Faster Numerical Univariate Polynomial Root-Finding by Means of Subdivision Iterations

Abstract

Root-finding for a univariate polynomial is four millennia old and still highly important for Computer Algebra and various other fields. Subdivision root-finders for a complex univariate polynomial are known to be highly efficient and practically promising. The recent one by Becker et al.  [2] competes for user’s choice and is nearly optimal for dense polynomials represented in monomial basis, but  [18] proposes and analyzes further significant acceleration, which becomes dramatic for polynomials admitting their fast evaluation (e.g., sparse ones). Here and in the companion paper  [19], we present some of these results and algorithms.

Appendix A.  Discrete Fourier transform (DFT)

DFT($\mathbf{p}$) outputs the vector of the values $p(\zeta ^j)=\sum _{i=0}^{d-1}p_i\zeta ^{ij}$ of a polynomial $p(x)=\sum _{i=0}^{d-1}p_ix^i$ on the set $\{1,\zeta ,\ldots ,\zeta ^{d-1}\}$. The fast Fourier transform (FFT) algorithm, for $d=2^h$ recursively splits p(x):

$$\begin{aligned} p(x)&=p_0(y)+xp_1(y), \text{ where } y=x^2, \\ p_0(y)&=p_0+p_2x^2+\cdots +p_{d-2}x^{d-2},~ p_1(y)&=x(p_1+p_3x^2+\cdots +p_{d-1}x^{d-2}). \end{aligned}$$

This reduces DFT$_d$ for p(x) to two DFT$_{d/2}$ (for $p_0(y)$ and $p_1(y)$) at a cost of d multiplications of $p_1(y)$ by x, for $x=\zeta ^i$, $i=0,1,\ldots ,d-1$, and of the pairwise addition of the d output values to $p_0(\zeta ^{2i})$. Since $\zeta ^{i+d/2}=-\zeta ^i$ for even d, we perform multiplication only d/2 times, that is, $f(d)\le 2f(d/2)+1.5d$ if f(k) ops are sufficient for DFT$_k$. Recursively we obtain the following estimate.

Theorem 3

For $d=2^h$ and a positive integer h, the DFT$_d$ only involves $f(d)\le 1.5dh=1.5d\log _2d$ arithmetic operations.

Inverse DFT is the converse problem of interpolation to a polynomial p(x) from its values at the dth roots of unity. At the cost of performing d divisions, this task can be reduced to DFT (see, e.g.,  [15, Theorem 2.2.2]).