Superfast solution of Toeplitz systems based on syzygy reduction

Houssam Khalil (1, 2) , Bernard Mourrain (1) , Michelle Schatzman (2)

1 GALAAD - Geometry, algebra, algorithms
2 ICJ - Institut Camille Jordan

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 module is generated by two elements and the solution of a Toeplitz system T u=g can be reinterpreted as the remainder of a vector depending on g, by these two generators. We obtain these generators and this remainder with computational complexity O(n log^2 n) for a Toeplitz matrix of size nxn.

Mots clés: syzygies of polynomials; Toeplitz matrices; polynomial division; syzygies of polynomials.

Domaines: Géométrie algébrique [math.AG]

Houssam Khalil, Bernard Mourrain, Michelle Schatzman. Superfast solution of Toeplitz systems based on syzygy reduction. Linear Algebra and its Applications, 2013, 438 (9), pp.3563-3575. ⟨10.1016/j.laa.2013.01.015⟩. ⟨hal-00780556⟩