Limitations of Newton-Raphson

Newton-Raphson iteration x(n+1) = x(n) − f(x(n)) / f′(x(n)) is fast (quadratic convergence), but it has these limitations:

1. f′(x) must be known and calculable. The derivative must be derived analytically and evaluated at each step, which costs extra work for complicated functions.
2. Division by zero. If f′(x(n)) = 0 (or very small) at some iterate, the tangent is horizontal (or nearly so) and the next value is undefined or thrown far away.
3. Needs a good initial guess. If x0 is far from the root, the method may diverge, oscillate between two values, or converge to a different root.
4. Convergence not guaranteed. It converges only if |f(x)·f″(x)| / [f′(x)]² < 1 near the root.
5. Slow convergence at multiple roots. If the root is repeated (f′(root) = 0), convergence becomes only linear.
6. Poor behaviour near an inflection point or a local maximum/minimum of f, where the tangent leads away from the root.
7. Cannot use a bracketing interval, so there is no automatic guarantee that the root stays inside a known interval, unlike bisection.

Example of failure

For f(x) = x^(1/3) the iteration gives x(n+1) = −2x(n), so it diverges from every starting value except the root itself.