期刊
BIT NUMERICAL MATHEMATICS
卷 58, 期 3, 页码 613-660出版社
SPRINGER
DOI: 10.1007/s10543-018-0703-3
关键词
Cubature; Approximation; Convexity; Best constants; Error estimates
资金
- Russian Science Foundation [17-71-10135]
- Russian Science Foundation [17-71-10135] Funding Source: Russian Science Foundation
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 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.
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