Solving structured 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 such 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 present consequences for Hermite–Padé approximation and bivariate interpolation.