On solution and perturbation estimates for the nonlinear matrix equation   $$X-A^{*}e^{X}A=I$$

Journal of the Egyptian Mathematical Society

Table 1 Summary of Results for Example 4 for different $\delta$ with $X_{0}=I$ and $Z_{0}=0$

$\delta$	Iterations allowed	Iterations performed	residual = $\Vert X-A^{*}e^{X}A-I\Vert _{F}$
1	1000	Over 1000	3.36$\times 10^{+02}$
0.1	1000	Over 1000	5.32$\times 10^{+00}$
0.01	1000	Over 1000	8.37 $\times 10^{-02}$
0.001	1000	1	2.03 $\times 10^{-12}$

\(\delta\)	Iterations allowed	Iterations performed	residual = \(\Vert X-A^{*}e^{X}A-I\Vert _{F}\)
1	1000	Over 1000	3.36\(\times 10^{+02}\)
0.1	1000	Over 1000	5.32\(\times 10^{+00}\)
0.01	1000	Over 1000	8.37 \(\times 10^{-02}\)
0.001	1000	1	2.03 \(\times 10^{-12}\)