TR-2013010: Transformations of Matrix Structures Work Again II

Matrices with the structures of Toeplitz, Hankel, Vandermonde and Cauchy types are omnipresent in modern computations in Sciences, Engineering and Signal and Image Processing. The four matrix classes have distinct features, but in [P90] we showed that Vandermonde and Hankel multipliers transform all these structures into each other and proposed to employ this property in order to extend any successful algorithm that inverts matrices of one of these four classes to inverting matrices with the structures of the three other types. The power of this approach was widely recognized later, when novel numerically stable algorithms solved nonsingular Toeplitz linear systems of equations in quadratic (versus classical cubic) arithmetic time based on transforming Toeplitz into Cauchy matrix structures. More recent papers combined such a transformation with a link of the Cauchy matrices to the Hierarchical Semiseparable matrix structure, which is a specialization of matrix representations employed by the Fast Multipole Method. This produced numerically stable algorithms that approximated the solution of a nonsingular Toeplitz linear system of equations in nearly linear arithmetic time. We first revisit the successful method of structure transformation, covering it comprehensively. Then we analyze the latter efficient approximation 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 numerical approximation algorithms for these tasks 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.