## 2 Proposed Order Reduction Methodology
### Adaptive Scheme
Since the moments at \(f_{\min}\), \(f_{\rm mid}\), and \(f_{\max}\) have already been added to the subspace of the projection basis, the next expansion point added to the subspace is chosen at the middle point between 1, 3 (\(f_{\min}\), \(f_{\rm mid}\)) by Equation (8), with the serial number of the new point marked as 1-3. If the transformation matrix constructed by 1, 1-3, 3, and 2 does not satisfy the accuracy tolerance, a new point is chosen, i.e., the middle point between 3 and 2, with the serial number of the new point marked as 3-2. It can be seen that the middle point between every two adjacent points is used, with the serial numbers corresponding to the points as 1, 1-3, 3, 3-2, 2. If the error is still larger than \(tol^{\rm s}\), the middle point between every two adjacent points is chosen, i.e., the middle point between 1 and 1-3, 1-3 and 3, 3 and 3-2, and 3-2 and 2. The iterative procedure is repeated until the error is smaller than the predefined value. Moreover, an orthogonalization operation is applied to the so-far generated projection basis to minimize its size by discarding its redundant vectors. Although the orthogonalization and compression procedures increase computational cost, a significant improvement in performances is obtained in the projection size and reliability of the whole reduction process. Algorithms 1-3 give the pseudocodes of the proposed MOR methodology. Algorithm 1 is the main loop of the proposed MOR scheme for 3-D transient field problems. Algorithm 2 gives the block-Arnoldi-based moment-matching methodology at a single frequency point. Algorithm 3 presents the error estimator used in the scheme. The new Krylov subspace, a combination of the different subspaces for each expansion point, is formulated as  
\[\begin{array}{ll}\mathbf{U}_{m}&={\rm span}\left\{\underbrace{\mathbf{P}(s_{1}),\ (( \mathbf{A}(s_{1}))^{-1}\mathbf{A})\mathbf{P}(s_{1}),\ldots,((\mathbf{A}(s_{1}))^{-1}\mathbf{A})^{q_{ 1}}\mathbf{P}(s_{1})}_{\text{point }s=s_{1}},\right.\\ &&\qquad\left.\cdots\underbrace{\mathbf{P}(s_{k}  
where, \(s_{1},\cdots,s_{k}\) are the expansion points used to obtain the ROM, and \(q_{1},\cdots q_{k}\) are the number of moments matched at the corresponding expansion point. ```
Input: The range of the broadband frequency (\(f_{\text{min}}\),\(f_{\text{max}}\)), coefficient matrices of NTM model (**A**,**B**,**P**), global error tolerance \(tol\)\(\tilde{s}\) and local error tolerance \(tol\)\(\tilde{s}\). 1. \(f_{s}=f_{\text{min}}\)
2. [\(\mathbf{Q}_{0}\)] = _S-POINT_ (\(f_{s}\), \(tol^{s}\))
3. \(error^{s}=EST-ERROR(f_{\text{min}}\), \(f_{\text{max}}\), \(\mathbf{Q}_{0}\), \(\boldsymbol{A}\), \(\boldsymbol{B}\), \(\boldsymbol{P}\))
4. **while**\(error^{s}>tol\)\(\tilde{s}\)do \(f_{s}\leftarrow\) set new expansion point \(N_{s}\gets N_{s}+1\)
5. [\(\boldsymbol{Q}_{1}\)] = _S-POINT_ (\(f_{s}\), \(tol^{s}\))
6. [\(\boldsymbol{A}=\mathbf{Q}_{0}{}^{T}\mathbf{A}\mathbf{Q}_{0}\tilde{\boldsymbol{ B}}=\mathbf{Q}_{0}{}^{T}\mathbf{B}\mathbf{Q}_{0}\tilde{\boldsymbol{P}}=\mathbf{Q}_{0}{}^{T}\boldsymbol {P}\) Output: The coefficient matrices of a reduced system(\(\boldsymbol{\widetilde{A}}\),\(\boldsymbol{\widetilde{B}}\),\(\boldsymbol{\widetilde{P}}\))
```  
**Algorithm 1** Main loop of the proposed scheme for 3-D transient field problems  
```
Input: The current expansion point \(f_{s}\), local error tolerance \(tol^{s}\) and coefficient matrices of the high-fidelity model (\(\boldsymbol{\widetilde{A}}\),\(\boldsymbol{\widetilde{B}}\),\(\boldsymbol{\widetilde{P}}\))
1. \(r=1\)
2. \(s=f_{s}\)
3. **while**\(error^{1}>tol^{s}\)\(\tilde{s}\)
4. \(\mathbf{Q}_{0}=blockArnoldi(s,r,\boldsymbol{A},\boldsymbol{B},\boldsymbol{P})\)
5. \(error^{1}\) = EST-ERROR (\(f_{s}\),\(\mathbf{Q}_{0}\),\(\boldsymbol{A}\),\(\boldsymbol{B}\),\(\boldsymbol{P}\))
6. \(r\gets r+1\)
7.