## 4 Proposed Methods
### Update of \(\mathbf{X}\)
Based on (23), the objective of updating \(\bar{\mathbf{z}}\) is to obtain MMSE, while updating \(\bar{\mathbf{x}}\) maps \(\bar{\mathbf{z}}\) onto the discrete set \(\mathcal{X}\). The update of \(\mathbf{\eta}\) aims to align \(\bar{\mathbf{x}}\) and \(\bar{\mathbf{z}}\) in the consensus phase. However, since \(\bar{\mathbf{x}}\) is discrete and \(\bar{\mathbf{z}}\) is continuous, the consensus adjustment in Equation (23c) may not be effective. To resolve this, we set \(\mathbf{\eta}=\mathbf{0}\) and rewrite Equation (23) as  
\[\bar{\mathbf{z}}^{(k+1)} =\left(\bar{\mathbf{C}}^{H}\bar{\mathbf{C}}+c\mathbf{\mathrm{I}}_{2 LB}\right) ^{-1}(c\mathbf{x}^{(k)}+\bar{\mathbf{C}}^{H}\bar{\mathbf{d}}) \tag{24a}\] \[\bar{\mathbf{x}}^{(k+1)} =g_{\mathcal{X}}(\bar{\mathbf{z}}^{(k+1)}) \tag{24b}\]  
Within each updating iteration, there remains a complex matrix inversion operating, which brings a significant computational burden. We propose an adaptively tuned DPR algorithm where \(\mathbf{\mathrm{E}}=(\bar{\mathbf{C}}^{H}\bar{\mathbf{C}}+c\mathbf{\mathrm{I}}_{2 LB})^{-1}\). Based on linear estimation theory [39; 25], we ensure that the estimation in Equation (24a) achieves MMSE. Since \(\bar{\mathbf{x}}\) is unknown during iteration, \(c\) can be updated using the estimated error variance by  
\[\frac{1}{c^{(k)}}=\frac{\left\|\bar{\mathbf{d}}-\bar{\mathbf{C}}\bar{\mathbf{x}}^{(k)} \right\|_{2}^{2}}{\mathrm{Tr}(\bar{\mathbf{C}}^{H}\bar{\mathbf{C}})} \tag{25}\]  
In large-scale MU-MIMO systems, \(\mathbf{\mathrm{E}}\) exhibits the property of a strictly diagonally dominant (SDD) [39]. Using a scaled identity matrix, we can approximate \(\mathbf{\mathrm{E}}\). This step can be listed as  
\[\bar{\mathbf{z}}^{(k+1)} =\arg\min_{\bar{\mathbf{z}}}L(\bar{\mathbf{z}},\bar{\mathbf{x}}^{(k)},\mathbf{\eta} ^{(k)}) \tag{22a}\]0  
where \(\mathbf{\Xi}=\bar{\mathbf{C}}^{H}\bar{\mathbf{C}}\). Step \((a)\) is based on \(\sum\limits_{i=1}^{2 LB}\mathbf{\Xi}_{ii}\gg\left\|\bar{\mathbf{d}}-\bar{\mathbf{C}}\bar{ \mathbf{x}}^{(k)}\right\|_{2}^{2}\), which yields \(c\gg 1\). Based on the conclusion of Equation (26), to reduce the computational complexity associated with matrix inversion, we approximate \(\mathbf{\mathrm{E}}\) as follows: \(\mathbf{\mathrm{E}}\approx\frac{1}{c}\mathbf{\mathrm```
0: H, S, \(P_{T}\), \(n_{max}\), \(k_{max}\), weighting factor \(0\leq\delta\leq 1\), \(n=0\). 0:\(\bar{\mathbf{x}}=\bar{\mathbf{x}}^{(n)}\). 1:while\(n<n_{max}\)do
2: Update \(\rho^{(n+1)}\) based on Equation (15). 3: Update \(\mathbf{X}_{0}^{(k+1)}\) based on Equation (16). 4:\(k=0\). 5:while\(k<k_{max}\)do
6: Update \(c^{(k+1)}\) based on Equation (25). 7: Update \(\bar{\mathbf{z}}^{(k+1)}\) based on Equation (24a). 8: Update \(\bar{\mathbf{x}}^{(k+1)}\) based on Equation (24b). 9:\(k=k+1\). 10:endwhile
11:\(\bar{\mathbf{x}}^{(n+1)}=\bar{\mathbf{x}}^{(k+1)}\). 12:\(n=n+1\). 13:endwhile
```

**Algorithm 1** DPR algorithm for solving Equation (19). ### Complexity Analysis

In this section, we analyze the computational complexity of the large-scale MIMO DFRC precoding algorithm from the perspective of real-valued multiplications, particularly when \(B\gg U\). Algorithm 1 summarizes the proposed method to thus solve the three sub-problems mentioned above. We update the three variables iteratively, noting that the computational cost scales linearly with the number of iterations. Specifically, updating the precoding factor \(\rho\) using Equation (15) requires a computational effort of \(\mathcal{O}(BUL+BL^{2})\) per iteration. The reference radar precoding matrix \(\mathbf{X}_{0}\) is updated through four matrix multiplications, resulting in a complexity of \(\mathcal{O}(BL^{2}+B^{2}L+B^{3})\). In the DPR algorithm, each iteration involves updating the primary variable based on Equation (25), which has a complexity of \(\mathcal{O}(B^{2}L^{3}(U+B))\). Additionally, the update of \(\bar{\mathbf{z}}\) according to Equation (24a) requires \(\mathcal{O}(L^{2}(U+B)B)\) operations. The projection step for \(\bar{\mathbf{x}}\) is negligible in terms of computational cost.