On the construction of high-order integration formulae for the adaptive quadrature method

Abstract

The adaptive quadrature method requires a fixed integration formula with an error estimator as the fundamental tool. In such a quadrature method, the integral interval is halved after integration by the formula when the estimated error is too large. Then the same formula is applied to both two halved intervals. This halving process is continued until the calculated value attains enough accuracy. Usually relatively low order formulae have been used for such a quadrature method. One of the reasons is their “reusability” of function values after having the interval. We point out that the property of reusability in the adaptive method can be interpreted into the notion in semi-dynamical systems. With this interpretation, we propose the method for listing up all the symmetric interpolatory integration formulae having above reusability, and give several high-order integration formulae. All formulae have positive weights and thus are stable.