that is greater than $2M/\pi$ but finally at late times approaches $2M/\pi$. In this view, the Lloyd’s bound [@Lloyd:limit; @Margolus:1997ih] is approached from above for all values of parameter $\mathfrak{a}$.

### Numerical results for the Schwarzschild black hole

In this subsection we present the numerical analysis of the complexity growth for the four-dimensional Schwarzschild metric. The critical time in this case can be found through the relations , and , $$\mu_c = -\omega \log( \omega ).$$ $r_M$ can be calculated using where for $d=3$ we have $$r_{M} + \omega ~\log ( r_{M} - \omega ) = -\frac{\mu+\chi }{2}.$$ Using this equation we find $r_M$ as $$r_{M} = \omega ~\left( 1 + W\left( \frac{-1}{\omega ~e^{1+ \frac{\mu + \chi}{2\omega}}}\right ) \right),$$ where $W$ is the Lambert $W$ function. The rate of complexity growth at late time can be read from by putting $d=3$, $$\dot{\mathcal{C}} =\frac{1}{\pi}\left( 2 M - \frac{r_{M}}{2 G_{N}} f(r_M) \log \frac{-\mathfrak{a}^2}{f(r_M)}\right).$$ For a fixed value of horizon parameter $\omega$ but different values of $\mathfrak{a}$ the complexity growth with respect to the boundary time is plotted in figure 3. For all values of the parameters, the rate of the complexity growth at late time approaches Lloyd’s bound [@Lloyd:limit; @Margolus:1997ih] from above.

