Solving a system of complex matrix equations using a gradient-based method and its application in image restoration

Abstract

This study presents some new iterative algorithms based on the gradient method to solve general constrained systems of conjugate transpose matrix equations for both real and complex matrices. In addition, we analyze the convergence properties of these methods and provide numerical techniques to determine the solutions. Then we prove that the optimal parameters of the new algorithm satisfy a constrained optimization problem. The effectiveness of the proposed iterative methods is demonstrated through various numerical examples employed in this study and compared the results by some existing algorithms.

Appendix A

The proof of Theorem 3.8: Initially, we define the error matrix as follows:

$$\begin{aligned} \xi (\varsigma )=L(\varsigma )-L^{\star }. \end{aligned}$$

Employing the procedure outlined in Algorithm 4 will yield the subsequent outcome

$$\begin{aligned}&\xi (\varsigma +1)=\xi (\varsigma )-\omega _{1}\left( \sum _{\tau =1}^{\gamma } A_{\tau }^HZ_{\tau }(\varsigma )B_{\tau }^H+\sum _{\tau =1}^{\gamma } C_{\tau }^T\overline{Z_{\tau }(\varsigma )}D_{\tau }^T \right) \nonumber \\&\qquad \qquad \qquad \qquad \qquad -\omega _{2}\bigg (\sum _{\tau =1}^{\gamma } V^HZ_{\tau }(\varsigma )B_{\tau }^H+\sum _{\tau =1}^{\gamma } A_{\tau }^HZ_{\tau }(\varsigma )W^H +\sum _{\tau =1}^{\gamma } V^T\overline{Z_{\tau }(\varsigma )}D_{\tau }^T+\sum _{\tau =1}^{\gamma } C_{\tau }^T\overline{Z_{\tau }(\varsigma )}W^T\bigg ), \end{aligned}$$

(6.1)

where

$$\begin{aligned} Z_{\tau }(\varsigma )=A_{\tau }\xi (\varsigma )B_{\tau }+C_{\tau }\overline{\xi (\varsigma )}D_{\tau }. \end{aligned}$$

By considering the definition of the Frobenius norm, we obtain:

