YBC 7289 (4)#
ニュートン法#
\[
f'(x_{n}) = \frac{f(x_{n}) - 0}{x_{n} - x_{n+1}}
\]
\[
x_{n+1} = x_{n} - \frac{f(x_{n})}{f'(x_{n})}
\]
\(f(x) = x^2-2\)のとき、\(f'(x)=2 x\)
\[\begin{split}
\begin{align}
x_{n+1} & = x_{n} - \frac{{x_{n}}^2-2}{2x_{n}} \\
& = \frac{1}{2}\left(x_{n}+\frac{2}{x_{n}}\right)
\end{align}
\end{split}\]
def f(x):
return (x+2/x)/2
x, c = 2, 6
for i in range(c):
x = f(x)
print("x = {0:.10f}, x**2 = {1:.10f}".format(x, x**2))
x = 1.5000000000, x**2 = 2.2500000000
x = 1.4166666667, x**2 = 2.0069444444
x = 1.4142156863, x**2 = 2.0000060073
x = 1.4142135624, x**2 = 2.0000000000
x = 1.4142135624, x**2 = 2.0000000000
x = 1.4142135624, x**2 = 2.0000000000