Transformations of Matrix Structures Work Again II (1311.3729v1)

Abstract: Matrices with the structures of Toeplitz, Hankel, Vandermonde and Cauchy types are omnipresent in modern computation. The four classes have distinct features, but in 1990 we showed that Vandermonde and Hankel multipliers transform all these structures into each other and proposed to employ this property to extend any successful algorithm that inverts matrices of one of these classes to inverting matrices of the three other classes. The power of this approach was widely recognized later, when novel numerically stable algorithms solved nonsingular Toeplitz linear systems of equations at first in quadratic and then in nearly linear (versus classical cubic) arithmetic time by means of transforming Toeplitz into Cauchy matrix structures and then linking it to the Hierarchical Semiseparable matrix structure, a specialization of matrix representations employed by the Fast Multipole Method. We first cover the method of structure transformation comprehensively, then analyze the latter algorithms for Toeplitz linear systems and extend them to approximate the products of Vandermonde and Cauchy matrices by a vector and the solutions of Vandermonde and Cauchy linear systems of equations where they are nonsingular and well conditioned. We decrease the arithmetic cost of the known algorithms from quadratic to nearly linear, and similarly for the computations with the matrices of a more general class having structures of Vandermonde and Cauchy types and for polynomial and rational evaluation and interpolation. We also accelerate a little further the known numerical approximation algorithms for a nonsingular Toeplitz or Toeplitz-like linear system by employing distinct transformations of matrix structures, and we briefly discuss some natural research challenges, particularly some promising applications of our techniques to high precision computations.