0.85^2/3計算

1.請教0.85^2/3 計算過程
　設 0.85^2/3 = X
　0.85² = X³
　X³ = 0.7225

　0.9³ = 0.729
　0.89³ = 0.704969
　0.897³ = 0.721734

　所以 X ≒ 0.897


2. (1/1000)^1/2 計算過程
　設 (1/1000)^1/2 = X
　0.001^1/2 = X
　0.001 = X²

　0.03² = 0.0009
　0.031² = 0.000961
　0.0316² = 0.00099856

　所以 X ≒ 0.0316

逼近法之牛頓迭代:
x^k = b, 取 x* 為試誤值,
用 x(new) = x* - (x*^k-b)/(kx*^{k-1}) = [(k-1)x* + b/x*^{k-1}]/k

如 (1/1000)^{1/2} = (0.001)^{1/2}
取 x* = 0.1, 則 x(new) = (0.1 + 0.001/0.1)/2 = 0.055
取 x* = 0.055, 得 x(new) = (0.055+0.001/0.055)/2 = 0.0366
繼續迭代計算, 依次得
0.03196
0.0316246
0.0316228
0.0316228

2015-06-07 04:21:52 補充：
x^3 = 0.85^2 = 0.7225
取初值 x* = 0.9
x(new) = (0.9*2+0.7225/0.9^2)/3 = 0.897325
(0.897325*2+0.7225/0.897325^2)/3 = 0.897317
(0.897317*2+0.7225/0.897317^2)/3 = 0.897317

2015-06-07 04:31:32 補充：
平方根也可以利用 (a+b)^2 = a^2 + b*(2a+b) 的公式, 即所謂 "直式算法".
立方根也有類似算法, 只是繁複了些.

直式算法之過程:
0.001 ≒ 0.03^2
0.001 = 0.03^2 + 0.0001 ≒ 0.03^2 + 0.001(0.03*2+0.001) = 0.031^2
0.001 = 0.031^2 + 0.000039 ≒ 0.031^2 + 0.0006(0.031*2+0.0006) = 0.0316^2

在 知識+ 找 "直式開平方根" 或許可找到真正直式計算之例.

計算機?

還是 Taylor's series...