Abstract
Given a black box oracle that evaluates a univariate polynomial p(x) of a degree d, we seek its zeros, aka the roots of the equation \(p(x)=0\). At FOCS 2016, Louis and Vempala approximated within \(1/2^b\) an absolutely largest zero of such a real-rooted polynomial at the cost of the evaluation of Newton’s ratio \(\frac{p(x)}{p'(x)}\) at \(O(b\log (d))\) points x and then extended this algorithm to approximation of an absolutely largest eigenvalue of a symmetric matrix at a record Boolean cost. By applying distinct approach and techniques we obtain much more general results at the same computational cost. Our use of Cauchy integrals and randomization is non-trivial and pioneering in this field. Somewhat surprisingly, the Boolean complexity of the accelerated versions of our algorithms in [25, 26] reached below the known lower bounds on the Boolean complexity of polynomial root-finding.
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Notes
- 1.
Hereafter we frequently refer to them just as roots or zeros.
- 2.
- 3.
We count roots with their multiplicities and can readily extend our study to various other convex domains on the complex plane such as a disc or a polygon.
- 4.
Up to small poly-logarithmic factors these estimates reach lower bound for approximation of even a single zero of p(x), but as we specify in Sect. 7, the algorithms of [25, 26] greatly accelerate those of [18, 21] for approximation of all \(m=o(d)\) zeros of p that lie in a disc reasonably well isolated from the \(d-m\) external zeros.
- 5.
The complexity of a subdivision root-finder is proportional to the number of roots in a region, while MPSolve is about as fast and slow for all roots as for their fixed subset. MPSolve implements Ehrlich-Aberth’s root-finding iterations, which empirically converge to all d roots very fast right from the start but with no formal support and so far only under an initialization that operates with the coefficients of p (see [28]).
- 6.
Both bounds \(r_{d-\ell +1}\le \sigma \) and \(r_{d-\ell +1}>1\) can hold simultaneously, but as soon as an \(\ell \)-test verifies any of them, it stops without checking if the other bound also holds.
- 7.
- 8.
Given the coefficients of p(x) one can fix \(q:=2^k\) for \(k=\lceil \log _2\lfloor \log _{\theta }(4d+2)\rfloor \rceil \) and then evaluate p(x) and \(p'(x)\) at all qth roots of unity by applying FFT.
- 9.
- 10.
We call a complex point c a tame root for a fixed error tolerance \(\epsilon \) if it is covered by an isolated disc \(D(c,\epsilon )\). Given such a disc \(D(c,\rho )\), we can readily compute \(\#(D(c,\rho ))\) by applying Corollary 6.
- 11.
This recipe detects output errors of any root-finder at the very end of computations. In the case of subdivision root-finders we can detect the loss of a root earlier – whenever we notice that at a subdivision step the indices of all suspect squares sum to less than m.
- 12.
We can also replace d by m in Theorem 23 if we only seek approximation of NIR(x) for \(|x|=1\) within \(1/d^{O(1)}\); actually we need such approximations within relative error \(1/d^{O(1)}\), which implies an absolute error bound \(1/d^{O(1)}\) if \(1/\textrm{NIR}(x)=1/d^{O(1)}\).
- 13.
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Pan, V.Y., Go, S., Luan, Q., Zhao, L. (2023). Fast Cauchy Sum Algorithms for Polynomial Zeros and Matrix Eigenvalues. In: Mavronicolas, M. (eds) Algorithms and Complexity. CIAC 2023. Lecture Notes in Computer Science, vol 13898. Springer, Cham. https://doi.org/10.1007/978-3-031-30448-4_24
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