Transformations of Matrix Structures Work Again

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.