/10.1007/978-3-642-04107-5_15
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Online ISBN: 978-3-642-04107-5
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