# Structure Approximation-Based Preconditioning for Solving Tempered Fractional Diffusion Equations
## 4 Spectral Analysis
Considering  
\[K_{i}^{-1}(G-\tau(G))G_{i}^{-1} =K_{i}^{-1}(E_{1}+F_{1})G_{i}^{-1}\] \[=K_{i}^{-1}E_{1}G_{i}^{-1}+K_{i}^{-1}F_{1}G_{i}^{-1}\] \[=K_{i}^{-1}E_{1}G_{i}^{-1}+(\hat{K}_{i}+\hat{K}_{i})F_{1}G_{i}^{-1}\] \[=(K_{i}^{-1}E_{1}+\hat{K}_{i}F_{1})G_{i}^{-1}+\hat{K}_{i}F_{1}G_{i }^{-1},\]  
we derive  
\[\|(K_{i}^{-1}E_{1}+\hat{K}_{i}F_{1})G_{i}^{-1}\|_{\infty} \leq(\|K_{i}^{-1}\|_{\infty}\cdot\|E_{1}\|_{\infty}+\|\hat{K}_{i} \|_{\infty}\cdot\|F_{1}\|_{\infty})\cdot\|G_{i}^{-1}\|_{\infty}\] \[\leq\varepsilon(\|K_{i}^{-1}\|_{\infty}+\|F_{1}\|_{\infty})\cdot \|G_{i}^{-1}\|_{\infty}\]  
and  
\[\|\sum_{i=1}^{N}e_{i}e_{i}^{\tau}(K_{i}^{-1}E_{1}+\hat{K}_{i}F_{1 })G_{i}^{-1}\|_{\infty} =\max_{1\leq i\leq N}\|e_{i}^{\tau}(K_{i}^{-1}E_{1}+\hat{K}_{i}F_{ 1})G_{i}^{-1}\|_{1}\] \[=\max_{1\leq i\leq N}\|(K_{i}^{-1}E_{1}+\hat{K}_{i}F_{1})G_{i}^{- 1}\|_{\infty}\] \[\leq\varepsilon\cdot\max_{1\leq i\leq N}(\|K_{i}^{-1}\|_{\infty} +\|F_{1}\|_{\infty})\cdot\|G_{i}^{-1}\|_{\infty}.\]  
Since \(\|K_{i}^{-1}\|_{\infty}\) and \(\|F_{1}\|_{\infty}\) are limited due to Lemma 2, it follows that \(\|G_{i}^{-1}\|_{\infty}\) is also bounded. In other words, there is a constant \(c_{2}>0\) such that  
\[\left\|\sum_{i=1}^{N}e_{i}e_{i}^{\tau}(K_{i}^{-1}E_{1}+\hat{K}_{i}F_{1})G_{i}^ {-1}\right\|_{\infty}\leq c_{2}\cdot\varepsilon.\]  
Next, we focus on the matrix product \(\hat{K}_{i}F_{1}G_{i}^{-1}\). We assume the block dimensions of \(\hat{K}_{i}\) align with those of \(F_{1}\); otherwise, the smaller block is extended accordingly. Through structural analysis of \(\hat{K}_{i}\) and \(F_{1}\), the subsequent outcomes are derived:  
\[\hat{K}_{i}F_{1}G_{i}^{-1}= \begin{pmatrix}0&0&\ldots&\ldots&0&+\\ 0&0&\ldots&\ldots&0&+\\ 0&0&\ldots&\ldots&0&0\\ \vdots&\vdots&&&\vdots&\vdots\\ 0&0&\ldots&\ldots&0&0\\ +&0&\ldots&\ldots&0&0\\ +&0&\ldots&\ldots&0&0\end{pmatrix} ,\quad G_{i}^{-1}= \begin{pmatrix}+&+&\ldots& \ldots&+&+\\ +&+&\ldots&\ldots&+&+\\ 0&0&\ldots&\ldots&0&0\\ \vdots&\vdots&&&\vdots&\vdots\\ 0&0&\ldots&\ldots&0&0\\ +&+&\ldots&\ldots&+&+\\ +&+&\ldots&\ldots&+&+\end{pmatrix} ,\]  
where '\(+\)' denotes non-negative entries. Therefore, we have  
\[\text{rank}\left(\sum_{i=1}^{N}e_{i}e_{i}^{\tau}\hat{K}_{i}F_{1}G_{i}^{-1} \right)\leq 4\varsigma,\]  
where \(\varsigma\) is the dimension of the blocks in \(\hat{K}_{i}F_{1}G_{i}^{-1}\). Consequently,  
\[\sum_{i=1}^{N}e_{i}e_{i}^{\tau}\Big{(}K_{i}^{-1}-K_{i}^{-1}\Big{)} =\sum_{i=1}^{N}e_{i}e_{i}^{\tau}\Big{(}K_{i}^{-1}E_{1}+\hat{K}_{i} F_{1}\Big{)}G_{i}^{-1}+\sum_{i=1}^{N}e_{i}e_{i}^{\tau}\hat{K}_{i}F_{1}G_{i}^{-1}\] \[=E_{2}+F_{2},\]  
where \(E_{2}\) and \(F_{2}\) are matrices of small norm and low rank, respectively. Finally, we focus on the approximation property of \(P_{3}\) and A.  
Theorem 4.2: _Let \(P_{3},P_{2},P_{1}\) and \(A\) be defined as shown previously. Supposing \(l\) is sufficiently smaller than \(N\), there exists \(N_{2}\) such that for \(N>N_{2}\), it holds that_  
\[P_{3}^{-1}-A^{-1}=E+F,\]  
_where \(E\) and \(F\) are of small norm and of low rank, respectively._  
Proof: By Theorem 4.2 in [25], we know that \(P_{3}^{-1}-P_{2}^{-1}=E_{3}-F_{3}\), where \(E_{3}\) is of small norm, satisfying \(\|E_{3}\|_{\infty}<c_{3}\) and \(F_{3}\) is of small rank. Due to Lemma 3, we know  
\[\|P_{1}^{-1}-A^{-1}\|_{\infty}\leq\max_{1\leq i\leq N}\Delta(x_{i},N_{1})+ \varepsilon=c_{1}(N_{1}-1)h+\varepsilon.\]  
Let \(N_{2}\) be an integer such that \((N_{1}-1)h<\varepsilon\), we have  
\[P_{3}^{-1}-A^{-1} =P_{3}^{-1}-P_{2}^{-1}+P_{2}^{-1}-P_{1}^{-1}+P_{1}^{-1}-A^{-1}\] \[=E_{3}+F_{3}+E_{2}+F_{2}+P_{1}^{-1}-A^{-1}\] \[=E_{3}+E_{2}+(P_{1}^{-1}-A^{-1})+F_{3}+F_{2}\] \[\triangleq E+F,\]  
where \(E=E_{3}+E_{2}+(P_{1}^{-1}-A^{-1})\) and \(F=F_{3}+F_{2}\). As  
\[\|E\|_{\infty}=\|E_{3}\|_{\infty}+\|E_{2}\|_{\infty}+\|P_{1}^{-1}-A^{-1}\|_{ \infty}<(c_{1}+c_{2}+c_{3}+1)\cdot\varepsilon,\]  
then the conclusion follows.