Best initial guess for Newton-Raphson method to find a real root of the cubic polynomial

Question

What initial guess to use for Newton-Raphson method to find a real root of the cubic polynomial $$ x^3 + px^2 +qx +r $$ so, that the Newton-Raphson method converges fast. It's given, the cubic polynomial has only one real root with high probability. Any leads would be appreciated.

Answer 1

Set $f(x)=x^3+px^2+qx+d$. If $f$ is monotonic, then N-R will work quickly, so assume $f$ admits a local max and min. Since $f$ has only one real root, the local extrema will either both be positive or both be negative, and they will occur at

\begin{align*} x_{max}=\frac{-p-\sqrt{p^2-3q}}{3} &&\text{and}&& x_{min}=\frac{-p+\sqrt{p^2-3q}}{3} \end{align*}

If both extrema are positive, choose an $x$-value less than $x_{max}$.

If both extrema are negative, choose an $x$-value greater than than $x_{min}$.

Answer 2

You don't need to guess to find a good initial estimate. With bracketing methods, such as the Newt-safe method, it suffices to use initial points $x_\mathrm L$ and $x_\mathrm R$ where $\operatorname{sign}(f(x_\mathrm L))\ne\operatorname{sign}(f(x_\mathrm R))$. Convergence is then guaranteed within this interval.

In this specific case, you can further improve upon the initial bounds by finding the local extrema as explained in Bonnaduck's answer, or for example by inserting an initial estimate of the root between $x_\mathrm L$ and $x_\mathrm R$ that is likely closer than either of the bounds, such as $x_0=0$ or $x_0=-\sqrt[3]r$.