# Implicit A-Stable Peer Triplets for ODE Constrained Optimal Control Problems
## 2 Materials and Methods
### Construction of Peer Triplets
#### 2.4.3 A-Stable Methods
For a first candidate with nearly minimal \(err_{s}+\delta\|A^{-1}B\|_{\infty}\), the damping factor \(\gamma\doteq 0.89\) was again too close to one to ensure superconvergence in numerical tests, see Remark 2. However, further searches nearby minimizing the damping factor found a better standard method with \(|\lambda_{2}|=0.60\). Its nodes are \(\mathbf{c}^{\mathsf{T}}=\left(\frac{1}{50},\frac{3}{5},1,\frac{41}{85}\right)\), the diagonal parameter \(\theta=3.34552931287687520\) is the largest zero of the cubic equation \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Starting Method**} & \multicolumn{3}{c}{**End Method**} \\ \cline{2-9}
**Triplet** & **Blocks** & \(\mu_{\theta}\) & \(\varrho(BA_{0}^{-1})\) & **Blocks** & \(\mu_{N}\) & \(\varrho(A_{N}^{-1}B_{N})\) & \(\varrho(B_{N}A^{-1})\) \\ \hline AP4o43 bdf & 3+1 & 5.47 & 1 & 1+3 & 3.81 & 1 & 1.15 \\ AP4o43 df & 3+1 & 6.27 & 1 & 1+3 & 4.40 & 1 & 1.03 \\ AP4o43 dig & 4 & 0.99 & 1 & 4 & 0.89 & 1.001 & 1 \\ AP4o43 sl & 3+1 & 1.88 & 1 & 4 & 0.72 & 1 & 1.03 \\ AP4o43 die & 3+1 & 3.80 & 1 & 1+3 & 0.66 & 2.6 & 1.98 \\ \hline AP3o32f & 1+1+1 & 1.50 & 1.02 & 1+2 & 0.94 & 1 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Properties of the boundary methods of Peer triplets. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(s\) & **Triplet** & **Nodes** & \(\alpha\) & \(\|A^{-1}B\|_{\infty}\) & \(|\lambda_{2}|\) & \(err_{s}\) & **Remarks** \\ \hline
4 & AP4o43 bdf & BDF4 & 73.35\({}^{\circ}\) & 5.79 & 0.099 & 0 & singly-implicit \\  & AP4o43 dif & \([0,1]\) & 84.0\({}^{\circ}\) & 2.01 & 0.26 & 0.0025 & diag-implicit \\  & AP4o43 dig & \([0,1]\) & 90\({}^{\circ}\) & 24.5 & 0.798 & 0.0260 & \(c_{3}=1\) \\  & AP4o43 sil & \([0,1]\) & 90\({}^{\circ}\) & 32.2 & 0.60 & 0.0230 & \(c_{3}=1\), sing.impl. \\  & AP4o43 die & \([0,2]\) & 90\({}^{\circ}\) & 6.08 & 0.66 & 0.0135 & nodes alternate \\ \hline
3 & AP3o32f & \([0,1]\) & 90\({}^{\circ}\) & 15.3 & 0.91 & 0.0170 & nodes alternate \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Properties of the standard methods of Peer triplets. \[112673616\,\theta^{3}+106686908\,\theta^{2}-2102637319\,\theta+1621264295=0.\]  
Further properties of the standard method are A-stability, nodes in \((0,1)\) with \(c_{3}=1\), norm \(\|A^{-1}B\|_{\infty}=32.2\), and an error constant \(err_{4}=0.0230\). The end method \((A_{N},B_{N},K_{N})\) has full matrices \(A_{N},K_{N}\), see Table 2.  
For the sake of completeness, we also present an A-stable diagonally-implicit three-stage method, since in the previous paper [13] we could not find such methods with reasonable parameters. After relaxing the order conditions by using superconvergence, such methods exist. Applying all conditions for forward order \(s=3\) and adjoint order \(s-1=2\), there remains a five-parameter family depending on the node differences \(d_{1}=c_{2}-c_{1}\), \(d_{3}=c_{3}-c_{2}\) and three elements of \(A\) or \(K\). A-stable methods exist in all four corners of the square \([-\frac{1}{2},\frac{1}{2}]^{2}\) in the \((d_{1},d_{3})\)-plane, the smallest errors \(err_{3}\) were observed in the second quadrant. The method AP3o32f with \((d_{1},d_{3})=(-\frac{5}{2\gamma},\frac{5}{2})\) has a node vector with \(c_{3}=1\). The coefficients can be found in Appendix A.6. The characteristic data are \(err_{3}\leq 0.017\), \(\|A^{-1}B\|_{\infty}\leq 15.3\), \(|\lambda_{2}|=0.91\). The starting method is of standard form with lower triangular \(A_{0}\) and diagonal \(K_{0}\). The main properties of the newly developed Peer triplets are summarized in Table 3 for the standard methods and Table 2 for the boundary methods.