Selection of regions and rules

We consider cubature rules of precision or degree d of the form

Bn

w(x) f(x) dV = IN[f] + RN[f]        (1)

where B_n is a region in n-dimensional Euclidean space E_n, x := ( x₁, ..., x_n ) is a point in E_n, dV is an element of volume in E_n, w is a weight function,

IN f :=

N i = 1

wi f(xi),     wi 0,     i = 1, ..., N        (2)

is a cubature rule and R_N f is the remainder which has the property that R_N f = 0 if f is a linear combination of monomials of the form

n j = 1

xji

with

n j = 1

ij d

and

RN f 0 for some such monomial with

n j = 1

ij = d + 1.

The regions B_n which we include in our compilation are the three bounded regions C_n, the hypercube, S_n, the hypersphere and T_n, the simplex and the entire space E_n. Associated with the bounded regions is the weight function w(x) 1 while with E_n, we associate two weight functions, exp(-r) and exp(-r²) where

r2 :=

n j = 1

xj2 .

As in [Str71], E_n with these two weight functions is denoted by E_n^r and E_n^r2, respectively. The definitions of the bounded regions are given by

Cn     : -1 xj 1,   j = 1, ..., n

Sn     : r2 1

Tn     :

n j = 1

xj 1,     xj 0,     j = 1, ..., n .

A more detailed description of the regions we consider is given here.

As a general rule we do not include cubature rules of degree 2m - 1 which use more than mⁿ points since the (Cartesian, Spherical or Conical) product Gaussian rule of degree 2m - 1 requires mⁿ points and is usually superior to any other cubature rule of the same degree. This also holds true for the top of an embedded sequence since there is a very good way to generate an embedded sequence starting with a product Gaussian rule and generating cubature rules of lower degree in an optimal manner [CH89]. Exceptions to this rule are cubature rules with properties that some users may desire, such as equal weights, points on the boundary or more symmetry.