$$\begin{aligned}{} & {} \Vert \xi (\varsigma +1)\Vert _F^2 =\Vert \xi (\varsigma )\Vert _F^2-\omega _{1}\textrm{trace} \bigg (\sum _{\tau =1}^{\gamma } \xi (\varsigma )B_{\tau }Z_{\tau }(\varsigma )^HA_{\tau }+\sum _{\tau =1}^{\gamma } A_{\tau }^HZ_{\tau }(\varsigma )B_{\tau }^H\xi (\varsigma )^{H}\bigg )\\{} & {} -\omega _{1}\textrm{trace} \bigg (\sum _{\tau =1}^{\gamma } \xi (\varsigma )\overline{D_{\tau }}Z_{\tau }(\varsigma )^T\overline{C_{\tau }}+\sum _{\tau =1}^{\gamma } C_{\tau }^T\overline{Z_{\tau }(\varsigma )}D_{\tau }^T\xi (\varsigma )^H\bigg )\\{} & {} -\omega _{2}\textrm{trace} \bigg (\sum _{\tau =1}^{\gamma } \xi (\varsigma )B_{\tau }Z_{\tau }(\varsigma )^HV+\sum _{\tau =1}^{\gamma } V^HZ_{\tau }(\varsigma )B_{\tau }^H\xi (\varsigma )^H\bigg ) -\omega _{2}\textrm{trace} \bigg (\sum _{\tau =1}^{\gamma } \xi (\varsigma )WZ_{\tau }(\varsigma )^HA_{\tau }+\sum _{\tau =1}^{\gamma } A_{\tau }^HZ_{\tau }(\varsigma )W^H\xi (\varsigma )^{H}\bigg )\\{} & {} -\omega _{2}\textrm{trace} \bigg (\sum _{\tau =1}^{\gamma } \xi (\varsigma )\overline{D_{\tau }}Z_{\tau }(\varsigma )^T\overline{V}+\sum _{\tau =1}^{\gamma } V^T\overline{Z_{\tau }(\varsigma )}D_{\tau }^T\xi (\varsigma )^{H}\bigg ) -\omega _{2}\textrm{trace} \bigg (\sum _{\tau =1}^{\gamma } \xi (\varsigma )\overline{W}Z_{\tau }(\varsigma )^T\overline{C_{\tau }}+\sum _{\tau =1}^{\gamma } C_{\tau }^T\overline{Z_{\tau }(\varsigma )}W^T\xi (\varsigma )^{H}\bigg )\\{} & {} +\omega _{1}^{2}\Vert \sum _{\tau =1}^{\gamma } A_{\tau }^HZ_{\tau }(\varsigma )B_{\tau }^H+\sum _{\tau =1}^{\gamma } C_{\tau }^T\overline{Z_{\tau }(\varsigma )}D_{\tau }^T\Vert _F^2\\{} & {} +\omega _{2}^{2}\Vert \sum _{\tau =1}^{\gamma } V^HZ_{\tau }(\varsigma )B_{\tau }^H+\sum _{\tau =1}^{\gamma } A_{\tau }^HZ_{\tau }(\varsigma )W^H+\sum _{\tau =1}^{\gamma } V^T\overline{Z_{\tau }(\varsigma )}D_{\tau }^T+\sum _{\tau =1}^{\gamma } C_{\tau }^T\overline{Z_{\tau }(\varsigma )}W^T \Vert _F^2\\{} & {} +\omega _{1}\omega _{2}\textrm{trace}\bigg ( \bigg ( \sum _{\tau =1}^{\gamma } V^HZ_{\tau }(\varsigma )B_{\tau }^H+\sum _{\tau =1}^{\gamma } A_{\tau }^HZ_{\tau }(\varsigma )W^H +\sum _{\tau =1}^{\gamma } V^T\overline{Z_{\tau }(\varsigma )}D_{\tau }^T+\sum _{\tau =1}^{\gamma } C_{\tau }^T\overline{Z_{\tau }(\varsigma )}W^T\bigg )\\{} & {} \times \left( \sum _{\tau =1}^{\gamma } A_{\tau }^HZ_{\tau }(\varsigma )B_{\tau }^H+\sum _{\tau =1}^{\gamma } C_{\tau }^T\overline{Z_{\tau }(\varsigma )}D_{\tau }^T \right) ^{H}\bigg ) +\omega _{1}\omega _{2}\textrm{trace}\bigg ( \left( \sum _{\tau =1}^{\gamma } A_{\tau }^HZ_{\tau }(\varsigma )B_{\tau }^H+\sum _{\tau =1}^{\gamma } C_{\tau }^T\overline{Z_{\tau }(\varsigma )}D_{\tau }^T \right) \\{} & {} \qquad \qquad \qquad \qquad \qquad \qquad \times \bigg (\sum _{\tau =1}^{\gamma } V^HZ_{\tau }(\varsigma )B_{\tau }^H+\sum _{\tau =1}^{\gamma } A_{\tau }^HZ_{\tau }(\varsigma )W^H +\sum _{\tau =1}^{\gamma } V^T\overline{Z_{\tau }(\varsigma )}D_{\tau }^T+\sum _{\tau =1}^{\gamma } C_{\tau }^T\overline{Z_{\tau }(\varsigma )}W^T\bigg )^{H}\bigg ) \end{aligned}$$

