## 3 Efficient Modal Modification Analysis Method
### Neumann Series Expansion- and Reduced Basis-Based Preconditioner Method
With the help of the preconditioner and Equation (14), one obtains  
\[\left(\mathbf{I}_{N}-\zeta_{i}\mathbf{B}_{m}^{-1}\Delta\mathbf{A}\right) \mathbf{\Phi}_{i}=\zeta_{i}\mathbf{B}_{m}^{-1}\mathbf{A}\mathbf{\Phi}_{i} \tag{16}\]  
where  
\[\zeta_{i}=\frac{1}{\lambda_{mi}} \tag{17}\]  
Next, we will use the Neumann series expansion to approximate the modified mode shapes. The process of obtaining an \(r\)-order Neumann basis procedure is shown in Algorithm 1. By using the Neumann series expansion, one obtains  
\[\mathbf{\Phi}_{mi}=\zeta_{i}\sum_{k=0}^{\infty}\left(\zeta_{i}\mathbf{B}_{m}^ {-1}\Delta\mathbf{A}\right)^{k}\mathbf{B}_{m}^{-1}\mathbf{A}\mathbf{\Phi}_{i} \tag{18}\]  
On the condition that the mode shapes do not change significantly, it may be convenient to approximate the eigenvectors with known mode shapes:  
\[\mathbf{\Phi}_{mi}\approx\sum_{k=0}^{\infty}\zeta_{i}^{-(k+1)}\mathbf{r}_{i} ^{(k)} \tag{19}\]  
where  
\[\left\{ \begin{array}{ll}\mathbf{r}_{i}^{(0)}=\mathbf{B}_{m}^{-1}\mathbf{A} \mathbf{\Phi}_{i}\\ \mathbf{r}_{i}^{(k)}=\mathbf{B}_{m}^{-1}\Delta\mathbf{A}\mathbf{r}_{i}^{(k-1) },\quad\forall k\geq 1\end{array} \right. \tag{20}\]  
As stated before, the calculation of the basis vectors requires the inverse of the matrix \(\mathbf{B}_{m}\), whose operation count is trivial due to the fact that it is constructed as a diagonal matrix. Thus, we define the \(r\)-order Neumann series expansion-based Krylov subspace as [48]  
\[\mathcal{R}_{r}(\mathbf{A},\mathbf{B};\mathbf{\Phi}_{i})=\text{span}\left\{ \mathbf{r}_{i}^{(0)},\mathbf{r}_{i}^{(1)},\dots,\mathbf{r}_{i}^{(n-1)}\right\} \tag{21}\]  
It is clear that the Neumann series expansion-based subspace \(\mathcal{R}_{r}(\mathbf{A},\mathbf{B};\mathbf{\Phi}_{i})\) is constructed as a linear combination of the traditional subspace \(\mathcal{K}_{n}(\mathbf{A},\mathbf{B};\mathbf{\Phi}_{i})\). Note that \(\mathbf{B}_{m}\) is a diagonal matrix, and the inverse process could be easy since the basis vectors can be efficiently calculated by only sparse matrix multiplications using a simple iterative procedure. However, the coefficients of each basis vector are still unknown and, therefore, need to be determined based on a preconditioner technique. Next, we introduce the reduced-basis-based preconditioner technique to solve these coefficients. ```
Input:\(\mathbf{A},\Delta\mathbf{A},\mathbf{B}_{m},\mathbf{\Phi}_{i}r\) Output:\(\mathcal{R}_{r}(\mathbf{A},\mathbf{B};\mathbf{\Phi}_{i})=\text{span}\Big{\{}\mathbf{r}_{i}^{(0)}, \mathbf{r}_{i}^{(1)},\ldots,\mathbf{r}_{i}^{(n-1)}\Big{\}}\)
1\(\mathbf{r}_{i}^{(0)}=\mathbf{B}_{m}^{-1}\mathbf{A}\mathbf{\Phi}_{i}\) for\(k=2,3,...,r\)do
2\(\mathbf{r}_{i}^{(k)}=\mathbf{B}_{m}^{-1}\Delta\mathbf{A}\mathbf{r}_{i}^{(k-1)}\)
3 end for
```  
**Algorithm 1**Neumann basis procedure  
With the proposed \(r\)-order Neumann subspace basis vectors, the modified mode shape vector can be rewritten as  
\[\mathbf{\Phi}_{mi}{=}\sum_{k=0}^{r-1}\alpha_{i}^{(k)}\mathbf{r}_{i}^{(k)}=\mathbf{ R}_{i}\mathbf{\alpha}_{i} \tag{22}\]  
where \(\mathbf{R}_{i}\) denotes the matrix of the explicit Neumann series expansion-based Krylov basis of \(\mathcal{R}_{r}(\mathbf{A},\mathbf{B};\mathbf{\Phi}_{i})\) and  
\[\mathbf{R}_{i}=\left[ \begin{array}{cccc}\mathbf{r}_{i}^{(0)},&\mathbf{r}_{ i}^{(1)},&\ldots,&\mathbf{r}_{i}^{(r-1)}\end{array} \right],\quad\mathbf{\alpha}_{i}= \left[ \begin{array}{cccc}\alpha_{i}^{(0)},&\alpha_{i}^{(1)},&\ldots,&\alpha_ {i}^{(r-1)}\end{array} \right]^{\text{T}} \tag{23}\]  
Since the approximate eigenvectors should satisfy the eigenproblem in Equation (15), the unknown coefficients \(\mathbf{\alpha}_{i}\) can be determined by substituting the approximate eigenvectors back into Equation (15). However, note that the solution of the unknown coefficients of \(\mathbf{\alpha}_{i}\) may still be time-consuming, since the dimensions of Equation (15) are still \(N_{s}\times N_{s}\). To reduce the dimension of the system and maintain the robustness of the system's eigenvalue problem, a reduced-basis preconditioner is used here.