smooth function of $g$, the obtained optimal $g$ and $\mu$ are sufficiently accurate. Using these parameter values, $E_g$ was obtained as the average of the results of eight M.C. calculations each with $5 \times 10^7$ steps at $\Delta = 0.01,0.04$ and $0.08$ for the $d$-wave. The standard deviations were less than $0.00011$. At other points, the numbers of M.C. calculations and steps were different but their error bars were within $0.00015$ with the total M.C. steps greater than $2\times 10^8$. The diamond shows the normal state value, $-0.73585 \pm 0.00024$, obtained from 20 M.C. calculations each with $10^7$ steps, using the Gutzwiller-projected normal state wave function. In the employed normal state $(\pi/6, 7\pi/10)$ and $(-\pi/6,
-7\pi/10)$ are fully occupied but $(\pi/6, -7\pi/10)$ and $(-\pi/6,
7\pi/10)$ are unoccupied in the component of the trial wave function prior to the Gutzwiller projection.

Clearly, $E_g/N_s$ is minimum at a finite value of $\Delta \approx 0.08$ in the case of the $d$-wave gap parameter. The optimal parameter values at $\Delta = 0.08$ are $g = 0.3037$ and $\mu = - 0.4263$. The least squares fit of a parabola to $E_g/N_s$ as a function of $\Delta$ is of good quality, as seen in Fig. \[fig:DEg-D\] 