$$\begin{aligned}{} & {} =\Vert \xi (\varsigma )\Vert _F^2-\omega _{1}\textrm{trace} \bigg (\sum _{\tau =1}^{\gamma } \xi (\varsigma )B_{\tau }Z_{\tau }(\varsigma )^HA_{\tau }+\sum _{\tau =1}^{\gamma } A_{\tau }^HZ_{\tau }(\varsigma )B_{\tau }^H\xi (\varsigma )^{H}\bigg ) \nonumber \\{} & {} -\omega _{1}\textrm{trace} \bigg (\sum _{\tau =1}^{\gamma } \xi (\varsigma )\overline{D_{\tau }}Z_{\tau }(\varsigma )^T\overline{C_{\tau }}+\sum _{\tau =1}^{\gamma } C_{\tau }^T\overline{Z_{\tau }(\varsigma )}D_{\tau }^T\xi (\varsigma )^H\bigg ) \nonumber \\{} & {} -\omega _{2}\textrm{trace} \bigg (\sum _{\tau =1}^{\gamma } \xi (\varsigma )B_{\tau }Z_{\tau }(\varsigma )^HV+\sum _{\tau =1}^{\gamma } V^HZ_{\tau }(\varsigma )B_{\tau }^H\xi (\varsigma )^H\bigg ) -\omega _{2}\textrm{trace} \bigg (\sum _{\tau =1}^{\gamma } \xi (\varsigma )WZ_{\tau }(\varsigma )^HA_{\tau }+\sum _{\tau =1}^{\gamma } A_{\tau }^HZ_{\tau }(\varsigma )W^H\xi (\varsigma )^{H}\bigg ) \nonumber \\{} & {} -\omega _{2}\textrm{trace} \bigg (\sum _{\tau =1}^{\gamma } \xi (\varsigma )\overline{D_{\tau }}Z_{\tau }(\varsigma )^T\overline{V}+\sum _{\tau =1}^{\gamma } V^T\overline{Z_{\tau }(\varsigma )}D_{\tau }^T\xi (\varsigma )^{H}\bigg ) -\omega _{2}\textrm{trace} \bigg (\sum _{\tau =1}^{\gamma } \xi (\varsigma )\overline{W}Z_{\tau }(\varsigma )^T\overline{C_{\tau }}+\sum _{\tau =1}^{\gamma } C_{\tau }^T\overline{Z_{\tau }(\varsigma )}W^T\xi (\varsigma )^{H}\bigg ) \nonumber \\{} & {} +\bigg \Vert \omega _{1}\left( \sum _{\tau =1}^{\gamma } A_{\tau }^HZ_{\tau }(\varsigma )B_{\tau }^H+\sum _{\tau =1}^{\gamma } C_{\tau }^T\overline{Z_{\tau }(\varsigma )}D_{\tau }^T\right) \nonumber \\{} & {} +\omega _{2}\left( \sum _{\tau =1}^{\gamma } V^HZ_{\tau }(\varsigma )B_{\tau }^H+\sum _{\tau =1}^{\gamma } A_{\tau }^HZ_{\tau }(\varsigma )W^H +\sum _{\tau =1}^{\gamma } V^T\overline{Z_{\tau }(\varsigma )}D_{\tau }^T+\sum _{\tau =1}^{\gamma } C_{\tau }^T\overline{Z_{\tau }(\varsigma )}W^T \right) \bigg \Vert _F^2 \nonumber \\{} & {} =\Vert \xi (\varsigma )\Vert _F^2-\omega _{1}\textrm{trace} \bigg (\sum _{\tau =1}^{\gamma } Z_{\tau }(\varsigma )^HA_{\tau }\xi (\varsigma )B_{\tau }+\sum _{\tau =1}^{\gamma } B_{\tau }^H\xi (\varsigma )^{H}A_{\tau }^HZ_{\tau }(\varsigma )\bigg ) \nonumber \\{} & {} -\omega _{1}\textrm{trace} \bigg (\sum _{\tau =1}^{\gamma } \xi (\varsigma )\overline{D_{\tau }}Z_{\tau }(\varsigma )^T\overline{C_{\tau }}+\sum _{\tau =1}^{\gamma } C_{\tau }^T\overline{Z_{\tau }(\varsigma )}D_{\tau }^T\xi (\varsigma )^H\bigg ) \nonumber \\{} & {} -\omega _{2}\textrm{trace} \bigg (\sum _{\tau =1}^{\gamma } Z_{\tau }(\varsigma )^HV\xi (\varsigma )B_{\tau }+\sum _{\tau =1}^{\gamma } B_{\tau }^H\xi (\varsigma )^HV^HZ_{\tau }(\varsigma )\bigg ) -\omega _{2}\textrm{trace} \bigg (\sum _{\tau =1}^{\gamma } Z_{\tau }(\varsigma )^HA_{\tau }\xi (\varsigma )W+\sum _{\tau =1}^{\gamma } W^H\xi (\varsigma )^{H}A_{\tau }^HZ_{\tau }(\varsigma )\bigg ) \nonumber \\{} & {} -\omega _{2}\textrm{trace} \bigg (\sum _{\tau =1}^{\gamma } \xi (\varsigma )\overline{D_{\tau }}Z_{\tau }(\varsigma )^T\overline{V}+\sum _{\tau =1}^{\gamma } V^T\overline{Z_{\tau }(\varsigma )}D_{\tau }^T\xi (\varsigma )^{H}\bigg ) -\omega _{2}\textrm{trace} \bigg (\xi (\varsigma )\overline{W}Z_{\tau }(\varsigma )^T\overline{C_{\tau }}+\sum _{\tau =1}^{\gamma } C_{\tau }^T\overline{Z_{\tau }(\varsigma )}W^T\xi (\varsigma )^{H}\bigg ) \nonumber \\{} & {} +\bigg \Vert \omega _{1}\left( \sum _{\tau =1}^{\gamma } A_{\tau }^HZ_{\tau }(\varsigma )B_{\tau }^H+\sum _{\tau =1}^{\gamma } C_{\tau }^T\overline{Z_{\tau }(\varsigma )}D_{\tau }^T\right) \nonumber \\{} & {} \qquad \qquad \qquad \qquad \qquad +\omega _{2}\left( \sum _{\tau =1}^{\gamma } V^HZ_{\tau }(\varsigma )B_{\tau }^H+A_{\tau }^HZ_{\tau }(\varsigma )W^H +{\sum _{\tau =1}^{\gamma }} {V^{T}}{\overline{Z_{\tau }(\varsigma )}} {D_{\tau }^{T}} +{\sum _{\tau =1}^{\gamma }} {C_{\tau }^{T}} {\overline{Z_{\tau }(\varsigma )}} {W^{T}} \right) \bigg \Vert _{F}^{2}. \end{aligned}$$

