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Testing the Formulae

We tested our computed integration formulae on six different functions described in Table 1.

j tex2html_wrap_inline1658 tex2html_wrap_inline1660
1 tex2html_wrap_inline1662 tex2html_wrap_inline1664
2 tex2html_wrap_inline1666 tex2html_wrap_inline1668
3 tex2html_wrap_inline1670 tex2html_wrap_inline1672
4 tex2html_wrap_inline1674 tex2html_wrap_inline1676
5 tex2html_wrap_inline1678 tex2html_wrap_inline1680
6 tex2html_wrap_inline1682 0
Table tex2html_wrap1704N-node formula is now defined as
Figure 7 and 8 show these integration errors for all computed formulae and all test functions. A detailed list of integration errors is contained in Tables 6 and 7 (see appendix). Since the machine precision is about tex2html_wrap_inline1688, the computed formulae have a "treshold" at about tex2html_wrap_inline3117. Without a doubt, the errors will get smaller when the numerical integration is done on a machine with higher machine precision (or which simulates more accurate floating point numbers by software).

Figure tex2html_wrap1705tex2html_wrap_inline1692, tex2html_wrap_inline1694 and tex2html_wrap_inline1696

Figure tex2html_wrap1706tex2html_wrap_inline1698, tex2html_wrap_inline1700 and tex2html_wrap_inline1702

Unfortunately, to the best of our knowledge, comparable numerical results for integration formulae do not seem to be available in the literature.

next up previous
Next: Conclusions Up: Computational Results Previous: The Weights

Joerg Fliege
Thu Dec 23 19:39:35 CET 1999