Abstract
Among numerical quadrature formulas the simplest one is the trapezoidal rule or the mid-point rule. Gregory's/Bickley's formula can be regarded as the trapezoidal/mid-point rule with correction expressed in terms of differences. These formulas are classical and simple, but their truncation errors have not yet been analyzed thoroughly. In this paper, we introduce Gregory's and Bickley's formulas with integral remainders, which are the direct application of Markoff's formula to the Euler-Maclaurin summation formula and to the second Euler-Maclaurin summation formula, respectively. Based on these remainders, we can get the error bound of these formulas. The round-off errors are also discussed. Some numerical examples are also shown.
