A deterministic algorithm to compute approximate roots of polynomial systems in polynomial average time

We describe a deterministic algorithm that computes an approximate root of n complex polynomial equations in n unknowns in average polynomial time with respect to the size of the input, in the Blum-Shub-Smale model with square root. It rests upon a derandomization of an algorithm of Beltr\'an and Pardo and gives a deterministic affirmative answer to Smale's 17th problem. The main idea is to make use of the randomness contained in the input itself

A stable, polynomial-time algorithm for the eigenpair problem

Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG geförderten) Allianz- bzw. Nationallizenz frei zugänglich.This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.We describe algorithms for computing eigenpairs (eigenvalue-eigenvector pairs) of a complex n×n matrix A. These algorithms are numerically stable, strongly accurate, and theoretically efficient (i.e., polynomial-time). We do not believe they outperform in practice the algorithms currently used for this computational problem. The merit of our paper is to give a positive answer to a long-standing open problem in numerical linear algebra.DFG, BU 1371/2-2, Geglättete Analyse von Konditionszahle

Rigid continuation paths I. Quasilinear average complexity for solving polynomial systems

How many operations do we need on the average to compute an approximate root of a random Gaussian polynomial system? Beyond Smale's 17th problem that asked whether a polynomial bound is possible, we prove a quasi-optimal bound $\text{(input size)}^{1+o(1)}$. This improves upon the previously known $\text{(input size)}^{\frac32 +o(1)}$ bound. The new algorithm relies on numerical continuation along \emph{rigid continuation paths}. The central idea is to consider rigid motions of the equations rather than line segments in the linear space of all polynomial systems. This leads to a better average condition number and allows for bigger steps. We show that on the average, we can compute one approximate root of a random Gaussian polynomial system of~$n$ equations of degree at most $D$ in $n+1$ homogeneous variables with $O(n^5 D^2)$ continuation steps. This is a decisive improvement over previous bounds that prove no better than $\sqrt{2}^{\min(n, D)}$ continuation steps on the average

The Polynomial Eigenvalue Problem is Well Conditioned for Random Inputs

We compute the exact expected value of the squared condition number for the polynomial eigenvalue problem, when the input matrices have entries coming from the standard complex Gaussian distribution, showing that in general this problem is quite well conditioned.The first author's work was partially supported by Agencia Nacional de Investigación e Innovación (ANII), Uruguay, and by CSIC group 618, Universidad de La República, Uruguay. The second author's work was partially supported by MTM2017-83816-P and MTM2017-90682-REDT from Spanish Ministry of Science MICINN and by 21.SI01.64658 from Universidad de Cantabria and Banco de Santander