Transformations of Matrix Structures Work Again

TL;DR

This paper revisits and extends a powerful matrix transformation method that enables efficient algorithms for structured matrices, leading to nearly linear time solutions for polynomial evaluation and interpolation.

Contribution

The paper comprehensively revisits the Vandermonde and Hankel multipliers method and specializes it to accelerate algorithms for Vandermonde and Cauchy matrices, achieving near-linear time complexity.

Findings

Demonstrated numerically stable algorithms for matrix inversion

Achieved nearly linear arithmetic time for polynomial evaluation

Extended the method to various structured matrices

Abstract

In 1989 we proposed to employ Vandermonde and Hankel multipliers to transform into each other the matrix structures of Toeplitz, Hankel, Vandermonde and Cauchy types as a means of extending any successful algorithm for the inversion of matrices having one of these structures to inverting the matrices with the structures of the three other types. Surprising power of this approach has been demonstrated in a number of works, which culminated in ingeneous numerically stable algorithms that approximated the solution of a nonsingular Toeplitz linear system in nearly linear (versus previuosly cubic) arithmetic time. We first revisit this powerful method, covering it comprehensively, and then specialize it to yield a similar acceleration of the known algorithms for computations with matrices having structures of Vandermonde or Cauchy types. In particular we arrive at numerically stable approximate multipoint polynomial evaluation and interpolation in nearly linear arithmetic time.