(6.2)

It is readily apparent that

$$\begin{aligned}&\textrm{trace} \bigg (\sum _{\tau =1}^{\gamma } \xi (\varsigma )\overline{D_{\tau }}Z_{\tau }(\varsigma )^T\overline{C_{\tau }}+\sum _{\tau =1}^{\gamma } C_{\tau }^T\overline{Z_{\tau }(\varsigma )}D_{\tau }^T\xi (\varsigma )^H\bigg ) \nonumber \\&= \textrm{trace} \bigg (\sum _{\tau =1}^{\gamma } Z_{\tau }(\varsigma )^H{C_{\tau }}\overline{\xi (\varsigma )}{D_{\tau }}+\sum _{\tau =1}^{\gamma } D_{\tau }^H\xi (\varsigma )^TC_{\tau }^H{Z_{\tau }(\varsigma )}\bigg ), \end{aligned}$$

(6.3)

$$\begin{aligned}&\textrm{trace} \bigg (\sum _{\tau =1}^{\gamma } \xi (\varsigma )\overline{D_{\tau }}Z_{\tau }(\varsigma )^T\overline{V}+\sum _{\tau =1}^{\gamma } V^T\overline{Z_{\tau }(\varsigma )}D_{\tau }^T\xi (\varsigma )^{H}\bigg ) \nonumber \\&=\textrm{trace} \bigg (\sum _{\tau =1}^{\gamma } Z_{\tau }(\varsigma )^H{V}\overline{\xi (\varsigma )}D_{\tau }+\sum _{\tau =1}^{\gamma } D_{\tau }^H\xi (\varsigma )^{T}V^H{Z_{\tau }(\varsigma )}\bigg ), \end{aligned}$$

(6.4)

and

$$\begin{aligned}&\textrm{trace} \bigg (\sum _{\tau =1}^{\gamma } \xi (\varsigma )\overline{W}Z_{\tau }(\varsigma )^T\overline{C_{\tau }}+\sum _{\tau =1}^{\gamma } C_{\tau }^T\overline{Z_{\tau }(\varsigma )}W^T\xi (\varsigma )^{H}\bigg ) \nonumber \\&=\textrm{trace} \bigg (\sum _{\tau =1}^{\gamma } Z_{\tau }(\varsigma )^H{C_{\tau }}\overline{\xi (\varsigma )}{W}+\sum _{\tau =1}^{\gamma } W^H\xi (\varsigma )^TC_{\tau }^H{Z_{\tau }(\varsigma )}\bigg ). \end{aligned}$$

(6.5)

Merging these relationships with equation (6.2) yields:

$$\begin{aligned}&\Vert \xi (\varsigma +1)\Vert _F^2=\Vert \xi (\varsigma )\Vert _F^2-\omega _{1}\textrm{trace} \bigg (\sum _{\tau =1}^{\gamma } Z_{\tau }(\varsigma )^HA_{\tau }\xi (\varsigma )B_{\tau }+\sum _{\tau =1}^{\gamma } B_{\tau }^H\xi (\varsigma )^{H}A_{\tau }^HZ_{\tau }(\varsigma )\bigg ) \nonumber \\&-\omega _{1}\textrm{trace} \bigg (\sum _{\tau =1}^{\gamma } Z_{\tau }(\varsigma )^H{C_{\tau }}\overline{\xi (\varsigma )}{D_{\tau }}+\sum _{\tau =1}^{\gamma } D_{\tau }^H\xi (\varsigma )^TC_{\tau }^H{Z_{\tau }(\varsigma )}\bigg ) \nonumber \\&-\omega _{2}\textrm{trace} \bigg (\sum _{\tau =1}^{\gamma } Z_{\tau }(\varsigma )^HV\xi (\varsigma )B_{\tau }+\sum _{\tau =1}^{\gamma } B_{\tau }^H\xi (\varsigma )^HV^HZ_{\tau }(\varsigma )\bigg )\nonumber \\&-\omega _{2}\textrm{trace} \bigg (\sum _{\tau =1}^{\gamma } Z_{\tau }(\varsigma )^HA_{\tau }\xi (\varsigma )W+\sum _{\tau =1}^{\gamma } W^H\xi (\varsigma )^{H}A_{\tau }^HZ_{\tau }(\varsigma )\bigg ) \nonumber \\&-\omega _{2}\textrm{trace} \bigg (\sum _{\tau =1}^{\gamma } Z_{\tau }(\varsigma )^H{V}\overline{\xi (\varsigma )}D_{\tau }+\sum _{\tau =1}^{\gamma } D_{\tau }^H\xi (\varsigma )^{T}V^H{Z_{\tau }(\varsigma )}\bigg )\nonumber \\&-\omega _{2}\textrm{trace} \bigg (\sum _{\tau =1}^{\gamma } Z_{\tau }(\varsigma )^H{C_{\tau }}\overline{\xi (\varsigma )}{W}+\sum _{\tau =1}^{\gamma } W^H\xi (\varsigma )^TC_{\tau }^H{Z_{\tau }(\varsigma )}\bigg ) \nonumber \\ \nonumber \\&+\bigg \Vert \omega _{1}\left( \sum _{\tau =1}^{\gamma } A_{\tau }^HZ_{\tau }(\varsigma )B_{\tau }^H+\sum _{\tau =1}^{\gamma } C_{\tau }^T\overline{Z_{\tau }(\varsigma )}D_{\tau }^T\right) \nonumber \\&+\omega _{2}\left( \sum _{\tau =1}^{\gamma } V^HZ_{\tau }(\varsigma )B_{\tau }^H+\sum _{\tau =1}^{\gamma } A_{\tau }^HZ_{\tau }(\varsigma )W^H +\sum _{\tau =1}^{\gamma } V^T\overline{Z_{\tau }(\varsigma )}D_{\tau }^T+\sum _{\tau =1}^{\gamma } C_{\tau }^T\overline{Z_{\tau }(\varsigma )}W^T \right) \bigg \Vert _F^2 \nonumber \\&=\Vert \xi (\varsigma )\Vert _F^2-\omega _{1}\textrm{trace} \bigg (\sum _{\tau =1}^{\gamma } Z_{\tau }(\varsigma )^H\left( A_{\tau }\xi (\varsigma )B_{\tau }+ {C_{\tau }}\overline{\xi (\varsigma )}{D_{\tau }}\right) \bigg )\nonumber \\&-\omega _{1}\textrm{trace} \bigg (\sum _{\tau =1}^{\gamma }\left( B_{\tau }^H\xi (\varsigma )^{H}A_{\tau }^H + D_{\tau }^H\xi (\varsigma )^TC_{\tau }^H\right) {Z_{\tau }(\varsigma )}\bigg ) \nonumber \\&-\omega _{2}\textrm{trace} \bigg (\sum _{\tau =1}^{\gamma } Z_{\tau }(\varsigma )^HV\xi (\varsigma )B_{\tau }+\sum _{\tau =1}^{\gamma } B_{\tau }^H\xi (\varsigma )^HV^HZ_{\tau }(\varsigma )\bigg )\nonumber \\&-\omega _{2}\textrm{trace} \bigg (\sum _{\tau =1}^{\gamma } Z_{\tau }(\varsigma )^HA_{\tau }\xi (\varsigma )W+\sum _{\tau =1}^{\gamma } W^H\xi (\varsigma )^{H}A_{\tau }^HZ_{\tau }(\varsigma )\bigg ) \nonumber \\&-\omega _{2}\textrm{trace} \bigg (\sum _{\tau =1}^{\gamma } Z_{\tau }(\varsigma )^H{V}\overline{\xi (\varsigma )}D_{\tau }+\sum _{\tau =1}^{\gamma } D_{\tau }^H\xi (\varsigma )^{T}V^H{Z_{\tau }(\varsigma )}\bigg )\nonumber \\&-\omega _{2}\textrm{trace} \bigg (\sum _{\tau =1}^{\gamma } Z_{\tau }(\varsigma )^H{C_{\tau }}\overline{\xi (\varsigma )}{W}+\sum _{\tau =1}^{\gamma } W^H\xi (\varsigma )^TC_{\tau }^H{Z_{\tau }(\varsigma )}\bigg ) \nonumber \\&+\bigg \Vert \omega _{1}\left( A_{\tau }^HZ_{\tau }(\varsigma )B_{\tau }^H+C_{\tau }^T\overline{Z_{\tau }(\varsigma )}D_{\tau }^T\right) \nonumber \\&\qquad \qquad \qquad \qquad +\omega _{2}\left( \sum _{\tau =1}^{\gamma } V^HZ_{\tau }(\varsigma )B_{\tau }^H+\sum _{\tau =1}^{\gamma } A_{\tau }^HZ_{\tau }(\varsigma )W^H +\sum _{\tau =1}^{\gamma } V^T\overline{Z_{\tau }(\varsigma )}D_{\tau }^T+\sum _{\tau =1}^{\gamma } C_{\tau }^T\overline{Z_{\tau }(\varsigma )}W^T \right) \bigg \Vert _F^2. \end{aligned}$$

