Publications on numerical integration

Color index

  1. B. Vandewoestyne and R. Cools, On obtaining higher order convergence for smooth periodic functions, Journal of Complexity, Vol 24/3, 328-340, 2008.
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  2. J. Van Deun and R. Cools, Integrating products of Bessel functions with an additional exponential or rational factor, Computer Physics Communications, Vol 178/8: 578-590, 2008.
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  3. B. Vandewoestyne and R. Cools, On obtaining higher order convergence for smooth periodic functions, Journal of Complexity, 2007.

  4. R. Cools and D. Nuyens, Construction of copy rules

  5. B. Vandewoestyne, R. Cools and T. Warnock, On obtaining quadratic and cubic error convergence using weighted Kronecker-sequences, Computing, Vol 80: 75-94, 2007.
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  6. G. Takhtamyshev, B. Vandewoestyne and R. Cools, Quasi-random integration in high dimensions, Mathematics and Computers in Simulation, Vol 73 (5): 309-319, 2007.
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  7. J. Van Deun and R. Cools, Algorithm 858: Computing infinite range integrals of an arbitrary product of Bessel functions, ACM Trans. Math. Software, Vol 32 (4): 580-596, 2006.
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  8. J. Van Deun and R. Cools, A stable recurrence for the incomplete gamma function with imaginary second argument, Numerische Mathematik, Vol 104: 445-456, 2006.
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  9. T. Pillards, B. Vandwoestyne and R. Cools, Minimizing the L2 and L∞ star discrepancies of a single point in the unit hypercube, J. Comput. Appl. Math., Vol. 197 (1): 282-285, 2006.
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  10. R. Cools, F.Y. Kuo and D. Nuyens, Constructing embedded lattice rules for multivariate integration, SIAM J. Sci. Comput., Vol 28 (6): 2162-2188, 2006.
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  11. J. Gonzales, F. Tuerlinckx, P. De Boeck and R. Cools, Numerical integration in logistic-normal models, Computational Statistics and Data Analysis, Vol 51 (3): 1535-1548, 2006.
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  12. T. Pillards and R. Cools, Using Box-Muller with low discrepancy points, Computational Science and Its Applications - ICCSA 2006, Proceedings Part V (Gavrilova, M. et al, eds), vol 3984, Lecture Notes in Computer Science, pp. 780-788, 2006

  13. R. Cools, More about cubature formulas and densest lattice packings, East Journal on Approximations, 12(1): 37-42, 2006.

  14. D. Nuyens and R. Cools, Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces, Math. Comp, 75: 903-920, 2006.
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  15. B. Vandewoestyne and R. Cools, Good permutations for deterministic scrambled Halton sequences in terms of L2-discrepancy, J. Comput. Appl. Math., 189(1-2): 341-361, 2006.
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  16. D. Nuyens and R. Cools, Fast component-by-component construction of rank-1 lattice rules with a non-prime number of points, J. of Complexity, 22 (1), pp 4-28, 2006.
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  17. T. Pillards and R. Cools, A note on E. Thiémard's algorithm to compute bounds for the star discrepancy, J. of Complexity, 21(3), pp 320-323, 2005.
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  18. T. Pillards and R. Cools. Transforming Low-Discrepancy Sequences from a Cube to a Simplex, J. Comput. Appl. Math., Vol. 174 (1): 29-42, 2005.
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  19. R. Cools and H. Govaert. Five- and six-dimensional lattic rules generated by structured matrices, Journal of Complexity, 19(6): 715-729, 2003.
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  20. R. Cools and H.J. Schmid. On the (non)-existence of some cubature formulas: gaps between a theory and its applications, Journal of Complexity, 19(3): 403-405, 2003.
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  21. R. Cools. An encyclopedia of cubature formulas, Journal of Complexity, 19(3): 445-453, 2003.
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  22. R. Cools. Extrapolation and adaptivity in software for automatic numerical integration, Numer. Algorithms, 34: 259-269, 2003.

