Description of the Tables

In our compilation, we give individual tables with information about cubature rules for the following regions of particular dimension, C_n, T_n, S_n, E_n^-r, n = 2,3,4, E_n^-r2, n = 2,3,4,5 and one additional table for C_n, T_n, S_n, E_n^-r and E_n^-r2 of general dimension.

In most cases, we also add a separate table of embedded sequences of cubature rules. These sequences are either fully or partially embedded. In a fully embedded sequence, each cubature rule of higher degree includes among its integration points each point of all the cubature rules of lower degree in the sequence. In a partially embedded sequence, some points in a cubature rule of lower degree do not appear among the points of a cubature rule of higher degree. Although, in theory, any sequence of cubature rules can be considered to be a partially embedded sequence, in practice we require that the number of points not appearing in the cubature rule of higher degree be `small'. If we relax the condition in (2) that all weights w_i 0, then we can look upon a partially embedded sequence as a fully embedded sequence in which some weights vanish. In our tables, we take this approach and indicate partial embedding by the letter P in the column labeled N. We remark that sometimes cubature rules appearing in an embedded sequence are also included among the individual cubature rules when they are more efficient than some of the cubature rules included in the tables, where we define one cubature rule over a region B_n to be more efficient than another cubature rule over the same region if both are of the same degree and the first cubature rule uses fewer points than the second.

In our tables, we give the following information about cubature rules and embedded sequences in addition to the degree, d and the corresponding number of integration points, N, namely the quality and the references. The quality of a cubature rule is given by one or two letters which use a slight extension of the notation introduced in [LJ75]. The first letter gives information about the weights w_i. If all w_i are positive, we denote this by the letter P unless they are all equal in which case we use the letter E. If some w_i are negative, we indicate this by the letter N. Occasionally, only the points of a cubature rule are published and not the weights, since these can be evaluated quite easily by solving a system of linear equations. In this case, we replace the first letter with a question mark. The second letter only appears for bounded regions B_n and gives information about the location of the integration points x_i. If all x_i B, we indicate this by the letter I unless some of the x_i lie on the boundary of B_n in which case we use the letter B. If some x_i B, we write O. In the case of an embedded sequence, consisting of p cubature rules, p or p+1 letters are used depending on whether B_n is the entire space or not. In the latter case, the last letter give information about the integration points in the last cubature rule of the sequence, namely the one of highest degree. In all cases, the first p letters give information about the weights of all the cubature rules in the sequence starting with the cubature rule of lowest degree. In the quality column, we also indicate the number of cubature rules which have the same values of d and N and the same quality. If there is a one-parameter family of cubature rules with the same characteristics, we indicate this by the infinity symbol, .

We now give the guidelines for the entries in the references column which we have tried to keep as short as possible although we try to supply all useful references. As principal reference, we give the first appearance of the cubature rule in a journal or book even though it has previously appeared in an internal report or thesis. At times, this reference is supplemented by a second reference which corrects some deficiencies in the first reference or which derives the same cubature rule in a different context. Another case where additional references are included occurs when a later paper contains all of the cubature rules in an earlier paper plus many additional ones. In this case, we wish to spare the user the trouble of consulting two references when all the information he requires appears in the later paper. Internal reports and theses are only mentioned if the cubature rule did not appear in a journal or book. On inspecting the cubature rules in the tables, the reader may notice that in many cases, less efficient cubature rules were computed after the appearance of more efficient ones either in [Str71] or elsewhere. One possible explanation for this is that the less efficient cubature rule was computed as an example to illustrate a particular theory for the construction of cubature rules. The reader may also notice that we have included some cubature rules from [Str71] even though our explicit aim was to present cubature rules which did not appear in [Str71]. One reason for this is that in many cases the cubature rule in [Str71] is minimal and we wished the reader to have this information since we append an asterisk to all cubature rules known to contain the theoretically minimal number of points. Another reason is to avoid gaps in the tables as one proceeds from one value of d to another value d₁ > d + 2. Finally, a cubature rule from [Str71] may not be minimal among all cubature rules of a given degree but still it has some additional useful feature which is not present in other cubature rules. So, if [Str71] lists a formula that is PI and none of the known formulae with less nodes is PI, we include a reference to [Str71]. We do not give additional references to cubature rules that appeared in [Str71].

All the cubature rules in our tables are cubature rules for which either numbers or formulas are given in the references for the points x_i and, in most cases, for the weights w_i. Hence, we did not include the cubature rules in [Hue73] which are given as functions of the moments nor the cubature rules in [CH86a,Kea79] which are of considerable interest and which can be generated by the algorithm given there.