Abstract:
Newton–Cotes integration rules are the simplest methods in numerical integration. The main advantage
of using these rules in quadrature software is ease of programming. In practice, only the lower orders are
implemented or tested, because of the negative coefficients of higher orders. Most textbooks state it is
not necessary to go beyond Boole’s 5-point rule. Explicit coefficients and error terms for higher orders are
seldom given literature. Higher-order rules include negative coefficients therefore roundoff error increases
while truncation error decreases as we increase the number of points. But is the optimal one really Simpson
or Boole?
In this paper, we list coefficients up to 19-points for both open and closed rules, derive the error terms
using an elementary and intuitive method, and test the rules on a battery of functions to find the optimum
all-round one.
