How to split a tera-polynomial

Abstract

This article presents a new algorithm to compute all the roots of two families of polynomials that are of interest for the Mandelbrot set $\mathcal{M}$ : the roots of those polynomials are respectively the parameters $c\in\mathcal{M}$ associated with periodic critical dynamics for $f_c(z)=z^2+c$ (hyperbolic centers) or with pre-periodic dynamics (Misiurewicz-Thurston parameters). The algorithm is based on the computation of discrete level lines that provide excellent starting points for the Newton method. In practice, we observe that these polynomials can be split in linear time of the degree. This article is paired with a code library [Mandel] that implements this algorithm. Using this library and about 723 000 core-hours on the HPC center Roméo (Reims), we have successfully found all hyperbolic centers of period $\leq 41$ and all Misiurewicz-Thurston parameters whose period and pre-period sum to $\leq 35$. Concretely, this task involves splitting a tera-polynomial, i.e. a polynomial of degree $\sim10^{12}$, which is orders of magnitude ahead of the previous state of the art. It also involves dealing with the certifiability of our numerical results, which is an issue that we address in detail, both mathematically and along the production chain. The certified database is available to the scientific community. For the smaller periods that can be represented using only hardware arithmetic (floating points FP80), the implementation of our algorithm can split the corresponding polynomials of degree $\sim10^{9}$ in less than one day-core. We complement these benchmarks with a statistical analysis of the separation of the roots, which confirms that no other polynomial in these families can be split without using higher precision arithmetic.

Figures (22)

• Figure 1: The Mandelbrot set $\mathcal{M}$ with $\partial\mathcal{M}$ in black and in gray, the interior of $\mathcal{M}$.
• Figure 2: Newton's method to split a polynomial of degree $d=20$ with 4 starting points (black) per root. The trajectories (white) are Newton's iterations, which are stoped in case of divergence; the gray dots mark the critical points; the white disk are contracted to the roots (red). The classical choice for the starting points (left) leads to $O(d^2)$ Newton steps. A better choice, proposed in this article (right) optimizes the position of the starting points by combining their computation with that of a level curve and brings, in practice, the total number of steps to $O(d)$. See also RSS2017 for a similar idea, though implemented differently.
• Figure 3: The hyperbolic points $\mathop{\mathrm{Hyp}}\nolimits(n)$ for $n\leq18$ in green and the Misiurewicz-Thurston parameters $M_{\ell,n}=\mathop{\mathrm{Mis}}\nolimits(\ell,n)$ with $\ell+n\leq16$ in red in different parts of the Mandelbrot set $\mathcal{M}$, with $\operatorname{Re} c$ increasing from left to right between images.
• Figure 4: Attraction bassins and Julia set of the Newton method (left) and of the accelerated method with $\mu=2$ (right) for $P(z)=(z-1)^2(z+1)(z^2+1)$.
• Figure 5: Gridpoints-per-root ratio for $\mathbb{M}_r(d)$ as a function of $\log_2(d)$. Comparison with the one-circle grid (dashed line).

Theorems & Definitions (42)

• Theorem 1: folklore, included in HT15
• Theorem 2
• Remark 1
• Remark 2
• Theorem 3: Splitting algorithm of HSS2001
• Remark 3
• Remark 4
• Theorem 4
• proof
• Proposition 5