閉型4点近似積分公式の一般形と最高の精度を持つ公式について

Abstract

The celebrated first Simpson's rule is obtained approximately from the function-values at 3 points on the interval of integration. On the other hand, the second Simpson's rule is based on the values at 4 points. And hence the latter may be expected to be slightly more accurate than the former. But acctually, the above two rules have the same order of accuracy. In this paper, we start dealing with the function-valuese at random interior points First, we shall derive the general form of the symmetric 4 points numerical integration formula of closed type with the same accuracy as that of the first or the second Simpson'rule. Next we discuss widely the 4 points integration formula not necessarily symmetric. And finally we shall give the concrete form of the best formula having the highest order of accuracy in all the 4 points numerical integration formulas.