A primal-dual formulation for certifiable computations in Schubert calculus

Formulating a Schubert problem as the solutions to a system of equations in either Pl\"ucker space or in the local coordinates of a Schubert cell typically involves more equations than variables. We present a novel primal-dual formulation of any Schubert problem on a Grassmannian or flag manifold as a system of bilinear equations with the same number of equations as variables. This formulation enables numerical computations in the Schubert calculus to be certified using algorithms based on Smale's \alpha-theory.Comment: 21 page

The complexity and geometry of numerically solving polynomial systems

These pages contain a short overview on the state of the art of efficient numerical analysis methods that solve systems of multivariate polynomial equations. We focus on the work of Steve Smale who initiated this research framework, and on the collaboration between Stephen Smale and Michael Shub, which set the foundations of this approach to polynomial system--solving, culminating in the more recent advances of Carlos Beltran, Luis Miguel Pardo, Peter Buergisser and Felipe Cucker

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

Complexity of Sparse Polynomial Solving 2: Renormalization

Renormalized homotopy continuation on toric varieties is introduced as a tool for solving sparse systems of polynomial equations, or sparse systems of exponential sums. The cost of continuation depends on a renormalized condition length, defined as a line integral of the condition number along all the lifted renormalized paths. The theory developed in this paper leads to a continuation algorithm tracking all the solutions between two generic systems with the same structure. The algorithm is randomized, in the sense that it follows a random path between the two systems. The probability of success is one. In order to produce an expected cost bound, several invariants depending solely of the supports of the equations are introduced. For instance, the mixed area is a quermassintegral that generalizes surface area in the same way that mixed volume generalizes ordinary volume. The facet gap measures for each direction in the 0-fan, how close is the supporting hyperplane to the nearest vertex. Once the supports are fixed, the expected cost depends on the input coefficients solely through two invariants: the renormalized toric condition number and the imbalance of the absolute values of the coefficients. This leads to a non-uniform complexity bound for polynomial solving in terms of those two invariants. Up to logarithms, the expected cost is quadratic in the first invariant and linear in the last one.Comment: 90 pages. Major revision from the previous versio