  23. K. Kim, R. Cools, L.Gr. Ixaru. Extended quadrature rules for oscillatory integrands. Appl. Numer. Math., 46: 59-73, 2003.
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  24. R. Cools. Advances in multidimensional integration. J. Comp. Appl. Math., 149 (1): 1-12, 2002.
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  25. K.J. Kim, R. Cools and L. Ixaru. Quadrature rules using first derivatives for oscillatory integrands. J. Comp. Appl. Math., 140 (1-2): 479-497, 2002.
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  26. R. Cools and Juan Carlos Santos-Leon. Cubature formulas of a nonalgebraic degree of precision. Constructive Approximation, 18(2): 223-240, 2002.
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  27. R. Cools and E. Novak. Spherical product algorithms and the integration of smooth functions with one singular point. SIAM Journal on Numerical Analysis, 39: 1132--1145, 2001.
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  28. R. Cools and J.N. Lyness. Three- and four-dimensional K-optimal lattice rules of moderate trigonometric degree. Mathematics of Computation, 70: 1549-1567, 2001.
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  29. R. Cools and K.J. Kim. Rotation Invariant Cubature Formulas over the n-Dimensional Unit Cube. J. Comput. Appl. Math., 132: 15-32, 2001.
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  30. R. Cools, I. P. Mysovskikh and H.J. Schmid. Cubature Formulae and Orthogonal Polynomials. Journal of Computational and Applied Mathematics, 127: 121-152, 2001.
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  31. R. Cools and K.J. Kim. A survey of known and new cubature formulas for the unit disk. Korean Journal of Computational & Applied Mathematics, 7(3): 477-485, 2000.
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  32. J.N. Lyness and R. Cools. Notes on a search for optimal lattice rules. In M.V. Noskov, ed., Cubature formulae and their applications - Proceedings of the V International Conference, pages 259-273 Krasnoyarsk State Technical University, 2000.

  33. R. Cools. Monomial cubature rules since ``Stroud'': a compilation - part 2. J. Comput. Appl. Math., 112(1-2): 21--27, 1999.
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  34. R. Cools, E. Novak and K. Ritter. Smolyak's construction of cubature formulas of arbitrary trigonometric degree. Computing, 62(2): 147 -- 162, 1999.
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  35. P. Bekaert, R. Cools and Y. D. Willems. An Empirical Comparison of Monte Carlo Radiosity Algorithms. In V. Skala, editor, Proceedings of the 7-th International Conference in Central Europe on Computer Graphics, Visualization and Interactive Digital Media'99, pages 9-16, University of West Bohemia, Plzen, Czech Republic, 1999.

  36. R. Cools and B. Maerten. A hybrid subdivision strategy for adaptive integration routines. Journal of Universal Computer Science, 4(5): 485 -- 499, 1998.
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  37. R. Cools and A. Reztsov. Different quality indexes for lattice rules. J. Complexity, 13: 235--258, 1997.

  38. R. Cools. Constructing cubature formulae: the science behind the art In A. Iserles, editor, Acta Numerica, Vol 6, pages 1-54, Cambridge University Press, 1997.

  39. R. Cools. The approximation of low-dimensional integrals: available tools and trends. In A. Sydow, editor, Numerical Mathematics, 15th IMACS World Congress on Scientific Computation, Modelling and Applied Mathematics, Vol. 2, pages 469-474, Wissenschaft & Technik Verlag, Berlin, 1997.

  40. D. Laurie and L. Pluym and R. Cools. Design and Implementation of a C++ package for two-dimensional numerical integration. In L. M. Venter and R. R. Lombard, editors, South African Institute of Computer Science and Information Technology: Proceedings of the 1997 National Research and Development Conference, pages 162--168, Potchefstroom University for Christian Higher Education, Vanderbijlpark, 1997.

  41. B. Maerten and R. Cools. An interactive program to approximate double integrals: an easy to use interface for Cubpack++. SIGNUM Newsletter, ACM, 32(3): 2--8,1997.

  42. R. Cools, D. Laurie and L. Pluym. Algorithm 764: Cubpack++: A C++ package for automatic two-dimensional cubature. ACM Trans. Math. Software, 23: 1--15, 1997.
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  43. R. Cools and I.H. Sloan. Minimal cubature formulae of trigonometric degree. Math. Comp., 65: 1583--1600, 1996.
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  44. R. Cools and P. Dellaportes. The Role of Embedded Integration Rules in Bayesian Statistics. Statistics and Computing, 6: 245--250, 1996.

  45. J.N. Lyness and R. Cools A survey of numerical cubature over triangles. In W. Gautschi, editor, Mathematics of computation 1943-1993: A half-century of computational mathematics, Proceedings of Symposia in Applied Mathematics, Vol. 48, pages 127-150, American Mathematical Society, Providence, Rhode Island, 1994.

  46. P. Verlinden and R. Cools. Proof of a conjectured asymptotic expansion for the approximation of surface integrals. Math. Comp. 63:717-726, 1994.

  47. R. Cools and A. Haegemans. An imbedded family of cubature formulae for n-dimensional product regions. J. Comput. Appl. Math. 51:251-262, 1994.

  48. P. Verlinden and R. Cools. The algebraic construction of a minimal cubature formula of degree 11 for the square. In M.V. Noskov, editor, Cubature Formulas and their Applications(Russian), pages 13-23, Krasnoyarsk, 1994.

