The Gauss-Legendre rule is very popular in the scalar case, which is due in part to its optimality properties.
and [w.sub.k], [x.sub.k] are, respectively, the weights and nodes of the m-point Gauss-Legendre rule;
Figure 5.1 shows that the Gauss-Legendre rule (Algorithm 5.2) performs considerably better than the other rules, both in number of function evaluations and relative residual.
The results are depicted in Figure 6.2, which evidences once more the good performance of the Gauss-Legendre rule.
The Gauss-Legendre rule has proved to be once more the right choice to work out that approximation.
The same problem occurs with Gauss-Legendre rules, because the estimate (4.5) involves nth order derivatives of f (t).
According to (4.6), a possibility for estimating the number of nodes and weights in Gauss-Legendre rules is by requiring that [parallel]G(2m) - G(m)[parallel] satisfies a prescribed tolerance, where G(m) := [[SIGMA].sup.m.sub.i=1] [w.sub.i]f ([t.sub.i]).
7 show respectively for adjacent and common vertex cases, the relative error when calculating the source integral through (9) and the observer integral by means of generalized 2D Cartesian product rules based either on the DE formula or on the Gauss-Legendre rules. The reference solutions of the 4D integrals are computed by utilizing the analytical expressions in .
These matrix elements are computed by integrating (9) through 2D generalized Cartesian product rules based on the DE and Gauss-Legendre rules, for levels 0, 1, 2, 3 and 4 in both cases.