>>13748691Here's a rough approximation of how close x0 needs to be.
Assume x0 is close to a root r.
x0 = r + t (t small).
x1 = x0 - f(x0)/f'(x0)
Taylor expand f and f' around r
x1 = r + t - (f(r) + tf'(r) + (t^2)f''(r)/2 ...)/(f'(r) + tf''(r) + (t^2)f'''(r)/2 ...)
Use f(r)=0
= r + t - (0 + tf'(r) + (t^2)f''(r)/2 ...)/(f'(r) + tf''(r) + (t^2)f'''(r)/2 ...)
Figure out what you need to add to 1/(f'(r) + tf''(r) + (t^2)f'''(r)/2 ...) to get the numerator
= r + t - (t - (t^2)f''(r)/(2f'(r)) ...) ~ r + (t^2)f''(r)/(2f'(r))
The t^2 error is what is meant by "quadratic convergence"
Roughly, if |(t^2)f''(r)/(2f'(r))| < |t| then the error is decreasing
This gives the crude approximation that if |t| < |2f'(r)/f''(r)| then you would expect convergence to r.
