Local error estimates and regularity tests for the implementation of double adaptive quadrature

This article presents a device which is suitable for a practical and efficient implementation of Double Adaptive Quadrature. The device includes local error estimates and attempts to detect the presence of numerical difficulties in the integrand function. If a family of rules with suitable properties is chosen, then this can be achieved without affecting the overall computational cost. Extensive numerical testing has been performed on a comprehensive set of functions showing the effectiveness of the device and its efficiency.