Integration Rules and Null Rules

All rules used by the algorithm are fully symmetric (FS-) rules. For each possible choice of the parameter ``key'' a set of five FS-rules is used. In such a set there is one integration rule R of polynomial degree 2m+1

 

where H is the region of integration, the evaluation points and the corresponding weights, . In addition there are four null rules of structure

 

All rules in such a set are based on the same set of evaluation points . We have (with ) for at least one value of for every . The null rules have polynomial degree 2m-1 , 2m-1 , 2m-3 and 2m-5 respectively. The integration rule is used to estimate the integral over each subregion while the null rules are used to produce four estimates of the error

 

over each subregion. More details about the construction of the final error-estimate are given in the next section.

A rule for the cube is fully symmetric if ,whenever the rule contains a point with associated weight w, it contains all points that can be generated from by permutations and/or sign-changes of the coordinates with the same associated weight. An integration rule has polynomial degree d if it integrates exactly all monomials with and fails to integrate exactly at least one monomial of degree d+1. A null rule, see Lyness [19], of polynomial degree d will integrate to zero all monomials of degree and will fail to do so for at least one monomial of degree d+1. Note that for any value of then is a null rule too and



is a FS integration rule of degree 2m-1 . Thus each null rule may be thought of as the difference between the basic integration rule R[f] and an appropriate integration rule of lower degree.

A point x in the FS-set of evaluation points with for i = 1, 2, ..., r and for i = r+1, r+2, ..., n, x is a generator of all points in this FS-set which can be generated from x using the FS-property. In Table 1 we list all of the types of generators that are used in this algorithm.
Table 1. Types of n-dimensional generators.

From Table 1 we see that 7 different types of generators are used, which we have numbered from 0 to 6. is a structure parameter, see Mantel and Rabinowitz [22], which gives the number of different generators of type j, where has possible values 0 and 1 only.

Based on the observation that software, using one integration rule only, changes performance from one test problem to another, we decided to offer the user a choice of rule through an input parameter ``key'' in all dimensions. In dimensions 2 and 3 we offer three different sets of rules while in other dimensions we offer two sets of rules. In each dimension the main difference between the sets of rules we offer is the polynomial degree. In Table 2 we list the different options we have in this algorithm specifying in which dimension(s) the set of FS-rules of a certain structure is available. By the notation we give the degree d=2m+1 of the integration rule in the set, the value of the seven structure parameters and the number of points L for a given set of FS-rules in this algorithm. Table 2. Structure-parameters for the sets of FS-rules.


where



The set of rules for key=1 is based on a pair of rules constructed by Eriksen [12] while the rules for key=2 are based on a paper by Berntsen and Espelid [7, 8]. The set of rules for key is based on a paper by Genz and Malik [18]. All rules used in the algorithm use evaluation points inside the region of integration.

In Table 3 we give the number of points used by the different structures for key values 3 and 4 for all dimensions between 2 and 10.
Table 3.