# Structure Approximation-Based Preconditioning for Solving Tempered Fractional Diffusion Equations
## 4 Spectral Analysis  
In this section, we discuss the spectral properties of the preconditioned matrix \(P_{3}^{-1}A\). First, we review some off-diagonal decay property reasoning. **Lemma 2** ([25]). : _Let \(\beta\in(1,2)\) and \(A=(a_{i,j})\in\mathcal{L}_{\beta+1}\) be a matrix, which satisfies_  
\[|a_{i,j}|\leq\frac{c}{(1+|i-j|)^{\beta+1}}\]  
_for \(\beta>0\), and some constant \(c>0\). Then \(\exists\) a constant \(\omega\) s.t. \(\|L\|_{\infty}\leq\omega\)._  
**Lemma 3** ([25]). : _Let \(A\) and \(P_{1}\) be as defined in Section 3, \(d(x)\in C^{1}[x_{L},x_{R}]\). For any given \(\varepsilon>0\), there exists a constant \(c_{1}\) and an integer \(N_{1}\), such that for \(l\geq N_{1}\),_  
\[\|P_{1}^{-1}-A^{-1}\|_{\infty}\leq c_{1}\max_{ \begin{subarray}{c}\leq i\leq N \end{subarray} }\Delta(x_{i},l)+\varepsilon,\]  
_where_  
\[\Delta(x_{i},l)=\max_{ \begin{subarray}{c}i-l\leq k\leq i+l\end{subarray} }|x_{ k}-x_{i}|=(l-1)h.\]  
Next, we discuss the spectral properties of \(P_{3}^{-1}A\) by analyzing the approximation \(P_{3}^{-1}-A^{-1}\). Due to  
\[P_{3}^{-1}-A^{-1}=P_{3}^{-1}-P_{2}^{-1}+P_{2}^{-1}-P_{1}^{-1}+P_{1}^{-1}-A^{- 1},\]  
we will respectively analyze the properties of \(P_{3}^{-1}-P_{2}^{-1},P_{2}^{-1}-P_{1}^{-1}\) and \(P_{1}^{-1}-A^{-1}\). And the approximation property of \(P_{1}\) and A is given in Lemma 3 above. Then, we give the results of the approximation property of \(P_{2}^{-1}\) and \(P_{1}^{-1}\). **Theorem 1**. : _Let \(P_{1}\) and \(P_{2}\) be defined as in Section 3. The approximation \(P_{2}^{-1}\) to \(P_{1}^{-1}\) satisfies_  
\[P_{2}^{-1}-P_{1}^{-1}=E_{2}+F_{2},\]  
_where \(E_{2}\) and \(F_{2}\) are matrices of small norm and low rank, respectively, i.e., \(\|E_{2}\|<c_{2}\cdot\epsilon\) and \(rank(F_{2})\leq 4c_{5}\) for some positive constant \(c_{2}\), \(\epsilon\) and \(\varsigma\)._  
Proof. : We can write \(P_{2}^{-1}-P_{1}^{-1}\) as  
\[P_{2}^{-1}-P_{1}^{-1} =\sum_{i=1}^{N}e_{i}e_{i}^{T}\Big{(}G_{i}^{-1}-K_{i}^{-1}\Big{)}\] \[=\sum_{i=1}^{N}e_{i}e_{i}^{T}K_{i}^{-1}(K_{i}-G_{i})G_{i}^{-1}\] \[=\sum_{i=1}^{N}d_{i}e_{i}e_{i}^{T}K_{i}^{-1}(G-\tau(G))G_{i}^{-1},\]  
where \(\|G_{i}^{-1}\|_{\infty}<\eta^{-1}\); for detailed proof, refer to Lemma 4.7 in [25]. As shown in [15], the matrix difference \(G-\tau(G)\) admits a decomposition \(E_{1}+F_{1}\), where \(E_{1}\) coincides with \(G-\tau(G)\) within the upper-left and lower-right \((N-1)\times(N-1)\) submatrices and vanishes elsewhere, while satisfying \(rank(E_{1})\leq 2(N-1)\). Further, if we let \(\varepsilon>0\) be fixed, then we have \(\|F_{1}\|_{2}<\varepsilon\). Given that \(K_{i}^{-1}\) exhibits off-diagonal decay, it admits a splitting \(\hat{K}_{i}+\hat{K}_{i}\), where  
\[\hat{K}_{i}= \begin{bmatrix}*&\cdots&*&0&\cdots&0\\ \vdots&\ddots&&\ddots&\ddots&\vdots\\ *&&\ddots&&\ddots&0\\ 0&\ddots&&\ddots&&*\\ \vdots&\ddotsddots&&\ddots&\vdots\\ 0&\cdots&0&*&\cdots&*\end{bmatrix} \quad\text{and}\quad\hat{K}_{i}= \begin{bmatrix}0& \cdots&0&*&\cdots&*\\ \vdots&\ddots&&\ddots&\ddots&\vdots\\ 0&&\ddots&&\ddots&*\\ *&\ddots&&\ddots&&0\\ \vdots&\ddots&\ddots&&\ddots&\ddots&\vdots\\ *&\cdots&*&0&\cdots&0\end{bmatrix} ,\]  
where \"\(*\)\" denotes the nonzero entries.