Computing real minima of positive univariate polynomials (of degree $\le 6$) Let $p : \mathbb R \rightarrow \mathbb R$ be a non-constant positive polynomial with $\deg(p) \leq 6$. We know that $p$ has between $1$ and $3$ minima. How can an approximation to the global minimum be found numerically?
This problem occurs naturally when trying to compute the closest point to a cubic spline.
The approaches I've seen so far all solve the generic quintic equation $p'(t) = 0$. But this gives up a lot of structure, since now we're computing (up to) $5$ roots instead of $3$. Are there any numerical methods more tailored to this kind of situation?
The kind of structure I'm thinking of: since we know that if $t$ is a minimum and $p''(t) \geq 0$ at any local minimum, we could factor the (non-constant, leading coefficient positive) quartic $p''(t)$ and obtain (up to) $3$ intervals where the roots must lie, where only one interval can be finite.

This doesn’t answer the question you asked, but it might solve the problem that motivated your question.
Here’s what I’d do to find the closest point on a cubic curve …

*

*Calculate $n$ points on the curve, and use these to construct a polyline approximation.

*Find the closest point on the polyline, and it’s corresponding parameter value, $t$.

*Do $m$ iterations of Newton-Raphson to polish the value of $t$.

Suggested values are: $n$ should be 3 to 10 times the degree of the curve, and $m$ should be 3 or 4. You’ll need to change those values to match your speed/accuracy/reliability requirements.
All of this should work fine on a GPU.
