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Cools, Ronald

An encyclopaedia of cubature formulas. (English) Zbl 1061.41020

J. Complexity 19, No. 3, 445-453 (2003).
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.

Cited in 81 Documents

MSC:

41A55 Approximate quadratures
65D30 Numerical integration
65D32 Numerical quadrature and cubature formulas
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\textit{R. Cools}, J. Complexity 19, No. 3, 445--453 (2003; Zbl 1061.41020)
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Digital Library of Mathematical Functions:

§3.5(x) Cubature Formulas ‣ §3.5 Quadrature ‣ Areas ‣ Chapter 3 Numerical Methods

Online 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:

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[8] Mysovskikh, I.P., Interpolatory cubature formulas, (1981), 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 Englewood Cliffs, NJ · Zbl 0379.65013
[11] Stroud, A.H.; Secrest, D., Gaussian quadrature formulas, (1966), 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.