KHALIL , Houssam

Several problems in applied mathematics require the solving of linear systems with very large sizes, and sometimes these systems must be solved multiple times. In such cases, the standard algorithms based on the Gauss elimination require O (n ^ 3) arithmetic operations to solve a system of size n, and it will be a handicap for the computation....

We present a new superfast algorithm for solving Toeplitz systems. This algorithm is based on a relation between the solution of such problems and syzygies of polynomials or moving lines. We show an explicit connection between the generators of a Toeplitz matrix and the generators of the corresponding module of syzygies. We show that this...

In this paper, we re-investigate the resolution of Toeplitz systems $T\, u =g$, from a new point of view, by correlating the solution of such problems with syzygies of polynomials or moving lines. We show an explicit connection between the generators of a Toeplitz matrix and the generators of the corresponding module of syzygies. We show that...

We study the decomposition of a multivariate Hankel matrix H_σ as a sum of Hankel matrices of small rank in correlation with the decomposition of its symbol σ as a sum of polynomial-exponential series. We present a new algorithm to compute the low rank decomposition of the Hankel operator and the decomposition of its symbol exploiting the...

We study the decomposition of a multi-symmetric tensor $T$ as a sum of powers of product of linear forms in correlation with the decomposition of its dual $T^*$ as a weighted sum of evaluations. We use the properties of the associated Artinian Gorenstein Algebra $A_\tau$ to compute the decomposition of its dual $T^*$ which is defined via a...

We present an algorithm for solving polynomial equations, which uses generalized eigenvalues and eigenvectors of resultant matrices. We give special attention to the case of two bivariate polynomials and the Sylvester or Bezout resultant constructions. We propose a new method to treat multiple roots, detail its numerical aspects and describe...

The Symmetric Tensor Approximation problem (STA) consists of approximating a symmetric tensor or a homogeneous polynomial by a linear combination of symmetric rank-1 tensors or powers of linear forms of low symmetric rank. We present two new Riemannian Newton-type methods for low rank approximation of symmetric tensor with complex...