## 3 Analysis of the Method
### Local Truncation Error
: The proof is similar to the one for Theorem 1, given above, and uses the identity:  
\[\mathcal{L}_{i}[z(t_{n});h]=\frac{h^{m+2}}{(m+2)!}\left[p_{i}^{m+2}-(m+2)\sum_ {j=0}^{m}\hat{\beta}_{i,j}p_{j}^{m+1}\right]\!z^{(m+2)}(t_{n})+\mathcal{O}(h^{ m+3}) \tag{13}\]2  
after considering the \(m+2\) grid points:  
\[\mathcal{L}_{i}[z(t_{n});h]=\frac{h^{m+2}}{(m+2)!}\left[p_{i}^{m+2}-(m+2)\sum_ {j=0}^{m}\hat{\beta}_{i,j}p_{j}^{m+1}\right]\!z^{(m+2)}(t_{n})+\mathcal{O}(h^{ m+3}) \tag{13}\]3  
By minimizing the local truncation error of the main method at \(t_{n+1}\) (i.e., \(p_{m}=1\)), it was shown in [11] that the optimal intra-step points are the roots of the polynomial:  
\[\mathcal{L}_{i}[z(t_{n});h]=\frac{h^{m+2}}{(m+2)!}\left[p_{i}^{m+2}-(m+2)\sum_ {j=0}^{m}\hat{\beta}_{i,j}p_{j}^{m+1}\right]\!z^{(m+2)}(t_{n})+\mathcal{O}(h^{ m+3}) \tag{13}\]4  
which is proportional to the first derivative of the shifted Legendre polynomials of degree \(m\) given by  
\[\mathcal{L}_{i}[z(t_{n});h]=\frac{h^{m+2}}{(m+2)!}\left[p_{i}^{m+2}-(m+2)\sum_ {j=0}^{m}\hat{\beta}_{i,j}p_{j}^{m+1}\right]\!z^{(m+2)}(t_{n})+\mathcal{O}(h^{ m+3}) \tag{13}\]5  
The shifted Legendre polynomials are defined in \([0,1]\) and have the property:  
\[\mathcal{L}_{i}[z(t_{n});h]=\frac{h^{m+2}}{(m+2)!}\left[p_{i}^{m+2}-(m+2)\sum_ {j=0}^{m}\hat{\beta}_{i,j}p_{j}^{m+1}\right]\!z^{(m+2)}(t_{n})+\mathcal{O}(h^{ m+3}) \tag{13}\]6Furthermore, \(P_{i}^{*}(t)\) has \(i\) distinct roots in \((0,1)\). By expanding (19), we obtain the polynomials that give optimal intra-step points with \(m=2,3,4,5\) as follows:  
\[\mathcal{L}_{i}[z(t_{n});h]=\frac{h^{m+2}}{(m+2)!}\left[p_{i}^{m+2}-(m+2)\sum_ {j=0}^{m}\hat{\beta}_{i,j}p_{j}^{m+1}\right]\!z^{(m+2)}(t_{n})+\mathcal{O}(h^{ m+3}) \tag{13}\]7 \[\mathcal{L}_{i}[z(t_{n});h]=\frac{h^{m+2}}{(m+2)!}\left[p_{i}^{m+2}-(m+2)\sum_ {j=0}^{m}\hat{\beta}_{i,j}p_{j}^{m+1}\right]\!z^{(m+2)}(t_{n})+\mathcal{O}(h^{ m+3}) \tag{13}\]8 \[\mathcal{L}_{i}[z(t_{n});h]=\frac{h^{m+2}}{(m+2)!}\left[p_{i}^{m+2}-(m+2)\sum_ {j=0}^{m}\hat{\beta}_{i,j}p_{j}^{m+1}\right]\!z^{(m+2)}(t_{n})+\mathcal{O}(h^{ m+3}) \tag{13}\]9 \[t_{n}<t_{n+p_{1}}<t_{n+p_{1}}<\cdots<t_{n+p_{m}}.\]0 \[t_{n}<t_{n+p_{1}}<t_{n+p_{1}}<\cdots<t_{n+p_{m}}.\]1  
As a consequence of Equation (21), the optimal roots (intra-step points) have the form  
\[t_{n}<t_{n+p_{1}}<t_{n+p_{1}}<\cdots<t_{n+p_{m}}.\]2  
when \(m\) is odd and  
\[t_{n}<t_{n+p_{1}}<t_{n+p_{1}}<\cdots<t_{n+p_{m}}.\]3  
when \(m\) is even. The truncation errors for overlapping and non-overlapping HBM algorithms with equally spaced and optimally spaced intra-step points are compared in Table 1 for \(m=3,4,5\). The table reveals that the truncation errors of the overlapping HBM are at least \(h\) orders of magnitude smaller than those of the non-overlapping HBM. This is consistent with the observations made in the preceding theorems regarding the explicit formulas for local truncation errors. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Equally Spaced**} & \multicolumn{3}{c}{**Optimal Points**} \\ \hline  & \(\mathcal{L}_{i}\) & **Non-Overlapping** & **Overlapping** & **Non-Overlapping** & **Overlapping** \\ \hline  & \(p_{1}\) & \(-\frac{19q^{(5)}(t_{n})k^{5}}{174900}\) & \(-\frac{11q^{(5)}(t_{n})k^{6}}{1049760}\) & \(-\frac{y^{(5)}(t_{n})k^{6}}{60000}\) & \(\frac{(4-3\sqrt{5})y^{(5)}(t