What to use as initial guess for Newton Raphson Method?

The root solving method Newton Raphson converges quickly to the estimated root value but requires a 'close' enough initial guess to converge. I have read that an initial value is often chosen by use of the bisection method, where it iterates until a low level of tolerance and it is fed as an initial guess into Newton's. However, the bisection method requires a change of signs along the function.

My question is what other method could you use to feed into Newton's if the function is never negative over its domain?

• Newton's method may run slowly if the function does not have non zero slope at a root. May 29, 2020 at 1:12
• If the function is never negative over its domain (assuming "nice" function), then it has no roots. The only possible zero would be at a horizontal tangent touching the x axis. May 29, 2020 at 2:59
• I am afraid that the last sentence is almost killing the question. May 29, 2020 at 4:34
• @ClaudeLeibovici A function can have roots but not necessarily be negative, a root is defined by f(x)=0; where 0 is not negative. So it still can have roots May 29, 2020 at 15:23

You describe a situation where your function $$f$$ is defined on $$\Omega \subseteq \mathbb{R}$$ and is nonnegative, i.e., $$f : \Omega \rightarrow [0,\infty)$$.
As you correctly observe, you cannot hope to apply the bisection method to $$f$$ in order to narrow the search for an initial guess.
However, there are at least two options. Any root of $$f$$ is necessarily a global minimum, of $$f$$, hence a root of $$f'$$.
1. If $$f$$ is at least three times differentiable and $$f'''$$ is continuous, then you can hope to apply Newton's method to the equation $$f'(x) = 0$$ and have quadratic convergence. In this case, you may be able to detect a sign change of $$f'$$ and apply bisection to narrow the search interval. This procedure will give you a list of candidates for roots of $$f$$, but you will of course have to verify them one by one.
2. Alternatively, you can attempt to locate a minimum of $$f$$ using a dedicated algorithm, such as the golden section search. Again, you will have to verify if the candidates are in fact roots of $$f$$. Restrictions apply to the application of the golden section search and while the convergence to a local minimum is assured it is only linear.