  49. R. Cools and P. Rabinowitz. Monomial cubature rules since ``Stroud'': a compilation. J. Comput. Appl. Math., 48:309-326, 1993.

  50. J. Berntsen, R. Cools and T.O. Espelid. Algorithm 720: An algorithm for automatic integration over a collection of 3-dimensional simplices. ACM Trans. Math. Software, 19:320-332, 1993.
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  51. R. Cools and H.J. Schmid. A new lower bound for the number of nodes in cubature formulae of degree 4n+1 for some circularly symmetric integrals. In H. Brass and G. Hämmerlin, editors, Numerical Integration IV, pages 57-66, Birkhäuser Verlag, 1993.

  52. M. Beckers and R. Cools. A relation between cubature formulae of trigonometric degree and lattice rules. In H. Brass and G. Hämmerlin, editors, Numerical Integration IV, pages 13-24, Birkhäuser Verlag, 1993.

  53. R. Cools. The art of constructing cubature formulae. In C.J. Wright, editor, Proceedings of the nineteenth South African Symposium on Numerical Mathematics, pages 15-33, SANUM & Dept. of Computer Science, University of Natal, Durban, 1993.

  54. P. Verlinden and R. Cools. On cubature formulae of degree 4k+1 attaining Möller's lower bound for integrals with circular symmetry. Numer. Math., 61:395-407, 1992.
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  55. R. Cools. A survey of methods for constructing cubature formulae. In T.O. Espelid and A. Genz, editors, Numerical Integration Recent Developments, Software and Applications, pages 1-24, Kluwer Academic Publishers, Dordrecht, 1992.

  56. R. Cools and A. Haegemans. CUBPACK: Progress Report. In T.O. Espelid and A. Genz, editors, Numerical Integration, Recent Developments, Software and Applications, pages 305-316, Kluwer Academic Publishers, Dordrecht, 1992.

  57. R. Cools and A. Haegemans. A lower bound for the number of function evaluations in an error estimate for numerical integration. Constr. Approx., 6:353-361, 1990.

  58. R. Cools and A. Haegemans. The construction of cubature formulae using continuation and bifurcation software. In D. Roose, B. De Dier, and A. Spence, editors, Continuation and Bifurcations: Techniques and Applications, pages 319-333, Kluwer Academic Publishers, Dordrecht, 1990.

  59. R. Cools and H.J. Schmid. Minimal cubature formulae of degree 2k - 1 for two classical functionals. Computing, 43:141-157, 1989.
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  60. R. Cools and A. Haegemans. On the construction of multi-dimensional embedded cubature formulae. Numer. Math., 55:735-745, 1989.
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  61. P. Verlinden, R. Cools, D. Roose, and A. Haegemans. The construction of cubature formulae for a family of integrals: a bifurcation problem. Computing, 40:337-346, 1988.

  62. R. Cools and A. Haegemans. Why do so many cubature formulae have so many positive weights ? BIT, 28:792-802, 1988.

  63. R. Cools and A. Haegemans. Construction of symmetric cubature formulae with the number of knots (almost) equal to Möller's lower bound. In H. Brass and G. Hämmerlin, editors, Numerical Integration III, pages 25-36. Birkhäuser Verlag, 1988.

  64. R. Cools and A. Haegemans. An embedded pair of cubature formulae of degree 5 and 7 for the triangle. BIT, 28:357-359, 1988.

  65. R. Cools and A. Haegemans. Another step forward in searching for cubature formulae with a minimal number of knots for the square. Computing, 40:139-146, 1988.

  66. R. Cools and A. Haegemans. Automatic computation of knots and weights of cubature formulae for circular symmetric planar regions. J. Comput. Appl. Math., 20:153-158, 1987.

  67. R. Cools and A. Haegemans. Construction of fully symmetric cubature formulae of degree 4k - 3 for fully symmetric planar regions. J. Comput. Appl. Math., 17:173-180, 1987.

  68. R. Cools and A. Haegemans. Construction of sequences of embedded cubature formulae for circular symmetric planar regions. In P. Keast and G. Fairweather, editors, Numerical Integration, pages 165-172. Reidel Publ. Comp., 1987.

  69. A. Haegemans and R. Cools. Construction of three-dimensional cubature formulae with points on regular and semi-regular polytopes. In P. Keast and G. Fairweather, editors, Numerical Integration, pages 153-163. Reidel Publ. Comp., 1987.

  70. R. Cools and A. Haegemans. Optimal addition of knots to cubature formulae for planar regions. Numer. Math., 49:269-274, 1986.
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