Carlos Beltrán

Carlos Beltrán, Jean-Pierre Dedieu, Gregorio Malajovich et al.

We define in the space of n by m matrices of rank n, n less or equal than m, the condition Riemannian structure as follows: For a given matrix A the tangent space of A is equipped with the Hermitian inner product obtained by multiplying the usual Frobenius inner product by the inverse of the square of the smallest singular value of A denoted sigma_n(A). When this smallest singular value has multiplicity 1, the function A -> log (sigma_n(A)^(-2)) is a convex function with respect to the condition Riemannian structure that is t -> log (sigma_n(A(t))^(-2)) is convex, in the usual sense for any geodesic A(t). In a more abstract setting, a function alpha defined on a Riemannian manifold (M,<,>) is said to be self-convex when log alpha (gamma(t)) is convex for any geodesic in (M,<,>). Necessary and sufficient conditions for self-convexity are given when alpha is C^2. When alpha(x) = d(x,N)^(-2) where d(x,N) is the distance from x to a C^2 submanifold N of R^j we prove that alpha is self-convex when restricted to the largest open set of points x where there is a unique closest point in N to x. We also show, using this more general notion, that the square of the condition number ||A|||_F / sigma_n(A) is self-convex in projective space and the solution variety.

Carlos Beltrán, Paul Breiding, Nick Vannieuwenhoven

We prove the existence of an open set of $n_1\times n_2 \times n_3$ tensors of rank $r$ on which a popular and efficient class of algorithms for computing tensor rank decompositions based on a reduction to a linear matrix pencil, typically followed by a generalized eigendecomposition, is arbitrarily numerically forward unstable. Our analysis shows that this problem is caused by the fact that the condition number of the tensor rank decomposition can be much larger for $n_1 \times n_2 \times 2$ tensors than for the $n_1\times n_2 \times n_3$ input tensor. Moreover, we present a lower bound for the limiting distribution of the condition number of random tensor rank decompositions of third-order tensors. The numerical experiments illustrate that for random tensor rank decompositions one should anticipate a loss of precision of a few digits.

Carlos Beltrán, Diego Armentano

We compute the exact 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.

Diego Armentano, Carlos Beltrán, Peter Bürgisser et al.

Smale's 17th problem asks for an algorithm which finds an approximate zero of polynomial systems in average polynomial time (see Smale 2000). The main progress on Smale's problem is Beltrán-Pardo (2011) and Bürgisser-Cucker (2010). In this paper we will improve on both approaches and we prove an important intermediate result. Our main results are Theorem 1 on the complexity of a randomized algorithm which improves the result of Beltrán-Pardo (2011), Theorem 2 on the average of the condition number of polynomial systems which improves the estimate found in Bürgisser-Cucker (2010), and Theorem 3 on the complexity of finding a single zero of polynomial systems. This last Theorem is the main result of Bürgisser-Cucker (2010). We give a proof of it relying only on homotopy methods, thus removing the need for the elimination theory methods used in Bürgisser-Cucker (2010). We build on methods developed in Armentano et al. (2015).

Diego Armentano, Carlos Beltrán, Peter Bürgisser et al.

We describe algorithms for computing eigenpairs (eigenvalue-eigenvector pairs) of a complex $n\times n$ 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.

Diego Armentano, Carlos Beltrán, Michael Shub

We describe two algorithms for the eigenvalue, eigenvector problem which, on input a Gaussian matrix with complex entries, finish with probability 1 and in average polynomial time.

Carlos Beltrán, Jean-Pierre Dedieu, Gregorio Malajovich et al.

In our previous paper [SIMAX 31 n.3 1491-1506(2010)], we studied the condition metric in the space of maximal rank matrices. Here, we show that this condition metric induces a Lipschitz-Riemann structure on that space. After investigating geodesics in such a nonsmooth structure, we show that the inverse of the smallest singular value of a matrix is a log-convex function along geodesics (Theorem 1). We also show that a similar result holds for the solution variety of linear systems (Theorem 31). Some of our intermediate results, such as Theorem 12, on the second covariant derivative or Hessian of a function with symmetries on a manifold, and Theorem 29 on piecewise self-convex functions, are of independent interest. Those results were motivated by our investigations on the com- plexity of path-following algorithms for solving polynomial systems.

Carlos Beltrán, Anton Leykin

Given a homotopy connecting two polynomial systems we provide a rigorous algorithm for tracking a regular homotopy path connecting an approximate zero of the start system to an approximate zero of the target system. Our method uses recent results on the complexity of homotopy continuation rooted in the alpha theory of Smale. Experimental results obtained with the implementation in the numerical algebraic geometry package of Macaulay2 demonstrate the practicality of the algorithm. In particular, we confirm the theoretical results for random linear homotopies and illustrate the plausibility of a conjecture by Shub and Smale on a good initial pair.