(6.6)

Now from the definition of Frobenius inner product and the fact that for two matrices $L_1$ and $L_2$, $\left\langle L_1, L_2 \right\rangle _{F}+ \left\langle L_2, L_1 \right\rangle _{F}$ is real and $-\left\langle L_1, L_2 \right\rangle _{F}- \left\langle L_2, L_1 \right\rangle _{F} \le \Vert L_{1}\Vert _F^2+\Vert L_2\Vert _F^2$ we obtain

$$\begin{aligned}&\Vert \xi (\varsigma +1)\Vert _F^2 =\Vert \xi (\varsigma )\Vert _F^2-2\omega _{1}\sum _{\tau =1}^{\gamma }\Vert Z_{\tau }(\varsigma )\Vert _F^2 -\omega _{2}\sum _{\tau =1}^{\gamma } \left\langle Z_{\tau }(\varsigma ),A_{\tau }\xi (\varsigma )V+ V\xi (\varsigma )B_{\tau }+{V}\overline{\xi (\varsigma )}D_{\tau }+C_{\tau }\overline{\xi (\varsigma )}{V}\right\rangle _{F} \\&-\omega _{2}\sum _{\tau =1}^{\gamma } \left\langle A_{\tau }\xi (\varsigma )W+ V\xi (\varsigma )B_{\tau }+{V}\overline{\xi (\varsigma )}D_{\tau }+C_{\tau }\overline{\xi (\varsigma )}{W},Z_{\tau }(\varsigma )\right\rangle _{F} \\&+\bigg \Vert \omega _{1}\left( \sum _{\tau =1}^{\gamma } A_{\tau }^HZ_{\tau }(\varsigma )B_{\tau }^H+\sum _{\tau =1}^{\gamma } C_{\tau }^T\overline{Z_{\tau }(\varsigma )}D_{\tau }^T\right) \\&\qquad \qquad \qquad \qquad \qquad \qquad +\omega _{2}\left( \sum _{\tau =1}^{\gamma } V^HZ_{\tau }(\varsigma )B_{\tau }^H+\sum _{\tau =1}^{\gamma } A_{\tau }^HZ_{\tau }(\varsigma )W^H +\sum _{\tau =1}^{\gamma } V^T\overline{Z_{\tau }(\varsigma )}D_{\tau }^T+\sum _{\tau =1}^{\gamma } C_{\tau }^T\overline{Z_{\tau }(\varsigma )}W^T \right) \bigg \Vert _F^2 \end{aligned}$$

