## 5 Entanglement and Numerical Test of DE-MPFSC Algorithm Stability
### Entanglement Degree \(\frac{1}{\xi}\) and Entanglement Degree Error \(\zeta\)  
**Lemma 4** ([20]).: _Let \(f_{\varepsilon}\) be a continuous differentiable function defined in a complete normed linear space and \(f_{\varepsilon}(v_{1},\cdots,v_{n})\in L_{2}(\mathbb{R}_{n}),M\in Sp(2n+P_{t}.\mathbb{R})\), \(M^{n}\) is a complete normed linear space. \(\{S_{i}^{n}|i=1,2,\cdots,n\}\) is a generalized \(n\) dimension complete normed linear subspace and \(M^{n}=S_{1}^{n}\cup S_{2}^{n}\cup\cdots\cup S_{n}^{n}\) forms an open cover of \(M^{n}\), then \(\forall\lambda_{i}^{t}\in\{\lambda_{i}^{t}|i=1,2,\cdots,n\},\exists S_{i}^{n},\lim_{t\to\infty}\lambda_{i}^{t}\sim\lambda_{i}\) or \(\lim_{t\to\infty}\lambda_{i}^{t}=(\lambda_{\varepsilon})_{i},\lambda_{i}\in S _{i}^{n}\). If \(\det(B)\
eq 0\), then the following formula,_  
\[\begin{split}\Delta_{v}^{2}\cdot\Delta_{x_{\beta}^{t}}^{2}& \geq(\frac{\sqrt{\lim_{t\to\infty}\lambda_{1}^{t}}+\cdots+\sqrt{ \lim_{t\to\infty}\lambda_{n}^{t}}}{2})^{2}\\ &=(\frac{\sqrt{(\lambda_{\varepsilon})_{1}}+\cdots+\sqrt{( \lambda_{\varepsilon})_{n}}}{2})^{2}.\end{split} \tag{33}\]  
The lemma is detailed in Reference [20], which theoretically shows that the convergence accuracy and convergence speed of the evolutionary individual in the iterative process represent an uncertain quantum relationship. The lemma is important for analyzing the distribution of the optimal point of the segmentation regions. The feasible points distributed in different regions force the internal and external penalty functions to search in different segmentation regions by the jumping function of probability. Let the region have two local optimal feasible points \(x_{1}\) and \(x_{2}\), which are distributed to satisfy \(\Delta_{v}^{2}\cdot\Delta_{x_{\beta}^{t}}^{2}\geq(\frac{\sqrt{\sum_{1}}+\sqrt {\sum_{2}}}{2})^{2}\) and \(\sqrt{\lim_{t\to\infty}x_{1}^{t}}=\sqrt{x_{1}}\), \(\sqrt{\lim_{t\to\infty}x_{2}^{t}}=\sqrt{x_{2}}\), where \(t\) is the number of individual iterations.  
The individual entanglement degree we define is \(\frac{1}{\xi}=\frac{v_{X(t)}}{x_{X(t)}}\), which is the numerical ratio between the convergence speed and convergence precision of individuals during population evolution. The higher the entanglement degree, the better the convergence stability of the algorithm or the faster convergence of the algorithm and the higher convergence precision. The entanglement degree error \(\zeta\) is the correlation of the entanglement effect, which is the efficiency of the entanglement decided by \(\frac{1}{\xi}\ln Cov(NP)=\zeta\ln E(NP-X_{i})\). The smaller the entanglement degree error, the smaller the fluctuation of the algorithm in the iterative process. Where, \(v_{X(t)}\) is the individual convergence speed, \(\iota(X(t))\) is the individual convergence precision, \(\frac{\ln Cov(NP)}{\ln E(NP-X_{i})}\) is the entanglement coefficient, \(\xi\) is the discrete degree between entangled individuals. About the entangled relationship between convergence precision and convergence speed, we can refer to the topological geometric relationship described by the author of Reference [20]. The effect of entanglement degree clearly shows the efficiency of algorithm convergence in the geometry.