## 3 Hybrid FE-SEA Modeling Parameter Identification Algorithm
### Iterative Improvement of Ill-Conditioned Matrix Parameters  
In general, the energy matrix will appear as an ill-conditioned matrix. Parameter iteration is carried out for the ill-conditioned matrix, and parameter correction of the energy matrix is realized. Assume that the general representation of the energy matrix is as follows [11]:  
\[Ax=B \tag{77}\]  
where \(A\) is the coefficient matrix, \(B\) is the diagonal matrix, and x is the solution vector. If \(\alpha\) is the weighting factor and \(E\) is the identity matrix of order n, then \(\alpha x\) is added to the left and right ends of Equation (77) according to the principal-component weighting method to obtain:  
\[(A+\alpha E)x=B+\alpha x \tag{78}\]  
In general, the ill-conditioned matrix of a system of linear equations is measured according to the number of conditions. The condition number of the coefficient matrix (\(A\)) is related to the ill-condition degree of the linear equations. For matrix \(A\), when the weighting factor \(\alpha>0\), the following relation holds (Cond (\(A+\alpha e\)) \(<\) Cond (\(A\))), where cond represents the number of conditions. By transforming Equation (77) into Equation (78), the number of conditions of the equation is reduced, and the ill-condition degree of the linear equations can be reduced. Since the left and right sides of Equation (78) contain a solution \(x\), the iterative relationship can be constructed as follows:  
\[(A+\alpha E)x^{(k+1)}=B+\alpha x^{(k)} \tag{79}\]  
If \(x^{(k+1)}=x^{(k)}+e^{(k)}\), then:  
\[(A+\alpha E)x^{(k)}+e^{(k)}=B+\alpha x^{(k)} \tag{80}\]  
\[(A+\alpha E)x^{(k)}(A+\alpha E)e^{(k)}=B+\alpha x^{(k)} \tag{81}\]  
\[(A+\alpha E)e^{(k)}=B-Ax^{(k)} \tag{82}\]  
If \(e^{(k)}\) is small enough, when \(\left\|e^{(k)}\right\|\)\(\angle\)\(\varepsilon\), \(x^{(k)}\) approximate solutions can be found, where \(x^{(k)}\) and \(x^{(k+1)}\) are the solutions obtained after iterating (\(k\)) times and iterating (\(k+1\)) times, respectively.\(\left\|e^{(k)}\right\|\) and \(\left\|x^{(k)}\right\|\) represent the norms of \(e^{(k)}\) and \(x^{(k)}\), respectively. \(\varepsilon\) is a constant term of a smaller order of magnitude. In order to improve the accuracy and stability of the calculation results, an error-correction method is introduced based on the principal-component-weighted iterative method. A new ill-conditioned matrix iteration method is presented. In general, the ill-conditioned matrix equation \(Ax=B\) is solved directly. The solution \(x\) appears easily in the ill-conditioned matrix, and the calculation error is large. When performing the calculation, enter matrix \(C\) first and rewrite Equation (77) as:  
\[ACy=B \tag{83}\]  
where matrix \(C\) is an \(N\)\(\times\)\(N\) nonsingular matrix. The vector y is solved according to Equation (83). The error can be corrected by Equation (84) to obtain a solution with less error. \[x=Cy \tag{84}\]  
The error is transferred to the intermediate variable y by Equation (84) to avoid solving \(x\). Improve the accuracy of solving \(x\). Then, add the variable \(\alpha x\) to both sides of the transformed matrix Equation (83) to obtain:  
\[(AC+\alpha E)y=B+\alpha y \tag{85}\]According to the iterative formula, the relationship is as follows:  
\[(AC+\alpha E)y^{(k+1)}=B+\alpha y^{(k)} \tag{86}\]  
The matrix Equation (86) is solved to obtain the intermediate variable \(y\). Finally, the exact solution \(x\) is obtained through Equation (84). The ill-conditioned matrix parameter iteration method combines the error-correction shift with the principal-component-weighted iteration method. It improves the traditional principal-component-weighted iterative method. Firstly, the matrix solution set \(X\) is converted to the matrix \(CY\), and the weighted iterative method of principal components is used to solve \(ACy=B\), thus obtaining the intermediate variable \(Y\). Then, the exact solution x is obtained from the matrix \(x=Cy\). This method can further improve the accuracy and stability of understanding. The specific steps are as follows:  
1.