When finding root, does Newton's method fail if the function is non-differentiable?

Question

According to wikipedia's description, the Newton's method finding a root presumes a differentiable function. Then, will it fail when encountering non-differentiable function? For example, can it find the zero-crossing point of a one-dimensional piecewise linear function?

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I have found that this is actually an active topic in modern Operation Research or related domains (e.g., see the paper "a nonsmooth version of Newton method" by L. Qi & J. Sun and the papers citing it therein).

Answer 1

Newton's method is an iterative procedure, generating a sequence of numbers $\{x_0, x_1, \dots\}$. In each case, we finding where the tangent line to the graph of the function at $(x_n, f(x_n))$ meets the $x$-axis in order to find $x_{n+1}$.

In many situations, this sequence converges to a root, but it can fail if

• $x_n$ for some $n$ is a non-differentiable point for $f$, or
• the sequence of approximations bounce around wildly without converging (for example, consider finding the root $x=0$ of $f(x) = x^{1/3}$, beginning with $x_0 \ne 0$.)

We cannot guarantee that Newton's method will fail for a piecewise linear function, but it could.

Answer 2

Methods that do not require the derivative to be available are the secant method, Aitken's method, and Steffensen's method.

As with all iterative methods, sometimes they work and sometimes they don't.