## 3 Methods
### Intermediate Complexity of Methods  
During iterative solution procedures, scenarios may arise where parametric circles lack intersection points, necessitating an analysis of complex roots. To address this, the concept of complex root distance is formalized in Definition 2. The algorithm strategically expands the solution space through intermediate complex-number embeddings, as detailed in Algorithm 1. Crucially, real-domain feasibility is enforced through a mapping operator that projects complex-valued iteration vectors back into real space. This mechanism exemplifies the Intermediate-Complexity Principle: while complex-number operations are permitted during transient computational phases, all terminal solutions are constrained to real-space coordinates.  
**Definition 1** (Distance in a Unitary Space).: _Let \(V_{u}\) be a unitary space over the complex field \(\mathbb{C}\) with an inner product \(\langle\cdot,\cdot\rangle:V_{u}\times V_{u}\to\mathbb{C}\) that satisfies the following properties:_  
1. _For all_ \(u,v,w\in V_{u}\) _and_ \(a,b\in\mathbb{C}\)_,_ \(\langle au+bv,w\rangle=a\langle u,w\rangle+b\langle v,w\rangle\) _(linearity)._
2. \(\langle u,v\rangle=\overline{\langle v,u\rangle}\) _(conjugate symmetry), where_ \(\overline{\langle v,u\rangle}\) _denotes the complex conjugate of_ \(\langle v,u\rangle\)_._
3. _For all_ \(u\in V_{u}\)_,_ \(\langle u,u\rangle\geq 0\) _and_ \(\langle u,u\rangle=0\) _if and only if_ \(u=0\) _(positive definiteness)._  
**Definition 2**.: _The distance \(d(u,v)\) between two vectors \(u,v\in V_{u}\) is defined as \(d(u,v)=\|u-v\|\), where the norm \(\|u\|\) of a vector \(u\) is given by \(\|u\|=\sqrt{\langle u,u\rangle}\). In the case of an \(n\)-dimensional unitary space \(V_{u}=\mathbb{C}^{n}\) with the standard inner product \(\langle u,v\rangle=\sum_{i=1}^{n}u_{i}\overline{v_{i}}\), the distance formula can be explicitly written as \(d(u,v)=\sqrt{\sum_{i=1}^{n}(u_{i}-v_{i})(\overline{u_{i}-v_{i}})}\)._  
```
1:Initialize \([x_{f}^{*},y_{f}^{*},\theta_{f}^{*}]_{0}\) (Initial guess)
2:Set \(\epsilon=5\) (Error tolerance)
3:Set \(k=0\) (iteration counter)
4:Set maximum number of iterations \(N_{max}\)
5:while\(k<N_{max}\) and \(V>\epsilon\)do
6:for all\(s_{2},s_{1},s_{0}\in\{+,-\}\)do
7: Compute \([\delta x_{f}^{*},\delta y_{f}^{*},\delta\theta_{f}^{*}]\)--Equation (16)
8: Update \([x_{f}^{*},y_{f}^{*},\theta_{f}^{*}]\leftarrow[x_{f}^{*},y_{f}^{*},\theta_{f}^ {*}]+[\delta x_{f}^{*},\delta y_{f}^{*},\delta\theta_{f}^{*}]\)
9: Compute \(V_{s_{2}s_{1}s_{0}}\)--Equation (14)
10:endfor
11:\(V=\min(V_{s_{2},s_{1}s_{0}})\)
12:\(k=k+1\)
13:endwhile
14:Return\(x_{f},y_{f},\theta_{f}\) and \(s_{2},s_{1},s_{0}\)
```  
**Algorithm 1** Iterative Optimization Solver for 8B-PGDM  
_Thus, the first-order approximation of the Lyapunov function derivative is given by_  
\[\dot{V}(x_{k})\approx\mathbf{J}_{V}(x_{k})(g(x_{k})-x_{k}) \tag{22}\]  
From the full differentiation of Equation (14), there is the form of the derivative of the Lyapunov function \(\dot{V}(\mathbf{x}_{k})\). Substituting Equation (16) into Equation (22) and still allowing \(\mathbf{x}_{k}\) to represent the vector \([x_{f}^{*},y_{f}^{*},\theta_{f}^{*}]_{k}]\) gives the derivative of the Lyapunov function, as shown in the following equation.  
\[\dot{V}(\mathbf{x}_{k}) \approx V_{x}^{d}\cdot\triangle V_{x}^{d}+V_{y}^{d}\cdot\triangle V _{y}^{d}+V_{x}^{o}\cdot\triangle V_{x}^{o}+V_{x}^{o}\cdot\triangle V_{x}^{o} \tag{23}\] \[\approx \left[\begin{array}{c}V_{x}^{d}\\ V_{y}^{d}\\ V_{x}^{o}\\ V_{x}^{o}\end{array}\right]\mathbf{J}_{V}(\mathbf{x}_{k})(g(\mathbf{x}_{k})- \mathbf{x}_{k})\] \[\approx -((V_{x}^{d})^{2}+(V_{y}^{d})^{2}+(V_{x}^{o})^{2}+(V_{x}^{o})^{2}) \leq 0\]  
At this point, the convergence has been proven.