where \(p_{0}(\tau)\ =\ 1\) and \(p_{1}(\tau)\ =\ \tau\) for all\(\tau\ \in\ [-1,\ 1].\) Moreover, suppose that\(\tau_{1}=\ -1\ <\ \tau_{2}\ <\ \ \ \ <\ \tau_{N}\ <\ 1\)are the LGR collocation points on the intervals\(\tau\ \in[-1,\ 1)\), which are the roots of polynomial\(p\left(.\right)=\ p_{N}\left(.\right)+\ p_{\text{N\ }-\ 1}\left(.\right)\).
Lemma 1 [18]. Suppose that\(\tau_{1}=\ -1\ <\ \tau_{2}\ <\ \ \ \ <\ \tau_{N}\ <\ 1\)are the LGR collocation points on the intervals\(\tau\ \in\ [-1,\ 1)\). Then, there exists a unique set of quadrature weights \(\left\{w_{k}\right\}_{k=1}^{N}\ \)such that for any polynomial \(q(.)\) of degree\(2(N\ -\ 1)\) or less, we have:
\begin{equation} \int_{-1}^{1}{q\left(t\right)\text{dt}}=\sum_{k=1}^{N}w_{k}q\left(\tau_{k}\right),\nonumber \\ \end{equation}
where \(w_{k}\) satisfies the following relation:
\begin{equation} w_{k}=\ \frac{1-\tau_{k}}{N^{2}\left(p_{N-1}\left(\tau_{k}\right)\right)^{2}},\ k=1,\ldots,N.\nonumber \\ \end{equation}
By using Lemma 1, the following approximation for the performance index of problem (22)-(25) is derived