An encyclopaedia of cubature formulas. (English) Zbl 1061.41020
Summary: About 13 years ago we started collecting published cubature formulas for the approximation of multivariate integrals over some standard regions. In this paper we describe how we make this information available to a larger audience via the World Wide Web.
MSC:
| 41A55 | Approximate quadratures |
| 65D30 | Numerical integration |
| 65D32 | Numerical quadrature and cubature formulas |
Digital Library of Mathematical Functions:
§3.5(x) Cubature Formulas ‣ §3.5 Quadrature ‣ Areas ‣ Chapter 3 Numerical MethodsOnline Encyclopedia of Integer Sequences:
Number of 2 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (10;0) and (01;1).a(0) = 0 and a(n) = (4*n^3 - 12*n^2 + 20*n - 9)/3 for n >= 1.
T(n,k) = binomial(n + k, n) - binomial(n + floor(k/2), n) + 1, square array read by descending antidiagonals (n >= 0, k >= 0).
a(n) = 2^n + 2*n^2 + 2*n + 1.
a(n) = (4*n^3 - 6*n^2 + 14*n + 3)/3.
a(n) = 2^n + 2*n^2 + 1.
a(n) = (4*n^3 + 12*n^2 - 4*n + 3)/3.
a(n) = (n^3 + 9*n + 14*n + 9)/3.
Square array read by descending antidiagonals (n >= 0, k >= 0): let b(n,k) = (n+k)!/((n+1)!*k!); then T(n,k) = b(n,k) if b(n,k) is an integer, and T(n,k) = floor(b(n,k)) + 1 otherwise.
a(n) = (4*n^3 - 6*n^2 + 20*n + 3)/3.
References:
| [1] | M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, Vol. 55, US Government Printing Office, Washington, DC, 1964.; M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, Vol. 55, US Government Printing Office, Washington, DC, 1964. · Zbl 0171.38503 |
| [2] | Bailey, D. H., A Fortran-90 based multiprecision system, ACM Trans. Math. Software, 21, 4, 379-387 (1995) · Zbl 0883.68017 |
| [3] | R. Cools, Constructing cubature formulae: The science behind the art, Acta Numerica, Vol. 6, Cambridge University Press, Cambridge, 1997, pp. 1-54.; R. Cools, Constructing cubature formulae: The science behind the art, Acta Numerica, Vol. 6, Cambridge University Press, Cambridge, 1997, pp. 1-54. · Zbl 0887.65028 |
| [4] | Cools, R., Monomial cubature rules since “Stroud”a compilation—part 2, J. Comput. Appl. Math., 112, 1-2, 21-27 (1999) · Zbl 0954.65021 |
| [5] | Cools, R.; Rabinowitz, P., Monomial cubature rules since “Stroud”a compilation, J. Comput. Appl. Math., 48, 309-326 (1993) · Zbl 0799.65027 |
| [6] | D.W. Lozier, B.R. Miller, B.V. Saunders, Design of a digital mathematical library for science, technology and education, in: Proceedings of the IEEE Forum on Research and Technology Advances in Digital Libraries; IEEE ADL ’99, Baltimore, MD, May 19, 1999.; D.W. Lozier, B.R. Miller, B.V. Saunders, Design of a digital mathematical library for science, technology and education, in: Proceedings of the IEEE Forum on Research and Technology Advances in Digital Libraries; IEEE ADL ’99, Baltimore, MD, May 19, 1999. |
| [7] | J. More, B. Garbow, K. Hillstrom, MINPACK, http://www.netlib.org:80/minpack/; J. More, B. Garbow, K. Hillstrom, MINPACK, http://www.netlib.org:80/minpack/ |
| [8] | Mysovskikh, I. P., Interpolatory Cubature Formulas (1981), Izdat. Nauka, Moscow: Izdat. Nauka, Moscow Leningrad, (Russian) · Zbl 0537.65019 |
| [9] | Radon, J., Zur mechanischen Kubatur, Monatsh. Math., 52, 286-300 (1948) · Zbl 0031.31504 |
| [10] | Stroud, A. H., Approximate Calculation of Multiple Integrals (1971), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0379.65013 |
| [11] | Stroud, A. H.; Secrest, D., Gaussian Quadrature Formulas (1966), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0156.17002 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
