Recurrence Width for Structured Dense Matrix Vector Multiplication

Matrix-vector multiplication is one of the most fundamental computing primitives that has been studied extensively. Given a matrix A ∈ FN×N and a vector b ∈ FN , it is known that in the worst-caseΘ(N 2) operations over F are needed to compute Ab. Many classes of structured dense matrices have been investigated which can be represented with O(N ) parameters, and for which matrix-vector multiplication can be performed with a sub-quadratic number of operations. One such class of structured matrices that admit near-linear matrix-vector multiplication are the orthogonal polynomial transforms whose rows correspond to a family of orthogonal polynomials. Other well known classes include the Toeplitz, Hankel, Vandermonde, Cauchy matrices and their extensions (e.g. confluent Cauchy-like matrices) which are all special cases of a displacement rank property. In this paper, we identify a notion of recurrence width t of matrices A so that such matrices can be represented with t 2N elements from F. For matrices with constant recurrence width we design algorithms to compute both Ab and AT b with a nearlinear number of operations. This notion of width is finer than all the above classes of structured matrices and thus computes near-linear matrix-vector multiplication for all of them using the same core algorithm. Technically, our work unifies, generalizes, and (we think) simplifies existing state-of-the-art results in structured matrix-vector multiplication. We consider generalizations and variants of this width to other notions that can also be handled by the same core algorithms. Finally, we show how applications in disparate areas such as multipoint evaluations of multivariate polynomials and computing linear sequences can be reduced to problems involving low recurrence width matrices.