Extrapolation and Adaptivity in Software for Automatic Numerical Integration on a Cube

Asymptotic expansions related to the integration of well-behaved functions on simplices and cubes have been known for several decades. Extensions of these results to classes of vertex and line singularities are also known. The nature of these expansions justifies extrapolation using the ε-algorithm of Wynn. In principle this requires a uniform subdivision of the region. This was implemented in QUADPACK for finite intervals and in TRIEX for triangles about 15 years ago. In this paper its incorporation in CUBPACK, a software package for automatic integration over a collection of cubes and simplices, is described and some results are reported. We also report on a special subdivision strategy that offers an alternative approach for higher-dimensional problems.

[1]  P. Wynn,et al.  On a device for computing the _{}(_{}) tranformation , 1956 .

[2]  Terje O. Espelid,et al.  Algorithm 698: DCUHRE: an adaptive multidemensional integration routine for a vector of integrals , 1991, TOMS.

[3]  J. N. Lyness,et al.  An error functional expansion for $N$-dimensional quadrature with an integrand function singular at a point , 1976 .

[4]  Elise de Doncker,et al.  Quadrature error expansions , 1993 .

[5]  Ronald Cools,et al.  Rotation invariant cubature formulas over the n -dimensional unit cube , 2001 .

[6]  Ann Haegemans,et al.  Algorithm 34 An algorithm for the automatic integration over a triangle , 2005, Computing.

[7]  Avram Sidi,et al.  Euler-Maclaurin expansions for integrals over triangles and squares of functions having algebraic/logarithmic singularities along an edge , 1983 .

[8]  R. Cools.  Monomial cubature rules since “Stroud”: a compilation—part 2 , 1999 .

[9]  Terje O. Espelid,et al.  DECUHR: an algorithm for automatic integration of singular functions over a hyperrectangular region , 2005, Numerical Algorithms.

[10]  Ronald Cools,et al.  Cubpack: Progress Report , 1992 .

[11]  Terje O. Espelid,et al.  Error estimation in automatic quadrature routines , 1991, TOMS.

[12]  J. N. Lyness,et al.  On quadrature error expansions. Part I , 1987 .

[13]  Alan Genz,et al.  An adaptive algorithm for numerical integration over an n-dimensional rectangular region , 1980 .

[14]  Alan Genz,et al.  An Adaptive Numerical Integration Algorithm for Simplices , 1989, Great Lakes Computer Science Conference.

[15]  Paul Van Dooren,et al.  An adaptive algorithm for numerical integration over the n-cube , 1976 .

[16]  R. Cools,et al.  Monomial cubature rules since “Stroud”: a compilation , 1993 .

[17]  Ronald Cools,et al.  Algorithm 824: CUBPACK: a package for automatic cubature; framework description , 2003, TOMS.

[18]  P. Verlinden,et al.  An error expansion for cubature with an integrand with homogeneous boundary singularities , 1993 .

[19]  Terje O. Espelid,et al.  An adaptive algorithm for the approximate calculation of multiple integrals , 1991, TOMS.

[20]  Alan Genz,et al.  A Package for Testing Multiple Integration Subroutines , 1987 .

[21]  Elise de Doncker,et al.  Algorithm 612: TRIEX: Integration Over a TRIangle Using Nonlinear EXtrapolation , 1984, TOMS.

[22]  P. Wynn,et al.  On a Device for Computing the e m (S n ) Transformation , 1956 .

[23]  Ronald Cools,et al.  A Hybrid Subdivision Strategy for Adaptive Integration , 1998, J. Univers. Comput. Sci..

[24]  Ian Robinson.  An Algorithm for Automatic Integration Over a Triangle Using Nonlinear Extrapolation , 1984, TOMS.

[25]  Terje O. Espelid.  On the construction of good fully symmetric integration rules , 1987 .