We describe algorithms for computing eigenpairs (eigenvalue--eigenvector) of a complex @math matrix @math . 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.
In the early 1990s Shub and Smale set up a program to understand the cost of solving square systems of complex polynomial equations using homotopy methods. In a collection of articles @cite_42 @cite_34 @cite_0 @cite_23 @cite_39 , known as the B 'ezout series , they put in place many of the notions and techniques that occur in this article. The B 'ezout series did not, however, conclusively settle the understanding of the cost above, and in 1998 Smale proposed it as the 17th in his list of problems for the mathematicians of the 21st century @cite_33 . The problem is not yet considered fully solved by the community but significant advances appear in @cite_5 @cite_29 @cite_4 .
@cite_4 @cite_33 @cite_29 @cite_42 @cite_39 @cite_0 @cite_23 @cite_5 @cite_34
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The 17th of the problems proposed by Steve Smale for the 21st century asks for the existence of a deterministic algorithm computing an approximate solution of a system of n complex polynomials in n unknowns in time polynomial, on the average, in the size N of the input system. A partial solution to this problem was given by Carlos Beltr an and Luis Miguel Pardo who exhibited a randomized algorithm doing so. In this paper we further extend this result in several directions. Firstly, we exhibit a linear homotopy algorithm that eciently implements a nonconstructive idea of Smale’s 17th Problem asks: “Can a zero of n complex polynomial equations in n unknowns be found approximately, on the average, in polynomial time with a uniform algorithm?”. We give a positive answer to this question. Namely, we describe a uniform probabilistic algorithm that computes an approximate zero of systems of polynomial equations f : Cn −→ Cn, performing a number of arithmetic operations which is polynomial in the size of the input, on the average. Abstract Generalized eigenvalue problems can be considered as a system of polynomials. The homotopy continuation method is used to find all the isolated zeros of the polynomial system which corresponds to the eigenpairs of the generalized eigenvalue problem. A special homotopy is constructed in such a way that there are exactly n distinct smooth curves connecting trivial solutions to desired eigenpairs. Since the curves followed by general homotopy curve following scheme are computed independently of one another, the algorithm is a likely candidate for exploiting the advantages of parallel processing to the generalized eigenvalue problems. This contribution is the numerically oriented companion article of the work [E. Cances, V. Ehrlacher, F. Legoll, B. Stamm and S. Xiang, arxiv preprint 1807.05131]. We focus here on the numerical resolution of the embedded corrector problem introduced in [E. Cances, V. Ehrlacher, F. Legoll and B. Stamm, CRAS 2015; E. Cances, V. Ehrlacher, F. Legoll, B. Stamm and S. Xiang, arxiv preprint 1807.05131] in the context of homogenization of diffusion equations. Our approach consists in considering a corrector-type problem, posed on the whole space, but with a diffusion matrix which is constant outside some bounded domain. In [E. Cances, V. Ehrlacher, F. Legoll, B. Stamm and S. Xiang, arxiv preprint 1807.05131], we have shown how to define three approximate homogenized diffusion coefficients on the basis of the embedded corrector problems. We have also proved that these approximations all converge to the exact homogenized coefficients when the size of the bounded domain increases. We show here that, under the assumption that the diffusion matrix is piecewise constant, the corrector problem to solve can be recast as an integral equation. In case of spherical inclusions with isotropic materials, we explain how to efficiently discretize this integral equation using spherical harmonics, and how to use the fast multipole method (FMM) to compute the resulting matrix-vector products at a cost which scales only linearly with respect to the number of inclusions. Numerical tests illustrate the performance of our approach in various settings.