Theory and Modern Applications

# Table 2 The RMS values (32) for $$p\in \{0,0.01\%,0.1\%\}$$ and $$\beta \in \{0,10^{-12},10^{-11},10^{-10}, 10^{-9},10^{-8},10^{-7}\}$$ for Tests 1 and 2

Test 1 Test 2
p β RMS(α) Minimum value of $$\mathbb{F}$$ or $$\mathbb{F}_{\beta }$$ p β RMS(α) Minimum value of $$\mathbb{F}$$ or $$\mathbb{F}_{\beta }$$
0 0 0.0868 $$\mathbb{F}=2.4\mbox{E}{-}30$$ 0 0 0.0872 $$\mathbb{F}=8.1\mbox{E}{-}30$$
0.01% 0 3.4204 $$\mathbb{F}=1.3\mbox{E}{-}29$$ 0.01% 0 3.3676 $$\mathbb{F}=8.5\mbox{E}{-}30$$
10−10 0.0790 $$\mathbb{F}_{\beta }=1.1\mbox{E}{-}7$$ 10−12 0.2368 $$\mathbb{F}_{\beta }=7.1\mbox{E}{-}8$$
10−9 0.0776 $$\mathbb{F}_{\beta }=4.2\mbox{E}{-}7$$ 10−11 0.1245 $$\mathbb{F}_{\beta }=2.3\mbox{E}{-}7$$
10−8 0.0982 $$\mathbb{F}_{\beta }=3.2\mbox{E}{-}6$$ 10−10 0.2728 $$\mathbb{F}_{\beta }=1.5\mbox{E}{-}6$$
0.1% 0 34.2297 $$\mathbb{F}=1.3\mbox{E}{-}29$$ 0.1% 0 33.9465 $$\mathbb{F}=2.1\mbox{E}{-}29$$
10−9 0.1573 $$\mathbb{F}_{\beta }=8.4\mbox{E}{-}6$$ 10−11 0.8454 $$\mathbb{F}_{\beta }=6.7\mbox{E}{-}6$$
10−8 0.1100 $$\mathbb{F}_{\beta }=1.1\mbox{E}{-}5$$ 10−10 0.5282 $$\mathbb{F}_{\beta }=8.6\mbox{E}{-}6$$
10−7 0.1701 $$\mathbb{F}_{\beta }=3.6\mbox{E}{-}5$$ 10−9 0.6936 $$\mathbb{F}_{\beta }=1.6\mbox{E}{-}5$$