October 4

Newton's method used the first two terms of a function's Taylor series, and omitted the others. Page 14 of the Course Notes provides a formula for the error you can expect from this omission, called the truncation error. If you omit all the terms from $f^{(n)}(x)h^n/n!$ on, your error is:

$$f^{(n)}(\theta)h^n/n!\qquad\qquad \theta \in [x,x+h]$$

Although this formula doesn't tell us exactly what $\theta$ is, it tells us its general neighbourhood. Often we can estimate the maximum size that $f^{(n)}(\theta)$ can be in the interval $[x,x+h]$, and get an idea of how bad our error can be. In the case of Newton's method, this tells us that if $x_k$ is distance $h$ from the root, then $x_{k+1}$ is some constant times $h^2$ from the root -- quadratic convergence.

However, when $h$ is big, you might not have convergence. You might experiment with the innocent function $f(x)= x^3 - 1$ in the interval $[-2, 0]$ to find a couple of points where Newton's method fails to converge.