On Matrices with Displacement Structure:

For matrices with displacement structure, basic operations like multiplication, in4 version, and linear system solving can all be expressed in terms of the following task: evaluate the 5 product AB, where A is a structured n × n matrix of displacement rank α, and B is an arbitrary 6 n × α matrix. Given B and a so-called generator of A, this product is classically computed with a 7 cost ranging from O(α2M (n)) to O(α2M (n) log(n)) arithmetic operations, depending on the type of 8 structure of A; here, M is a cost function for polynomial multiplication. In this paper, we first gen9 eralize classical displacement operators, based on block diagonal matrices with companion diagonal 10 blocks, and then design fast algorithms to perform the task above for this extended class of struc11 tured matrices. The cost of these algorithms ranges from O(αω−1M (n)) to O(αω−1M (n) log(n)), 12 with ω such that two n × n matrices over a field can be multiplied using O(nω) field operations. 13 By combining this result with classical randomized regularization techniques, we obtain faster Las 14 Vegas algorithms for structured inversion and linear system solving. 15