$$\begin{aligned}&\le \Vert \xi (\varsigma )\Vert _F^2-2\omega _{1}\sum _{\tau =1}^{\gamma }\Vert Z_{\tau }(\varsigma )\Vert _F^2 +\omega _{2}\sum _{\tau =1}^{\gamma } \Vert Z_{\tau }(\varsigma )\Vert _F^2 +\omega _{2}\sum _{\tau =1}^{\gamma } \Vert A_{\tau }\xi (\varsigma )W+ V\xi (\varsigma )B_{\tau }+{V}\overline{\xi (\varsigma )}D_{\tau }+C_{\tau }\overline{\xi (\varsigma )}{W}\Vert _F^2 \nonumber \\&+\sum _{\tau =1}^{\gamma }\bigg \Vert \omega _{1}\left( A_{\tau }^HZ_{\tau }(\varsigma )B_{\tau }^H+ C_{\tau }^T\overline{Z_{\tau }(\varsigma )}D_{\tau }^T\right) +\omega _{2}\left( V^HZ_{\tau }(\varsigma )B_{\tau }^H+ A_{\tau }^HZ_{\tau }(\varsigma )W^H + V^T\overline{Z_{\tau }(\varsigma )}D_{\tau }^T+ C_{\tau }^T\overline{Z_{\tau }(\varsigma )}W^T \right) \bigg \Vert _F^2 \nonumber \\&\le \Vert \xi (\varsigma )\Vert _F^2-2\omega _{1}\sum _{\tau =1}^{\gamma }\Vert Z_{\tau }(\varsigma )\Vert _F^2 +\omega _{2}\sum _{\tau =1}^{\gamma }\Vert Z_{\tau }(\varsigma )\Vert _F^2 +\omega _{2}\sum _{\tau =1}^{\gamma }\Vert A_{\tau }\xi (\varsigma )W+ V\xi (\varsigma )B_{\tau }+ C_{\tau }\overline{\xi (\varsigma )}{W}+{V}\overline{\xi (\varsigma )}D_{\tau }\Vert _F^2 \nonumber \\&+2\omega _{1}^{2}\sum _{\tau =1}^{\gamma }\Vert A_{\tau }^HZ_{\tau }(\varsigma )B_{\tau }^H+C_{\tau }^T\overline{Z_{\tau }(\varsigma )}D_{\tau }^T\Vert _F^{2}+2\omega _{2}^{2}\sum _{\tau =1}^{\gamma }\Vert A_{\tau }^HZ_{\tau }(\varsigma )W^H+V^HZ_{\tau }(\varsigma )B_{\tau }^H +C_{\tau }^T\overline{Z_{\tau }(\varsigma )}W^T +V^T\overline{Z_{\tau }(\varsigma )}D_{\tau }^T\Vert _F ^{2}. \end{aligned}$$

(6.7)

Then by using Lemmas 2.2 and 2.3, one can ascertain that:

$$\begin{aligned}&\sum _{\tau =1}^{\gamma } \Vert A_{\tau }\xi (\varsigma )W+ V\xi (\varsigma )B_{\tau }+C_{\tau }\overline{\xi (\varsigma )}{W}+{V}\overline{\xi (\varsigma )}D_{\tau }\Vert _F^{2} \le \nonumber \\&\qquad \qquad \qquad 4 \sum _{\tau =1}^{\gamma } \left( \left( \Vert A_{\tau }\Vert _2^{2}+\Vert C_{\tau }\Vert _2^{2}\right) \Vert W\Vert _2^{2} +\left( \Vert B_{\tau }\Vert _2^{2} +\Vert D_{\tau }\Vert _2^{2}\right) \Vert V\Vert _2^{2}\right) \Vert {\xi (\varsigma )}\Vert _F^{2}\le 4 \theta _{1}\gamma \Vert {\xi (\varsigma )}\Vert _F^{2}, \end{aligned}$$

(6.8)

$$\begin{aligned} \sum _{\tau =1}^{\gamma } \Vert A_{\tau }^HZ_{\tau }(\varsigma )B_{\tau }^H+C_{\tau }^T\overline{Z_{\tau }(\varsigma )}D_{\tau }^T\Vert _F^{2}\le 2 \sum _{\tau =1}^{\gamma } \left( \Vert A_{\tau }\Vert _2^{2}\Vert B_{\tau }\Vert _2^{2}+\Vert C_{\tau }\Vert _2^{2} \Vert D_{\tau }\Vert _2^{2} \right) \Vert Z_{\tau }(\varsigma )\Vert _F^{2}\le 2\theta _{2} \sum _{\tau =1}^{\gamma } \Vert Z_{\tau }(\varsigma )\Vert _F^{2}, \end{aligned}$$

