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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?

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  • $\begingroup$ Newton's method may run slowly if the function does not have non zero slope at a root. $\endgroup$ – copper.hat May 29 '20 at 1:12
  • $\begingroup$ 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. $\endgroup$ – herb steinberg May 29 '20 at 2:59
  • $\begingroup$ I am afraid that the last sentence is almost killing the question. $\endgroup$ – Claude Leibovici May 29 '20 at 4:34
  • $\begingroup$ @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 $\endgroup$ – notMyName May 29 '20 at 15:23
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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.
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