A Simultaneous Numerical Integration Routine for the Fast Calculation of Similar Integrations

In this paper, a fast and simultaneous integration routine that is tailored for obtaining the results of multiple numerical integrations is introduced. The routine uses the same nodes for multiple integrations defined for similar integrands. In the paper it is demonstrated by several examples that if the integrands of interest are similar on the integration path, then using the same nodes decreases the computational costs dramatically. The method is introduced by updating the popular Gauss-Kronrod quadrature rule, but the same steps given in the paper can be applied to any other numerical integration rule.