(6.9)

and

$$\begin{aligned}&\sum _{\tau =1}^{\gamma } \Vert A_{\tau }^HZ_{\tau }(\varsigma )V^H+V^HZ_{\tau }(\varsigma )B_{\tau }^H +C_{\tau }^T\overline{Z_{\tau }(\varsigma )}V^T+V^T\overline{Z_{\tau }(\varsigma )}D_{\tau }^T\Vert _F^{2} \le \nonumber \\&\qquad \qquad 4 \sum _{\tau =1}^{\gamma } \left( \left( \Vert A_{\tau }\Vert _2^{2}+\Vert C_{\tau }\Vert _2^{2}\right) \Vert W\Vert _2^{2} +\left( \Vert B_{\tau }\Vert _2^{2} +\Vert D_{\tau }\Vert _2^{2}\right) \Vert V\Vert _2^{2}\right) \Vert {Z_{\tau }(\varsigma )}\Vert _F^{2} \le 4\theta _{1} \sum _{\tau =1}^{\gamma } \Vert Z_{\tau }(\varsigma )\Vert _F^{2}. \end{aligned}$$

(6.10)

Given these inequalities and (6.7), it can be concluded that:

$$\begin{aligned} \Vert \xi (\varsigma +1)\Vert _F^2 \le \Vert \xi (\varsigma )\Vert _F^2+4 \theta _{1} \omega _{2}\gamma \Vert {\xi (\varsigma )}\Vert _F^{2} +(\omega _{2}-2\omega _{1} +4\theta _{2}\omega _{1}^2+8\theta _{1}\omega _{2}^2)\sum _{\tau =1}^{\gamma } \Vert {Z_{\tau }(\varsigma )}\Vert _F^{2}, \end{aligned}$$

(6.11)

where $\theta _{1}$ and $\theta _{2}$ are defined in (3.39) and (3.40), respectively. Now there are $\epsilon _{\tau }>0, \tau =1,..., \gamma ,$ such that

$$\begin{aligned} \Vert \xi (\varsigma )\Vert _F&\le \left( \dfrac{1}{\Vert A_{\tau }\Vert _2\Vert B_{\tau }\Vert _2+\Vert C_{\tau }\Vert _2\Vert D_{\tau }\Vert _2}+\epsilon _{\tau } \right) \Vert Z_{\tau }(\varsigma )\Vert _F\\&\le \left( \dfrac{1}{\Vert A_{\tau }\Vert _2\Vert B_{\tau }\Vert _2+\Vert C_{\tau }\Vert _2\Vert D_{\tau }\Vert _2}+\epsilon \right) \Vert Z_{\tau }(\varsigma )\Vert _F\\&\le \sqrt{ \theta _3} \Vert Z_{\tau }(\varsigma )\Vert _F \quad \tau =1,..., \gamma , \; \varsigma \ge 0, \end{aligned}$$

where $\epsilon =\max _{1\le \tau \le \gamma } \{\epsilon _{\tau }\}$. Hence we conclude

$$\begin{aligned} \gamma \Vert {\xi (\varsigma )}\Vert _F^{2} \le {\theta _{3}} \sum _{\tau =1}^{\gamma } \Vert Z_{\tau }(\varsigma )\Vert _F^2, \quad \tau =1,..., \gamma , \; \varsigma \ge 0. \end{aligned}$$

Now (6.11) yields:

$$\begin{aligned} \Vert \xi (\varsigma +1)\Vert _F^2 \le \Vert \xi (\varsigma )\Vert _F^2 +(\omega _{2}-2\omega _{1} +4\theta _{1} \theta _3 \omega _{2}+4\theta _{2}\omega _{1}^2+8\theta _{1}\omega _{2}^2)\sum _{\tau =1}^{\gamma } \Vert {Z_{\tau }(\varsigma )}\Vert _F^{2}. \end{aligned}$$

(6.12)

If the parameters $\omega _{1}$ and $\omega _{2}$ are selected according to (3.38), then it follows that:

$$\begin{aligned} \omega _{2}-2\omega _{1} +4\theta _{1} \theta _3 \omega _{2}+4\theta _{2}\omega _{1}^2+8\theta _{1}\omega _{2}^2<0. \end{aligned}$$

Consequences drawn from the convergence theorem of series indicate that:

$$\begin{aligned} \lim \limits _{\varsigma \longrightarrow \infty } \left( A_{\tau }\xi (\varsigma )B_{\tau }+C_{\tau }\overline{\xi (\varsigma )}D_{\tau }\right) =0,\; \tau =1,2,..., \gamma . \end{aligned}$$

Finally given that the matrix equation (3.31) has a unique solution, it can be concluded that $\lim \limits _{\varsigma \longrightarrow \infty } L(\varsigma )=L^{\star }$.