1841.sagews

Suppose we want to find a numerical approximation to $\sqrt{2}$.  Here's one way to approach it with Newton's method.

f(x) = x^2 - 2
N(x) = expand(x - f(x)/diff(f(x),x))
show(N(x))

\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{2} \, x + \frac{1}{x}

Now, we iterate.

xi = 1.0
for i in range(5):
    xi = N(xi)
    print(xi)

1.50000000000000 1.41666666666667 1.41421568627451 1.41421356237469 1.41421356237309