Gauss–Kronrod quadrature formula

Summary

A Gauss–Kronrod quadrature formula extends an $n$-point Gauss–Legendre rule by $n + 1$ new nodes into a rule with $2n + 1$ nodes. The old nodes are kept, and the new nodes and all weights are chosen to give the highest possible degree of exactness. The difference between the two results is a cheap estimate of the error of the integral. For this reason Gauss–Kronrod pairs are the core of many adaptive integration programs.