Solving toeplitz- and vandermonde-like linear systems with large displacement rank

Abstract

Linear systems with structures such as Toeplitz-, Vandermonde-or Cauchy-likeness can be solved in O˜(α 2 n) operations, where n is the matrix size, α is its displacement rank, and O˜denotes the omission of logarithmic factors. We show that for Toeplitz-like and Vandermonde-like matrices, this cost can be reduced to O˜(α ω−1 n), where ω is a feasible exponent for matrix multiplication over the base field. The best known estimate for ω is ω < 2.38, resulting in costs of order O˜(α 1.38 n). We also present consequences for Hermite-Padé approximation and bivariate interpolation.

Linear Algebra for Computing Gröbner Bases of Linear Recursive Multidimensional Sequences

“…In Sec. 6, we give two approaches based on the notion of displacement rank and the polynomial multiplication interpretation, we refer to [2] for both. If d i is the maximal degree of the polynomials in x i , then solving the mth multi-Hankel matrix can be done in O(M(2 m−1 d 1 ···d m )) operations in the base field.…”

“…This nice structure allows us to solve a linear system with a Hankellike matrix H, i.e. a small sum of Hankel matrices, in O(α ω−1 M(d) log d) operations if α = rank ϕ(H) and if d is the size of H, see [2].…”

“…1. Besides the change of basis in O(n 2d ), we need one call of BM inÕ(n d ) operations in K. Finally, each Hankel system can be solved in O(M(d) log d) operations (see[2]). …”

Résolution rapide des systèmes de Toeplitz bande par blocs de Toeplitz bandes

“…De plus, la résolution d'un système de Toeplitz biniveaux de taille k 2 m × k 2 m requiert O(k 3 2 m log 2 (k 2 m)) opérations (voir [7,9,4,3]). …”

Computing specified generators of structured matrix inverses

“…with D(x) the diagonal matrix whose entry (i, i) is the ith coefficient xi of vector x, and Zn,ϕ the n × n unit ϕ-circulant matrix having a ϕ in position (1, n), ones in positions (i + 1, i), and zeros everywhere else. When a structured matrix A ∈ K n×n is invertible, its inverse A −1 is known to be structured too, and some asymptotically fast algorithms are available for computing length-α generators for A −1 and linear system solutions, whose costs in terms of operations in K are in O˜(α 2 n) (see [19] and the references therein) and, since more recently, in O˜(α ω−1 n) (see [2,3]). (Here and hereafter the O˜notation hides all logarithmic factors.)…”