BIT Numerical Mathematics

, Volume 58, Issue 3, pp 613–660 | Cite as

Numerical integration using integrals over hyperplane sections of simplices in a triangulation of a polytope

  • Allal Guessab
  • Boris Semisalov


In this paper, we consider the problem of approximating a definite integral of a given function f when, rather than its values at some points, a number of integrals of f over some hyperplane sections of simplices in a triangulation of a polytope P in \(\mathbb {R}^d\) are only available. We present several new families of “extended” integration formulas, all of which are a weighted sum of integrals over some hyperplane sections of simplices, and which contain in a special case of our result multivariate analogues of the midpoint rule, the trapezoidal rule and the Simpson’s rule. Along with an efficient algorithm for their implementations, several illustrative numerical examples are provided comparing these cubature formulas among themselves. The paper also presents the best possible explicit constants for their approximation errors. We perform numerical tests which allow the comparison of the new cubature formulas. Finally, we will discuss a conjecture suggested by the numerical results.


Cubature Approximation Convexity Best constants Error estimates 

Mathematics Subject Classification

65D32 33C45 41A44 


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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques et de leurs Applications, UMR CNRS 4152Université de Pau et des Pays de l’AdourPauFrance
  2. 2.Novosibirsk State UniversityNovosibirskRussia
  3. 3.Institute of Computational Technologies SB RASNovosibirskRussia

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