白皮書1.3v與1.4v差異

1.3v與1.4v白皮書差異

白皮書內容主要的差異是數學式

1.3v數學式

$$\begin{gathered} \frac{dH(t)}{dt}\approx \frac{2\lambda }{L_0}K(t-h) \\ \approx \frac{1}{hexp(W(\frac{1}{2}))}exp(W(\frac{1}{2})\frac{t}{h}) \\ =\frac{W(\frac{1}{2})}{h}exp(W(\frac{1}{2})\frac{t}{h}) \end{gathered}$$

$H(t)\approx exp(W(\frac{1}{2})\frac{t}{h})\approx exp(0.352\frac{t}{h})$

1.4v數學式

$$\begin{gathered} \frac{dH(t)}{dt}\approx \frac{2\lambda }{L_0}K(t-h) \\ \approx \frac{1}{hexp(W(\frac{1}{2}))}exp(W(\frac{1}{2})\frac{t}{h}) \\ =\frac{2W(\frac{1}{2})}{h}exp(W(\frac{1}{2})\frac{t}{h}) \end{gathered}$$

$H(t)\approx 2exp(W(\frac{1}{2})\frac{t}{h})\approx 2exp(0.352\frac{t}{h})$

分析

整個微分方程的估算 , 是利用下列關係式

$$\begin{gathered} 1.2\lambda h=L_0 \\ 2.K(t)=exp(W(\frac{1}{2})\frac{t}{h}) \\ 3.\frac{c}{h}exp(c\frac{t}{h})\approx \frac{1}{2h}exp(c\frac{t}{h}-c) \end{gathered}$$

先把下列關係式 , 右邊用上列1,2代換掉

$\frac{dH(t)}{dt}\approx \frac{2\lambda }{L_0}K(t-h)$

可以得到

$\frac{dH(t)}{dt}\approx \frac{\frac{L_0}{h}}{L_0}exp(W(\frac{1}{2})\frac{t-h}{h})$

整理一下 , 得到

$$\begin{gathered} \frac{dH(t)}{dt}\approx \frac{1}{h}exp(W(\frac{1}{2})\frac{t}{h}-W(\frac{1}{2})\frac{h}{h}) \\ =\frac{1}{h}exp(W(\frac{1}{2})\frac{t}{h}-W(\frac{1}{2})) \end{gathered}$$

由指數律 , 得到

$\frac{dH(t)}{dt}\approx \frac{1}{h\ast exp(W(\frac{1}{2}))}exp(W(\frac{1}{2})\frac{t}{h})$

用指數律合併 , 在分號上下各乘以2 , 過程如下

$$\begin{gathered} \frac{1}{h\ast exp(W(\frac{1}{2}))}exp(W(\frac{1}{2})\frac{t}{h}) \\ =\frac{1}{h}exp(W(\frac{1}{2})\frac{t}{h}-W(\frac{1}{2})) \\ =\frac{2\ast 1}{2\ast h}exp(W(\frac{1}{2})\frac{t}{h}-W(\frac{1}{2})) \\ =2\ast \frac{1}{2h}exp(W(\frac{1}{2})\frac{t}{h}-W(\frac{1}{2})) \end{gathered}$$

再由3的關係式 , 有

$$\begin{gathered} 2\ast \frac{1}{2h}exp(W(\frac{1}{2})\frac{t}{h}-W(\frac{1}{2})) \\ \approx 2\ast \frac{W(\frac{1}{2})}{h}exp(W(\frac{1}{2})\frac{t}{h}) \\ =\frac{2W(\frac{1}{2})}{h}exp(W(\frac{1}{2})\frac{t}{h}) \\ \approx \frac{dH(t)}{dt} \end{gathered}$$

這是1.4v的權重增長式子

所以

$H(t)$積分後 , 就有

$H(t)\approx 2exp(W(\frac{1}{2})\frac{t}{h})\approx 2exp(0.352\frac{t}{h})$

結論

1.4v的權重增長公式修正了1.3v增長公式的計算錯誤

tags: IOTA