Gauss-Kronrod Quadrature
\[\begin{eqnarray} I & := & \int_{a}^{b} f(x) \ dx \nonumber \\ & \approx & \sum_{i=1}^{n} f(x_{i}) w(x_{i}) + \sum_{i=1}^{2 n} f(x_{i}) w(x_{i}) \nonumber \end{eqnarray}\]Let $F_{n + 2p -1} \in \Pi_{n + 2p -1}$ be polynomial of degree $n + 2p - 1$. There exist $Q_{n + p -1} \in \Pi_{n + p -1}$, $G_{n + p} \in \Pi_{n + p}$, and $c_{k} \in \mathbb{R}$ such that
\[\begin{eqnarray} F_{n + 2p -1}(x) = Q_{n+p-1}(x) + G_{n+p}(x) \sum_{k=0}^{p-1} c_{k} x^{k} . \end{eqnarray}\]$Q_{n+p-1}$ can be integrated by $(n + p)$ point Gauss quadrature. If $G_{n+p}$ has a property:
\[\int_{-1}^{1} G_{n+p}(x) x^{k} \ dx, \ k = 0, \ldots, p - 1 ,\]$F_{n + 2p - 1}$ can be integrated by $(n + p)$ points Gauss quadrature.
Kronrod consider that the case $p = n + 1$ for the $n$ pointgs Gauss quadrature. Instad of using a polynomial, Kronrod uses multiplication of a polynomial $K_{n+1}(x) \in \Pi_{n+1}$ and the Ledgendre polynoamial $P_{n}(x)$.
\[\begin{eqnarray} \int_{-1}^{1} K_{n+1}(x) P_{n}(x) x^{k} \ dx , k = 0, 1, \ldots, n \end{eqnarray}\]Kronrod determiens coefficients of $K_{n+1}$ and its zeros by substituting polynomial expression of $K_{n}$ into above equation. In particular, $G_{2n}$ has the same zeros as the $n$ Guass-Ledgendre quadrature.