SimplicialCubature-package {SimplicialCubature}

Numerical integration of functions over simplices

Description

Provides methods to evaluate integrals of the form

\int_S f(x) dx,

where S is a simplex (or a union of simplices) in n-space and f(x) is a function defined on S. The function f(x) may be vector valued and the simplices can be m-dimensional simplices, 1 <= m <=n. For example, if m=n-1, the package will evaluate a surface area integral; if m=1, the package will evaluate a line integral.

There are exact methods for polynomials and adaptive methods for integrating an arbitrary function. The two main functions are:

adaptIntegrateSimplex - integrate a general (possibly vector valued) function over a simplex using the method of Genz and Cools. It is expected that these methods will yield several significant digits, but not many digits of precision.

integrateSimplexPolynomial - integrate a single polynomial exactly over a simplex using either the Grundmann-Moller method or the Lasserre-Avrachenkov method.

The naming of the functions, arguments, and return values deliberately mimics that in the CRAN packages cubature (for integrating over hyper-rectangles) and SphericalCubature (for integrating over spheres and balls).

Please let me know if you find any mistakes. I will try to fix bugs when time is available.

Constructive comments for improvements are welcome; actually implementing any suggestions will be dependent on time constraints.

Version history:

• 1.0.1 original package

• 1.1 fix an incorrect test for dimension=1 in SimplexVolume; add functions SimplexSurfaceArea. Rename function UnitSimplex to UnitSimplexV to eliminate name conflict with function in package mvmesh.

• 1.2 miscellaneous small changes.

• 1.3 Eliminate internal function integrate.vector.fn and replace with a slightly different function adaptIntegrateVectorFunc. This new function is used in the new version of package SphericalCubature.

Author(s)

John P. Nolan, with R translations of adsimp code by Alan Genz

Maintainer: John P. Nolan <[e-mail]>

This research was supported by an agreement with Cornell University, Operations Research & Information Engineering, under contract W911NF-12-1-0385 from the Army Research Development and Engineering Command.

References

V. Baldoni, N. Berline, J. A. De Loera, M. Koppe, and M. Vergene, How to integrate a polynomial over a simplex, Mathematics of Computation, 80, 297-325 (2011)

A. Genz and R. Cools, An adaptive numerical cubature algorithm for simplices, ACM Trans. Math. Software, 29, 297-308 (2003)

A. Grundmann and H.M. Moller, Invariant Integration Formulas for the n-Simplex by Combinatorial Methods, SIAM Journal on Numerical Analysis, 15, 282-289 (1978)

J. B. Lasserre and E. E. Avrachenkov, The Multi-Dimensional Version of \int^b_a x^p dx, American Mathematical Monthly, 108, 151-154 (2001)

N. Konerth, Exact integration on simplices, Undergraduate Research Paper, Math/Stat Department, American University (2014). Online at http://aladinrc.wrlc.org/handle/1961/14863

See Also

adaptIntegrateSimplex, integrateSimplexPolynomial