The relaxed gradient based iterative algorithm for solving the generalized coupled complex conjugate and transpose Sylvester matrix equations

Figures & data

Figure 1. Comparison of convergence performance of RGI with different parameters μ for Example 5.1.

Table 1. Iterative number of the RGI algorithm with different μ for Example 5.1.

μ	${\mu }_1=5.1499e-06$	${\mu }_2=5.2559e-06$	${\mu }_3=5.0499e-06$	${\mu }_{\mathrm{opt}}=5.2499e-06$
IT	2184	3000	2227	2142

Figure 2. The convergence curves of the tested methods with $\delta =0.1$ (left) and $\delta =0.01$ (right) for Example 5.1.

Figure 3. The convergence curves of the tested methods with $\delta =0.001$ (left) and $\delta =0.0001$ (right) for Example 5.1.

Table 2. Numerical results of the tested methods with different δ for Example 5.1.

Algorithm	δ	0.1	0.01	0.001	0.0001
GI [23]	IT	2403	8937	16093	23252
${\mu }_{\mathrm{GI}}=4.5503e-06$	CPU	2.2139	8.4308	14.2259	20.4224
	ERR	0.0997	0.01	9.9968e-04	9.9996e-05
RGI	IT	2142	8238	15189	22151
${\mu }_{\mathrm{RGI}}=5.2499e-06$	CPU	1.9771	7.5984	11.5423	18.4224
	ERR	0.0995	0.01	9.9979e-04	9.9999e-05

Figure 4. The convergence curves of the tested methods with $\xi =0.1$ (left) and $\xi =0.01$ (right) for Example 5.2.

Figure 5. The convergence curves of the tested methods with $\xi =0.001$ (left) and $\xi =0.0001$ (right) for of Example 5.2.

Table 3. Numerical results of the tested methods with different ξ for Example 5.2.

Algorithm	ξ	0.1	0.01	0.001	0.0001
GI [23]	IT	3360	16003	30706	48950
${\mu }_{\mathrm{GI}}=4.5361e-05$	CPU	2.9696	13.7964	19.5881	41.3501
	ERR	0.1	0.01	9.9998e-04	9.9995e-05
RGI	IT	2594	11590	20780	30138
${\mu }_{\mathrm{RGI}}=1.0821e-04$	CPU	2.2231	10.0329	15.4115	26.0794
	ERR	0.0995	0.01	9.9999e-04	9.9982e-05

Beik FPA, Mahmoud MM. Gradient-based iterative algorithm for solving the generalized coupled Sylvester-transpose and conjugate matrix equations over reflexive (anti-reflexive) matrices. Trans Inst Meas Control. 2014;36:99–110. doi: 10.1177/0142331213482485

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