Transformations of matrix structures work again

Abstract

Matrices with the structures of Toeplitz, Hankel, Vandermonde and Cauchy types are omnipresent in modern computations in Sciences, Engineering, and Signal and Image Processing. These four matrix classes have distinct features, but in our 1990 paper in Mathematics of Computation we showed that Vandermonde and Hankel multipliers can be applied to transform each class into the others, and then we demonstrated how by using these transforms we can readily extend any successful matrix inversion algorithm from one of these classes to all the others. 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 the same 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 approximation algorithms for Toeplitz linear systems and extend them to approximate multiplication of Vandermonde and Cauchy matrices by a vector and to approximate solution of Vandermonde and Cauchy linear systems of equations provided that 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 having structures that generalize the structures of Vandermonde and Cauchy matrices 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 comment on some natural research challenges, particularly some promising applications of our techniques to high precision computations.